2.1 The Picard scheme

We begin with recollections on line bundles on smooth projective complex varieties and their moduli. A more precise, and much more general, account of this standard material can be found in Kleiman’s notes [Reference Kleiman22]. In what follows, X is a connected smooth projective complex variety.

Definition 2.2. The Picard functor of X

$$\begin{align*}T \longrightarrow \mathrm{Pic}_{\mathrm{abs}}(X \times T) / \mathrm{Pic}_{\mathrm{abs}}(T) \end{align*}$$

from complex schemes to abelian groups is represented by a scheme $\mathrm {Pic}(X)$ called the Picard scheme of X (relative to $\operatorname {\mathrm {Spec}}({\mathbb {C}})$ ).

Lemma 2.4 (Compare [Reference Kleiman22, Exercise 4.3])

There exists a (nonunique) algebraic line bundle $\mathcal P$ on $\mathrm {Pic}(X) \times X$ satisfying the following property: given any complex scheme T and a line bundle $\mathcal L$ on $T \times X$ , there exists a unique morphism $h \colon T \to \mathrm {Pic}(X)$ such that

$$\begin{align*}\mathcal L \cong ((h \times 1)^*\mathcal P) \otimes f^*\mathcal N \end{align*}$$

for $f \colon T \times X \to T$ the projection map and $\mathcal N$ some line bundle on T.

The Picard scheme $\mathrm {Pic}(X)$ only parameterises isomorphism classes of line bundles on X. One should think of the choice of a Poincaré line bundle as making compatible choices of representatives of those isomorphism classes.

We will need a further decomposition of the Picard scheme into components. To introduce it, we let $\mathcal O_X(1)$ be a very ample line bundle on X and write $\mathcal F(n) = \mathcal F \otimes \mathcal O_X(1)^{\otimes n}$ for any sheaf of $\mathcal O_X$ -modules $\mathcal F$ and $n \in \mathbb Z$ . Given a polynomial $P \in {\mathbb {Q}}[x]$ , let $\mathrm {Pic}^P(X)(-) \subset \mathrm {Pic}(X)(-)$ be the subfunctor of the Picard functor whose T-points are represented by the line bundles $\mathcal L$ on $X \times T$ such that

$$\begin{align*}\chi(X_t, \mathcal L_t^{-1}(n)) = P(n) \quad \text{ for all } t \in T \end{align*}$$

where $X_t$ and $\mathcal L_t$ denote the base change to t.

Passing to complex points, the picture is vastly simplified by Hodge theory as recalled in the following two results.

Proposition 2.8. The Picard space $\mathrm {Pic}(X)$ is a disjoint union of connected components indexed by the Néron–Severi group

$$\begin{align*}\mathrm{Pic}(X) = \bigsqcup_{\alpha \in \mathrm{NS}(X)} \mathrm{Pic}^\alpha(X). \end{align*}$$

Each space $\mathrm {Pic}^\alpha (X)$ is a torus, noncanonically isomorphic to $H^1(X; \mathcal O_X) / H^1(X; \mathbb Z)$ , and parameterises isomorphism classes of holomorphic line bundles on X with Chern class $\alpha $ .

2.2 Jet bundles

We recall the definition of jet bundles in algebraic geometry and explain the construction of the jet evaluation map which will be used throughout this article. The main reference for this section is [Reference Grothendieck17, Section 16.7]. In this section only, the full generality offered by schemes will be convenient, so we momentarily work in this setting.

Let $f \colon Y \to S$ be a morphism of schemes, $\Delta \colon Y \to Y \times _S Y$ be the diagonal and $\mathcal I$ be its ideal sheaf. We let $p_i \colon Y \times _S Y \to Y$ be the two projections for $i=1,2$ .

Definition 2.9. Let $\mathcal F$ be an $\mathcal O_Y$ -module. Its relative r-jet bundle is defined by

$$\begin{align*}J^r_{Y/S} \mathcal F := (p_1)_* \big( \mathcal O_{Y \times_S Y}/\mathcal I^{r+1} \otimes p_2^*\mathcal F\big). \end{align*}$$

The two projections $p_i$ give two morphisms of sheaves of rings $\mathcal O_Y \to J^r_{Y/S} \mathcal O_Y$ . We choose the one given by $p_1$ to define an $\mathcal O_Y$ -module structure. The other one, induced by $p_2$ , is denoted by

$$\begin{align*}d^r_{Y/S} \colon \mathcal O_Y \to J^r_{Y/S} \mathcal O_Y \end{align*}$$

and is called the jet map. In particular $J^r_{Y/S} \mathcal F = J^r_{Y/S} \mathcal O_Y \otimes _{\mathcal O_Y} \mathcal F$ where $J^r_{Y/S} \mathcal O_Y$ is seen as a right $\mathcal O_Y$ -module via $d^r_{Y/S}$ . We will also write $d^r_{Y/S} \colon \mathcal F \to J^r_{Y/S} \mathcal F$ for the tensor of the jet map with $\mathcal F$ .

The fibre of the jet bundle at a closed point $y \in Y$ with maximal ideal sheaf $\mathfrak m$ is $(J^r_{Y/S}\mathcal F)|_y \cong \mathcal F/ \mathfrak m^{r+1}\mathcal F$ . Intuitively, the jet map should be thought of as taking the r-th Taylor expansion of a function. In particular, as the Leibniz rule for differentiation shows, it is not a morphism of $\mathcal O_Y$ -modules when $r> 0$ . On the contrary, taking the derivative of a function commutes with multiplication by a constant. At the level of the relative jet bundles, the functions on S play the role of the scalars, and this fact is expressed by the following:

Lemma 2.10. The pushforward of the jet map

$$\begin{align*}f_* d^r_{Y/S} \colon f_*\mathcal F \longrightarrow f_*J^r_{Y/S} \mathcal F \end{align*}$$

is a morphism of $\mathcal O_S$ -modules.

Definition 2.11. Let $\mathcal F$ be an $\mathcal O_Y$ -module. The fibrewise jet evaluation map is the composition of the pushforward of the jet map followed by the counit:

$$\begin{align*}f^*f_*\mathcal F \longrightarrow f^*f_* J^r_{Y/S}\mathcal F \longrightarrow J^r_{Y/S}\mathcal F. \end{align*}$$

To explain the definition above, we assume that Y and S are complex varieties for the rest of this section. As we will alternate between two points of view on vector bundles (as sheaves or as spaces over the base), it will sometimes be convenient to be explicit about which viewpoint is adopted:

Definition 2.12. Let $\mathcal F$ be a vector bundle (i.e., a locally free sheaf of $\mathcal O_Y$ -modules of finite rank) on a complex variety Y. The total space of the associated geometric vector bundle is

$$\begin{align*}\mathbb V(\mathcal F) := \underline{\operatorname{\mathrm{Spec}}}_Y \big( \operatorname{\mathrm{Sym}}^* (\mathcal F^\vee) \big)^{\mathrm{an}} \end{align*}$$

where $(-)^{\mathrm {an}}$ denotes the analytification functor.

Suppose now that $\mathcal F$ is a vector bundle on Y such that $f_*\mathcal F$ is also a vector bundle on S. As sets, we have an identification

$$\begin{align*}\mathbb V(f_*\mathcal F) = \{ (s, \sigma) \mid s \in S, \ \sigma \in H^0(Y_s; \mathcal F|_{Y_s}) \} \end{align*}$$

where $Y_s = f^{-1}(s) \subset Y$ is the fibre above s. In particular, when $S = \operatorname {\mathrm {Spec}} {\mathbb {C}}$ is a point, this is the space of global sections $H^0(Y;\mathcal F)$ . In general, it should be thought of as a space of fibrewise sections.

Lemma 2.13. Under the above assumptions, the counit map $f^*f_* \mathcal F \to \mathcal F$ induces the evaluation map

$$ \begin{align*} \mathbb V(f^*f_*\mathcal F) \cong \mathbb V(f_*\mathcal F) \times_S Y &\longrightarrow \mathbb V(\mathcal F) \\ ((s,\sigma), y) &\longmapsto \sigma(y) \end{align*} $$

on geometric realisation.

Summarising the situation, we see that the fibrewise jet evaluation map

$$\begin{align*}\mathbb V(f_*\mathcal F) \times_S Y \longrightarrow \mathbb V(J^r_{Y/S}\mathcal F) \end{align*}$$

is given above a point $s \in S$ by

$$ \begin{align*} H^0(Y_s;\mathcal F|_{Y_s}) \times Y &\longrightarrow \mathbb V(J^r \mathcal F|_{Y_s}) \\ (\sigma, y) &\longmapsto j^r\sigma(y) \in \mathcal F|_{Y_s} \big/ \mathfrak m^{r+1}\mathcal F|_{Y_s} \end{align*} $$

where $\mathfrak m$ is the maximal ideal sheaf of $y \in Y_s$ . In other words, it takes the Taylor expansion of $\sigma $ at y up to order r.

2.3 Jet ampleness

Having now defined jet bundles, we state the crucial definition of jet ampleness of a line bundle on a smooth projective complex variety X.

Definition 2.14 (Compare [Reference Beltrametti, Di Rocco and Sommese2])

Let $k \geq 0$ be an integer. Let $x_1, \ldots , x_t$ be t distinct points in X and $(k_1,\ldots ,k_t)$ be a t-uple of positive integers such that $\sum _i k_i = k+1$ . Denote by $\mathcal O_X$ the structure sheaf of X and by $\mathfrak m_i$ the maximal ideal sheaf corresponding to $x_i$ . We regard the tensor product $\otimes _{i=1}^t \mathfrak m_i^{k_i}$ as a subsheaf of $\mathcal O_X$ under the multiplication map $\otimes _{i=1}^t \mathfrak m_i^{k_i} \to \mathcal O_X$ . We say that a line bundle $\mathcal L$ is k-jet ample if the evaluation map

$$\begin{align*}\Gamma\left(\mathcal L\right) \longrightarrow \Gamma\left(\mathcal L \otimes \left(\mathcal O_X / \otimes_{i=1}^t \mathfrak m_i^{k_i}\right)\right) \cong \bigoplus_{i=1}^t \Gamma\left(\mathcal L \otimes \left(\mathcal O_X / \mathfrak m_i^{k_i}\right)\right) \end{align*}$$

is surjective for any $x_1,\ldots ,x_t$ and $k_1,\ldots ,k_t$ as above.

The following proposition is the main tool to produce line bundles having a very high degree of jet ampleness.

We refer to Appendix A for how to compute $d(X, \alpha )$ in some special cases. Given an integer d, we also explain in Proposition A.7 how to find an $\alpha $ such that $d(X, \alpha ) \geq d$ .

2.4 The topology of hypersurfaces

It is well known that all smooth degree d complex hypersurfaces in $\mathbb P^n$ are diffeomorphic. As a way of justifying the study of the moduli space of hypersurfaces of a given Chern class, we observe that such hypersurfaces are also all diffeomorphic, provided the Chern class is ample enough. First, recall that ampleness is a numerical property:

We recall the following classical definition which is central in our work:

Remark 2.21. Given a nonsingular section $s \in \Gamma (X,\mathcal L)$ , its vanishing locus

$$\begin{align*}V(s) := \{ x \in X \mid s(x) = 0 \} \subset X \end{align*}$$

is a smooth hypersurface.

Any hypersurface H can be seen as a Weil divisor, hence a Cartier divisor (X is smooth), and therefore has an attached line bundle $\mathcal O_X(H)$ . If $H = V(s)$ with $s \in \Gamma (X,\mathcal L)$ , then $\mathcal O_X(H) \cong \mathcal L$ . The following bit of language will be convenient:

Proof. The proof is an adaptation in families of the classical proof for a single linear system. Let $\mathrm {Pic}^\alpha (X)$ be the connected component of the Picard space classifying isomorphism classes of line bundles of Chern class $\alpha $ , and let $\mathcal P$ be a Poincaré line bundle on $\mathrm {Pic}^\alpha (X) \times X$ . For $[\mathcal L] \in \mathrm {Pic}^\alpha (X)$ , we write $\mathcal P_{[\mathcal L]}$ for the line bundle on X which represents the isomorphism class $[\mathcal L]$ . Let $p \colon \mathrm {Pic}^\alpha (X) \times X \to \mathrm {Pic}^\alpha (X)$ be the projection. By cohomology and base change [Reference Hartshorne20, Theorem III.12.11], the sheaf $p_*\mathcal P$ is a vector bundle provided that $H^1(X, \mathcal P_{[\mathcal L]}) = 0$ for all $[\mathcal L] \in \mathrm {Pic}^\alpha (X)$ . This follows by the Kodaira vanishing theorem and the assumption that $\alpha - c_1(K_X)$ is ample. Let

$$\begin{align*}\mathbb V(p_*\mathcal P)^{\mathrm{ns}} \subset \mathbb V(p_*\mathcal P) = \{ ([\mathcal L], s) \mid [\mathcal L] \in \mathrm{Pic}^\alpha(X), \ s \in \Gamma(\mathcal P_{[\mathcal L]}) \} \end{align*}$$

be the subset of those sections that are nonsingular. It is the complement of the (fibrewise) discriminant which has complex codimension at least 1 by our second assumption on the ampleness of $\alpha $ . Hence $\mathbb V(p_*\mathcal P)^{\mathrm {ns}}$ is connected. The incidence variety

$$\begin{align*}\{ ([\mathcal L],s,x) \in \mathbb V(p_*\mathcal P)^{\mathrm{ns}} \times X \mid s(x) = 0 \} \longrightarrow \mathbb V(p_*\mathcal P)^{\mathrm{ns}} \end{align*}$$

is a smooth fibre bundle by Ehresmann’s lemma. The base is connected and therefore all the fibres are diffeomorphic.

3.1 Moduli functors and the Hilbert scheme

In this subsection, we define the moduli functor of smooth hypersurfaces in X and show that it is representable by an open subscheme of the Hilbert scheme of X. Under further assumptions on the ampleness of the first Chern class, we give an explicit construction of that scheme as an open subscheme of a projective bundle over the Picard scheme. Although the techniques we will use are known to algebraic geometers working with moduli, we have chosen to include those results for two main reasons: firstly to convince topologists that our space $\mathcal {M}_{\mathrm {hyp}}^\alpha $ (introduced in the introduction, and properly defined in this section) really is a moduli space with a universal property, and secondly to uncover precisely which assumptions on the first Chern class $\alpha $ are needed to construct the moduli space as a subspace of a projective bundle.

The reader only interested in $\mathcal {M}_{\mathrm {hyp}}^\alpha $ as a mere topological space can jump to the next subsection where we provide a point set model for the analytification of the moduli scheme, which will be thereafter used throughout the article.

In algebraic geometry, families of hypersurfaces are more commonly known as relative effective Cartier divisors. (See (33, Tag 056P).) Following [Reference Kleiman22, Part 3], we recall the definition of their moduli functor:

Definition 3.1. Let $P \in {\mathbb {Q}}[x]$ be a polynomial. The moduli functor of effective divisors with Hilbert polynomial P is the functor

$$ \begin{align*} \mathfrak M^P \colon \mathsf{Sch}_{\mathbb{C}}^{\mathrm{op}} &\longrightarrow \mathsf{Set} \\ T &\longmapsto \left\{ Z \subset X \times T\ \middle\vert \begin{array}{l} Z \to T \text{ is flat and proper and for all } t \in T \\ \text{the ideal sheaf } \mathcal I_{Z_t} \text{ is a line bundle and is } \\ \text{such that } \chi(X_t, \mathcal I_{Z_t}^{-1}(n)) = P(n) \end{array}\right\} \end{align*} $$

We define $\mathfrak M^{\mathrm {sm},P} \subset \mathfrak M^P$ to be the subfunctorFootnote ¹ where $Z_t$ is furthermore required to be smooth over the residue field $\operatorname {\mathrm {Spec}}(\kappa (t))$ at $t \in T$ .

When $\mathrm {Pic}(X)$ is a discrete space, families of hypersurfaces are represented by the more classical linear systems of divisors. We give an example to orient the reader:

Example 3.2 (Compare [Reference Kleiman22, Definition 3.12 and Theorem 3.13])

Let $X = \mathbb P^n$ and let $P' \in {\mathbb {Q}}[x]$ be the Hilbert polynomial of a hypersurface of degree $d \geq 1$ . Such a hypersurface has ideal sheaf $\mathcal O(-d)$ with Hilbert polynomial $P(m) = \chi (\mathbb P^n, \mathcal O(d+m)) = \chi (\mathbb P^n, \mathcal O(m+2d)) - P'(m+2d)$ .Footnote ² One can show that the Hilbert scheme is in this case the complete linear system

$$\begin{align*}\mathrm{Hilb}^{P'}(\mathbb P^n) = |\mathcal O(d)| = \mathbb P\big(H^0(\mathbb P^n,\mathcal O(d))\big). \end{align*}$$

Therefore $\mathrm {Hilb}^{P'}(\mathbb P^n)(-) = \mathfrak M^P(-)$ is represented by a complex projective space of complex dimension $\dim _{\mathbb {C}} H^0(\mathbb P^n,\mathcal O(d)) -1$ . The subfunctor $\mathfrak M^{\mathrm {sm},P}$ is represented by the complement of the discriminant hypersurface.

In general, we have the following general representability result:

Our goal is now to recall an explicit description of the scheme representing $\mathfrak M^P(-)$ under some ampleness conditions. For the remainder of this section, we fix a polynomial $P \in {\mathbb {Q}}[x]$ , denote by $\mathrm {Pic}^P(X) \subset \mathrm {Pic}(X)$ the subscheme defined in Proposition 2.6, and still write $\mathcal P$ for the restriction of the chosen Poincaré line bundle to it. Recall that $p \colon \mathrm {Pic}^P(X) \times X \to \mathrm {Pic}^P(X)$ denotes the first projection. Let

$$\begin{align*}\mathcal K := \ker\big( p^*p_* \mathcal P \to \mathcal P \big) \end{align*}$$

be the kernel sheaf of the counit of $p^* \dashv p_*$ . From the monoidality of $p^*$ and standard properties of the relative Proj construction, we have an isomorphism

$$\begin{align*}\underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X) \times X}\big(\operatorname{\mathrm{Sym}} p^*(p_*\mathcal P)^\vee\big) \cong \underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X)}\big(\operatorname{\mathrm{Sym}} (p_*\mathcal P)^\vee\big) \times X. \end{align*}$$

Using the surjection of sheaves $p^*(p_*\mathcal P)^\vee \twoheadrightarrow \mathcal K^\vee $ we thus obtain a closed immersion

$$\begin{align*}\underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X) \times X}\big(\operatorname{\mathrm{Sym}} \mathcal K^\vee\big) \subset \underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X)}\big(\operatorname{\mathrm{Sym}} (p_*\mathcal P)^\vee\big) \times X. \end{align*}$$

Definition 3.4. We define the universal family to be the morphism

$$\begin{align*}\mathcal U := \underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X) \times X}\big(\operatorname{\mathrm{Sym}} \mathcal K^\vee\big) \longrightarrow \underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X)}\big(\operatorname{\mathrm{Sym}} (p_*\mathcal P)^\vee\big) =: \mathcal M \end{align*}$$

obtained by projecting onto the first coordinate.

Definition 3.6. The morphism of functors

$$\begin{align*}\mathfrak M^P(T) \longrightarrow \mathrm{Pic}^P(X)(T), \quad (Z \subset X \times T) \longmapsto [\mathcal I_Z^{-1}] \end{align*}$$

is represented by the projection morphism

$$\begin{align*}\mathcal M = \underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X)}\big(\operatorname{\mathrm{Sym}} (p_*\mathcal P)^\vee\big) \longrightarrow \mathrm{Pic}^P(X) \end{align*}$$

which is usually called the Abel–Jacobi morphism.

We now explain how to obtain a scheme representing $\mathfrak M^{\mathrm {sm},P}(-)$ by removing the discriminant locus from $\mathcal M$ . We assume that the evaluation morphism $p^*p_* \mathcal P \to \mathcal P$ is surjective. Recall the jet evaluation morphism from Definition 2.11 and denote by $\mathcal Q$ the cokernel of its dual morphism:

$$\begin{align*}\mathcal Q := \mathrm{coker}\left(\big( J^1_{\mathrm{Pic}^P(X) \times X/ \mathrm{Pic}^P(X)} \mathcal P \big)^\vee \longrightarrow p^*(p_*\mathcal P)^\vee \right). \end{align*}$$

The composition

$$\begin{align*}\mathcal P^\vee \longrightarrow p^*(p_*\mathcal P)^\vee \longrightarrow \mathcal Q \end{align*}$$

of the dual of the evaluation morphism followed by the projection, is the zero morphism. Indeed it factors through $(J^1\mathcal P)^\vee $ via the dual of the projection morphism from the first jet bundle onto the zeroth jet bundle $J^0\mathcal P = \mathcal P$ . We thus obtain an induced surjective morphism $\mathcal K^\vee \twoheadrightarrow \mathcal Q$ .

Theorem 3.7. Assume that the counit morphism $p^*p_* \mathcal P \to \mathcal P$ is surjective, and that $H^1(X_t,\mathcal P_t) = 0$ for all $t \in \mathrm {Pic}^P(X)$ . Then the moduli functor $\mathfrak M^{\mathrm {sm},P}(-)$ is represented by an open subscheme of $\mathcal M$ , hence of the Hilbert scheme of X. More precisely, let $\pi \colon \mathcal U \to \mathcal M$ be the universal family and

$$\begin{align*}\mathcal Z := \underline{\operatorname{\mathrm{Proj}}}_{\mathrm{Pic}^P(X) \times X}\big(\operatorname{\mathrm{Sym}} \mathcal Q\big) \hookrightarrow \mathcal U \end{align*}$$

be the closed subscheme determined by the surjection $\mathcal K^\vee \twoheadrightarrow \mathcal Q$ . Then $\mathfrak M^{\mathrm {sm},P}(-)$ is represented by $\mathcal M \setminus \pi (\mathcal Z)$ .

We close this section with some general remarks about our assumptions in Theorem 3.7. We show that, although stated scheme-theoretically, they can be checked after analytification.

Proof. By upper semicontinuity of cohomology, the subscheme

$$\begin{align*}\left\{ t \in \mathrm{Pic}^P(X) \mid H^1(X_t, \mathcal P_t) = 0 \right\} \subset \mathrm{Pic}^P(X) \end{align*}$$

is open. Assume that its complement is nonempty. Then it contains a complex point: it is locally of finite type over $\operatorname {\mathrm {Spec}}({\mathbb {C}})$ and Hilbert’s Nullstellensatz applies. This cannot be the case by assumption.

We finally comment on the relation between Hilbert polynomials and Chern classes. For a holomorphic line bundle $\mathcal L$ on X, recall the Hirzebruch–Riemann–Roch theorem giving an equality

$$\begin{align*}\chi(X, \mathcal L) = \int_X \mathrm{ch}(\mathcal L) \ \mathrm{td}(X) \end{align*}$$

where $\mathrm {ch}(-)$ is the Chern character and $\mathrm {td}(X)$ is the Todd class of X. In particular, if $Z \subset X$ is an effective Cartier divisor, its Hilbert polynomial only depends on the first Chern class of its associated line bundle $\mathcal O_X(Z)$ . We thus obtain a numerical criterion:

Lemma 3.11. Let $P \in {\mathbb {Q}}[x]$ and let C be the collection of Chern classes

$$\begin{align*}C := \left\{ c_1(\mathcal L) \mid [\mathcal L] \in \mathrm{Pic}^P(X)({\mathbb{C}}) \right\}. \end{align*}$$

If $\alpha - c_1(K_X)$ is ample for all $\alpha \in C$ , then $H^1(X_t, \mathcal P_t) = 0$ for all $t \in \mathrm {Pic}^P(X)$ .

3.2 A convenient point-set model

In this section, we unravel the result of Theorem 3.7 and give an explicit point-set model for the moduli space of smooth hypersurfaces.

We begin with notations which we will use throughout the rest of the article. If $[\mathcal L] \in \mathrm {Pic}(X)$ is an isomorphism class of a line bundle, we write $\mathcal {P}_{[\mathcal {L}]}$ for the representative of that isomorphism class given by the restriction of $\mathcal P$ to $X\cong \{[\mathcal L]\} \times X \subset \mathrm {Pic}(X) \times X$ . If $\alpha \in \mathrm {NS}(X)$ we recall from Proposition 2.8 that $\mathrm {Pic}^\alpha (X) \subset \mathrm {Pic}(X)$ denotes the connected component parameterising line bundles of Chern class $\alpha $ , and we will write $\mathcal P_\alpha $ for the restriction of the Poincaré line bundle to that component.

Definition 3.12. The first jet bundle of $\mathcal P$ relative to the projection p (see Definition 2.9) is denoted

$$\begin{align*}J^1_p\mathcal P := J^1_{\mathrm{Pic}(X) \times X / \mathrm{Pic}(X)} \mathcal P. \end{align*}$$

When restricted to $\mathrm {Pic}^\alpha (X)$ for some $\alpha \in \mathrm {NS}(X)$ , we will write $J^1_p\mathcal P_\alpha := J^1_{\mathrm {Pic}^\alpha (X) \times X / \mathrm {Pic}^\alpha (X)}\mathcal P_\alpha $ .

Lemma 3.13. Let $K_X$ be the canonical sheaf of X. Let $\alpha \in \mathrm {NS}(X)$ be such that $\alpha - c_1(K_X)$ is ample. Then $p_* \mathcal P_\alpha $ is a vector bundle and the fibrewise jet evaluation map gives a map of vector bundles

$$\begin{align*}p^*p_* \mathcal P_\alpha \longrightarrow J^1_p\mathcal P_\alpha. \end{align*}$$

As sets, the geometric realisations are given by

$$\begin{align*}\mathbb V(p^*p_* \mathcal P_\alpha) = \mathbb V(p_*\mathcal P_\alpha) \times X = \left\{ ([\mathcal L], x, s) \mid [\mathcal L] \in \mathrm{Pic}^\alpha(X), \ x \in X, \ s \in \Gamma(\mathcal{P}_{[\mathcal{L}]}) \right\} \end{align*}$$

and

$$\begin{align*}\mathbb V(J^1_p\mathcal P_\alpha) = \left\{ ([\mathcal L], x, v) \mid [\mathcal L] \in \mathrm{Pic}^\alpha(X), \ x \in X, \ v \in J^1\mathcal{P}_{[\mathcal{L}]}|_x \right\}. \end{align*}$$

Under these identifications, the jet evaluation map is given by

$$\begin{align*}\mathrm{jev} \colon ([\mathcal L], x, s) \longmapsto ([\mathcal L], x, j^1s(x)). \end{align*}$$

Recall from Theorem 3.7 the scheme $\mathcal M \setminus \pi (\mathcal Z)$ representing the moduli functor of smooth hypersurfaces. After analytification, we may restrict the Abel–Jacobi map

(3) $$ \begin{align} \big( \mathcal M \setminus \pi(\mathcal Z) \big)^{\mathrm{an}} \longrightarrow \mathrm{Pic}(X)^{\mathrm{an}} \end{align} $$

to the connected component $\mathrm {Pic}^\alpha (X) \subset \mathrm {Pic}(X)$ (we will from now on drop the superscript “ $\mathrm {an}$ ”), recalled in Proposition 2.8, provided that $\alpha $ is ample enough:

Definition 3.14. Let $\alpha \in \mathrm {NS}(X)$ be such that $\alpha - c_1(K_X)$ is ample. The moduli of smooth hypersurfaces in X of Chern class $\alpha $ is defined to be the preimage of $\mathrm {Pic}^\alpha (X)$ under the Abel–Jacobi map (3). In other terms, we have a homeomorphism:

$$\begin{align*}\mathcal{M}_{\mathrm{hyp}}^\alpha \cong \big( \mathbb V(p_* \mathcal P_\alpha) \setminus \mathrm{proj}(\mathrm{jev}^{-1}(0)) \big) / {\mathbb{C}}^\times \end{align*}$$

where the scalars act fibrewise over $\mathrm {Pic}(X)$ , and $\mathrm {proj} \colon \mathbb V(p^*p_*\mathcal P) \to \mathbb V(p_*\mathcal P)$ is the projection induced by p.

Remark 3.15. As sets, we have an identification

$$\begin{align*}\mathcal{M}_{\mathrm{hyp}}^\alpha = \{ ([\mathcal L], [s]) \mid [\mathcal L] \in \mathrm{Pic}^\alpha(X), \ [s] \in \Gamma_{\mathrm{ns}}(\mathcal P_{[\mathcal L]})/{\mathbb{C}}^\times\}. \end{align*}$$

That is, it will be technically convenient to think of a smooth hypersurface as a tuple consisting of a line bundle $\mathcal L$ and a nonsingular global section of it (up to isomorphism and scaling action). However, the name of moduli space is justified by the previous section: we have a homeomorphism

$$\begin{align*}\mathcal{M}_{\mathrm{hyp}}^\alpha \cong \left\{ Z \subset X \text{ smooth hypersurface with } c_1(\mathcal O_X(Z)) = \alpha \right\} \subset \mathrm{Hilb}(X)^{\mathrm{an}}. \end{align*}$$

4.1 A topological counterpart

We begin with some generalities about the topology of continuous section spaces. Let $E \to A \times B$ be a fibre bundle on a topological space $A \times B$ . We denote a point of E as a tuple $(a,b,e)$ where $a \in A$ , $b \in B$ and $e \in E|_{(a,b)}$ is in the fibre. All mapping spaces are given the compact open topology.

Definition 4.1. The space of fibrewise sections of E over A is defined to be the subspace

$$ \begin{align*} \Gamma_{\mathcal C^0, \mathrm{fib}}(E \to A) := \left\{ (a, s) \mid a \in A, \ s \in \Gamma_{\mathcal C^0}(E|_{a \times B}) \right\} &\hookrightarrow \mathrm{Map}(B, E) \\ (a,s) &\mapsto [b \mapsto (a,b,s(b))]. \end{align*} $$

Postcomposition with the projection maps $E \to A \times B \to A$ gives a continuous map $\mathrm {Map}(B, E) \to \mathrm {Map}(B, A)$ which, when restricted to fibrewise sections, yields the projection map

$$\begin{align*}\Gamma_{\mathcal C^0, \mathrm{fib}}(E \to A) \longrightarrow A, \quad (a,s) \longmapsto a. \end{align*}$$

In particular, this projection map is continuous.

To lighten the notation, and as no confusion can arise, we will from now on drop the symbol $\mathbb V(-)$ when considering continuous sections of a vector bundle.

Definition 4.4. Taking fibrewise, over $\mathrm {Pic}(X)$ , continuous global sections of $J^1_p\mathcal P$ which are never vanishing, we obtain the space

$$ \begin{align*} \Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P \setminus 0) &:= \Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P \setminus 0 \to \mathrm{Pic}(X)) \\ &= \left\{ ([\mathcal L], s) \mid [\mathcal L] \in \mathrm{Pic}(X), \ s \in \Gamma_{\mathcal C^0}(J^1\mathcal{P}_{[\mathcal{L}]} \setminus 0) \right\}. \end{align*} $$

The group ${\mathbb {C}}^\times $ acts by multiplying the sections by scalars and we let $\Gamma _{\mathcal C^0, \mathrm {fib}}(J^1_p\mathcal P \setminus 0) / {\mathbb {C}}^\times $ be the quotient for that action.

4.2 The main theorem

By Remark 4.3, the fibrewise jet map followed by the inclusion of the space of holomorphic sections inside continuous sections gives rise to a continuous map

$$\begin{align*}j^1 \colon \mathcal{M}_{\mathrm{hyp}}^\alpha \longrightarrow \Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P_\alpha \setminus 0) / {\mathbb{C}}^\times. \end{align*}$$

Denote by $\mathbb P(J^1\mathcal O_X)$ the projectivisation of the first jet bundle of $\mathcal O_X$ on X. We will make use of the following:

It follows from the proposition that if $\mathcal L$ is a line bundle with Chern class $\alpha $ , the quotient map

$$\begin{align*}\Gamma_{\mathcal C^0}(J^1\mathcal L \setminus 0) \longrightarrow \Gamma_{\mathcal C^0}(\mathbb P(J^1\mathcal L)) \cong \Gamma_{\mathcal C^0}(\mathbb P(J^1\mathcal O_X)) \end{align*}$$

has image inside the connected component $\Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X))$ . Here we have used that $J^1\mathcal L \cong J^1\mathcal O_X \otimes \mathcal L$ and the fact that the projectivisation of a vector bundle is invariant under tensoring with a line bundle.

Now, let $\mathcal L_0$ be a chosen line bundle with Chern class $\alpha $ . By choosing an isomorphism of topological line bundles $\mathcal L_0 \cong \mathcal {P}_{[\mathcal {L}]}$ for each $[\mathcal L] \in \mathrm {Pic}^\alpha (X)$ , we obtain a map

(4) $$ \begin{align} \Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P_\alpha \setminus 0) / {\mathbb{C}}^\times \longrightarrow \Gamma_{\mathcal C^0}(\mathbb P(J^1\mathcal L_0)) \cong \Gamma_{\mathcal C^0}(\mathbb P(J^1\mathcal O_X)) \end{align} $$

which factors through $\Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X))$ . As any two choices of isomorphisms $\mathcal L_0 \cong \mathcal {P}_{[\mathcal {L}]}$ differ by a nonzero constant, we see that the map is indeed uniquely well-defined and continuous. The following is our main result:

Theorem 4.6. Let X be a smooth projective complex variety. Let $\alpha \in \mathrm {NS}(X)$ be such that $\alpha - c_1(K_X)$ is ample. The jet map

$$\begin{align*}j^1 \colon \mathcal{M}_{\mathrm{hyp}}^\alpha \longrightarrow \Gamma_{\mathcal C^0}^\alpha(\mathbb P(J^1\mathcal O_X)) \end{align*}$$

induces an isomorphism in integral homology in the range of degrees $* < \frac {d(X,\alpha ) - 3}{2}$ . (See Definition 2.17.)

5.1 The homology isomorphism

The jet map fits in the following diagram where the top row is its restriction to a fibre above an $[\mathcal L] \in \mathrm {Pic}^\alpha (X)$ :

The uppermost map was studied in [Reference Aumonier1] where the following result was proved:

Theorem 5.3 (Compare [Reference Aumonier1, Corollary 8.1])

Let $\mathcal L$ be a d-jet ample line bundle on a smooth projective complex variety X. Then the jet map

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times \longrightarrow \Gamma_{\mathcal C^0}(J^1\mathcal L \setminus 0) / {\mathbb{C}}^\times \end{align*}$$

induces an isomorphism in homology in the range of degrees $* < \frac {d-1}{2}$ .

Now, if both lower vertical maps were fibrations, a comparison of the associated Serre spectral sequences would prove Proposition 5.1. This is indeed the case for the map on the right-hand side. The other map is only a microfibration, which turns out to be sufficient for the argument to go through. We start by reviewing this technical notion popularised by Weiss in [Reference Weiss36].

Definition 5.4. A map $\pi \colon E \to B$ is called a Serre microfibration if for any $k \geq 0$ and any commutative diagram

there exists an $\varepsilon> 0$ and a map $h \colon [0,\varepsilon ] \times D^k \to E$ such that $h(0,x) = u(x)$ and $\pi \circ h(t,x) = v(t,x)$ for all $x \in D^k$ and $t \in [0, \varepsilon ]$ .

Contrary to the case of fibrations, the homotopy types of the fibres of a microfibration can vary. Nonetheless, we have the very useful comparison theorem of Raptis generalising a result of Weiss:

In the present situation, we only have access to Theorem 5.3 which provides an isomorphism in homology, rather than on homotopy groups. The remedy chosen here is to suspend the spaces so as to obtain simply connected spaces and then apply the homology Whitehead theorem.

Definition 5.7. For a map $p \colon E \to B$ , its fibrewise (unreduced) k ^th suspension is defined to be

$$\begin{align*}\Sigma^k_B E = \left( E \times [0,1] \times S^{k-1} \right) \bigg/ \big((e,0,s) \sim (e,0,s') \text{ and } (e,1,s) \sim (e',1,s) \text{ if } p(e) = p(e') \big). \end{align*}$$

The fibre of the natural map $\Sigma ^k_B p \colon \Sigma ^k_B E \to B$ induced by p is the unreduced k ^th suspension of the fibre of p (here modelled as the join with the sphere $S^{k-1}$ ):

$$\begin{align*}(\Sigma^k_B p)^{-1}(b) = \Sigma^k p^{-1}(b), \quad \forall b \in B. \end{align*}$$

Lemma 5.8. The map

$$\begin{align*}\Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P_\alpha \setminus 0) / {\mathbb{C}}^\times \longrightarrow \mathrm{Pic}^\alpha(X) \end{align*}$$

is a fibre bundle.

Proof. Let $U \subset \mathrm {Pic}^\alpha (X)$ be a small contractible open subset. A topological vector bundle being trivial over a contractible base, we obtain an isomorphism of vector bundles

$$\begin{align*}\psi \colon J^1_p \mathcal P_\alpha|_{U \times X} \overset{\cong}{\longrightarrow} U \times J^1 \mathcal P_{[\mathcal L_0]} \end{align*}$$

over $U \times X$ , with $[\mathcal L_0] \in U$ a chosen basepoint. The map

$$ \begin{align*} \left( \Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P_\alpha \setminus 0) / {\mathbb{C}}^\times \right)|_U &\longrightarrow U \times \Gamma_{\mathcal C^0}(J^1 \mathcal P_{[\mathcal L_0]} \setminus 0) \\ ([\mathcal L], s) &\longmapsto ([\mathcal L], \psi \circ s) \end{align*} $$

is then a homeomorphism over U exhibiting the local triviality of the fibre bundle.

We will say that a map $A \to B$ is homology m-connected if it induces an isomorphism on homology groups $H_i(A) \to H_i(B)$ for $i < m$ and a surjection when $i = m$ .

Lemma 5.9. Let $q \colon V \to B$ be a fibre bundle, and $p \colon U \to B$ be the restriction of a fibre bundle $E \to B$ to an open subset $U \subset E$ . Let $f \colon U \to V$ be a map over B and suppose that for every $b \in B$ , the restriction to the fibre

$$\begin{align*}f_b \colon p^{-1}(b) \longrightarrow q^{-1}(b) \end{align*}$$

is homology m-connected. Then $f \colon U \to V$ is homology m-connected.

Proof. For any $b \in B$ , the suspension of f on the fibre

$$\begin{align*}\Sigma^2 f_b \colon \Sigma^2 p^{-1}(b) \longrightarrow \Sigma^2 q^{-1}(b) \end{align*}$$

induces an isomorphism in homology in degrees $* \leq m+1$ and a surjective morphism in degree $* = m+2$ . As both spaces are simply connected, the homology Whitehead theorem implies that this map is $(m+2)$ -connected. We would like to apply Theorem 5.6 to $\Sigma ^2_B f$ , but $\Sigma ^2_B U \subset \Sigma ^2_B E$ is not open and it is unclear if $\Sigma ^2_B U \to B$ is a microfibration. We resolve the issue by enlarging slightly the space to a homotopy equivalent one. More precisely, let

$$\begin{align*}W = \big(\Sigma^2_B U\big) \cup \big( (E \times (0.5,1] \times S^1)) / \sim\big) \subset \Sigma^2_B E, \end{align*}$$

and denote by $E_b, W_b, U_b$ the fibres of the respective spaces above a point $b \in B$ . Using in each fibre the homotopy equivalence $\big ((E_b \times (0.5,1] \times S^1)/\sim \big ) \simeq S^1$ given by collapsing gives a homotopy equivalence

$$\begin{align*}(W, W_b) \overset{\simeq}{\longrightarrow} (\Sigma^2_B U, \Sigma^2 U_b) \end{align*}$$

for all $b \in B$ . Now, the fibrewise suspension of the fibre bundles $E \to B$ and $V \to B$ are fibre bundles. As $W \subset \Sigma ^2_B E$ is open, the restriction $W \to B$ is a microfibration. Applying Theorem 5.6 to the composite

$$\begin{align*}W \overset{\simeq}{\longrightarrow} \Sigma^2_B U \overset{\Sigma^2_B f}{\longrightarrow} \Sigma^2_B V \end{align*}$$

and using that the first map is a homotopy equivalence, we obtain that $\Sigma ^2_B f \colon \Sigma ^2_B U \to \Sigma ^2_B V$ is $(m+2)$ -connected. Hence it is homology $(m+2)$ -connected. Comparing the Mayer–Vietoris sequences of the fibrewise suspensions finally shows that $f \colon U \to B$ is homology m-connected.

5.2 The homotopy type of the space of fibrewise sections

To prove that the map of Proposition 5.2 is a weak equivalence, we will extend it to a morphism between fibrations with equal total spaces, and then compare the homotopy fibres. (See (7) below for the precise diagram.) For this latter part, we shall need to work with convenient point-set models for the homotopy fibres, and thus we begin by some basic algebraic topology to fix the notation.

For a pointed space $(A,a)$ , we let $P(A,a) = \mathrm {Map}_*(([0,1],0), (A,a))$ be the space of paths in A starting at a. We will write $\mathrm {cst}_*$ for the constant loop based at a point $*$ . For a map of pointed spaces $\pi \colon (E,e_0) \to (B,b_0)$ , we write

$$\begin{align*}H\pi := \{ (e, \gamma) \in E \times P(B,b_0) \mid \gamma(1) = \pi(e) \} \end{align*}$$

for its homotopy fibre.

5.2.1 The homotopy fibre of a homotopy fibre

Let $\pi \colon (E,e_0) \to (B,b_0)$ be a fibration between pointed spaces, $F = \pi ^{-1}(b_0)$ be the fibre, and $\Omega _{b_0} B$ be the loop space of B based at $b_0$ . Writing $i \colon F \hookrightarrow E$ for the inclusion, it is well known that $Hi$ and $\Omega _{b_0} B$ are homotopy equivalent:

Lemma 5.10. Let $\gamma \colon [0,1] \to B$ be a loop based at $b_0$ . Let $\alpha \colon [0,1] \to E$ be a lift of that loop starting at $e_0$ . The map on connected components

$$ \begin{align*} \pi_0(\Omega_{b_0} B) &\longrightarrow \pi_0(Hi) \\ [\gamma] &\longmapsto [(\alpha(1), \alpha)] \end{align*} $$

is a bijection.

Proof. It follows from basic manipulations and [Reference Dieck7, Note 4.7.1].

5.2.3 The proof of Proposition 5.2 the homotopy equivalence

For concreteness, we start by fixing basepoints. Let $[\mathcal L_0] \in \mathrm {Pic}^\alpha (X)$ . We will write $\mathcal P_{[\mathcal L_0]} = \mathcal L_0$ for brevity. We also pick a section $s_0 \in \Gamma _{\mathcal C^0}(J^1\mathcal L_0 \setminus 0)$ . We will use these as basepoints, as well as the images $[s_0] \in \Gamma _{\mathcal C^0}(J^1\mathcal L_0 \setminus 0) / {\mathbb {C}}^\times $ and $\mathbb P s_0 \in \Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X))$ .

Pointwise multiplication of maps gives $\mathrm {Map}(X, {\mathbb {C}}^\times )$ the structure of a topological group. By [Reference Crabb and Sutherland5, Proposition 2.6], there is a principal $\mathrm {Map}(X,{\mathbb {C}}^\times )$ -bundle:

(5) $$ \begin{align} \mathrm{Map}(X,{\mathbb{C}}^\times) \longrightarrow \Gamma_{\mathcal C^0}(J^1\mathcal L_0 \setminus 0) \longrightarrow \Gamma_{\mathcal C^0}^\alpha(\mathbb P(J^1\mathcal O_X)). \end{align} $$

There is also the subgroup ${\mathbb {C}}^\times \subset \mathrm {Map}(X, {\mathbb {C}}^\times )$ of the constant functions, and modding out fibrewise gives a principal bundle:

(6) $$ \begin{align} \mathrm{Map}(X,{\mathbb{C}}^\times) / {\mathbb{C}}^\times \longrightarrow \Gamma_{\mathcal C^0}(J^1\mathcal L_0 \setminus 0) / {\mathbb{C}}^\times \longrightarrow \Gamma_{\mathcal C^0}^\alpha(\mathbb P(J^1\mathcal O_X)). \end{align} $$

We obtain a commutative diagram of pointed spaces where each row is a fibration sequence

(7)

and the spaces $F_1$ and $F_2$ are defined as the respective homotopy fibres. Using the 5-lemma and the long exact sequence of homotopy groups associated to a fibration, Proposition 5.2 follows directly from the next lemma.

Lemma 5.12. Using the notation as above, the map induced on the homotopy fibres $F_1 \to F_2$ is a homotopy equivalence.

Proof. We already know that $F_1 \simeq \Omega _{[\mathcal L_0]} \mathrm {Pic}^\alpha (X)$ (cf. Lemma 5.8) and $F_2 \simeq \mathrm {Map}(X, {\mathbb {C}}^\times ) / {\mathbb {C}}^\times $ (cf. (Eq. 6)), which are both homotopy equivalent to the discrete space $H^1(X;\mathbb Z)$ . Therefore we only need to verify that the map $F_1 \to F_2$ induces a bijection on the set of connected components. We have a diagram of sets

where the right vertical map is induced from $F_1 \to F_2$ , the top map is explained in Lemma 5.10, the bottom map is induced by the inclusion of the fibre inside the homotopy fibre, and the dotted arrow is defined by composition. It suffices to show that this last arrow is a bijection.

To do so, we go through the composition and use the explicit descriptions of the maps recalled in the last section. Let $\gamma \in \Omega _{[\mathcal L_0]} \mathrm {Pic}^\alpha (X)$ be a loop. Using the fibre bundle of Lemma 5.8

$$\begin{align*}\Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p\mathcal P_\alpha \setminus 0) / {\mathbb{C}}^\times \longrightarrow \mathrm{Pic}^\alpha(X) \end{align*}$$

we choose a lift to a path

$$\begin{align*}\alpha \colon [0,1] \to \Gamma_{\mathcal C^0, \mathrm{fib}}(J^1_p \mathcal P_\alpha \setminus 0) / {\mathbb{C}}^\times \end{align*}$$

starting at $[s_0]$ and ending at some $[s_1']$ . The map is given according to Lemma 5.10 by

$$\begin{align*}\pi_0 \big( \Omega_{[\mathcal L_0]} \mathrm{Pic}^\alpha(X) \big) \longrightarrow \pi_0(F_1), \quad [\gamma] \longmapsto [([s_1'], \alpha)]. \end{align*}$$

Write $\mathbb P \alpha $ for the image of $\alpha $ in $\Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X))$ . Applying then gives

$$\begin{align*}[([s_1'], \alpha)] \longmapsto [([s_1'], \mathbb P\alpha)]. \end{align*}$$

Using the fibration (6)

$$\begin{align*}\Gamma_{\mathcal C^0}(J^1\mathcal L_0 \setminus 0)/{\mathbb{C}}^\times \longrightarrow \Gamma_{\mathcal C^0}^\alpha(\mathbb P(J^1\mathcal O_X)) \end{align*}$$

we lift the path $\mathbb P\alpha $ to a path $\beta $ in $\Gamma _{\mathcal C^0}(J^1\mathcal L_0 \setminus 0)/{\mathbb {C}}^\times $ starting at $[s_0]$ and ending at some point $[s_1]$ . Using the principal bundle structure of (6), there is a unique class of a map $[\varphi _1] \in \mathrm {Map}(X,{\mathbb {C}}^\times ) / {\mathbb {C}}^\times $ such that $[\varphi _1 \cdot s_1] = [s_1']$ . By Lemma 5.11

$$\begin{align*}[([s_1'], \mathbb P\alpha)] = [([\varphi_1 \cdot s_1], \mathbb P \alpha)] = [([\varphi_1 \cdot s_0], \mathbb P \mathrm{cst}_{s_0})]. \end{align*}$$

From the homotopy equivalence given by the inclusion of the strict fibre inside the homotopy fibre of the bundle (6):

$$\begin{align*}\mathrm{Map}(X, {\mathbb{C}}^\times) / {\mathbb{C}}^\times \overset{\simeq}{\longrightarrow} F_2, \quad [\varphi] \longmapsto [([\varphi \cdot s_0], \mathbb P \mathrm{cst}_{s_0})] \end{align*}$$

we see that the inverse of the map then sends

$$\begin{align*}[([\varphi_1 \cdot s_0], \mathbb P \mathrm{cst}_{s_0})] \longmapsto [\varphi_1]. \end{align*}$$

We see first that it is surjective: indeed the composition of and the inverse of is surjective as any $[\varphi ]$ is the image of $[([\varphi \cdot s_0], \mathrm {cst}_{[s_0]})]$ . Secondly we check that it is compatible with the group structures: on the source given by composition of loops, and on the target given by multiplication of maps. This is enough to finish the proof as both groups are isomorphic to $H^1(X;\mathbb Z)$ , so that any surjective morphism is in fact an isomorphism.

We check the compatibility with the group structures using the notations as above. Let $\gamma _1, \gamma _2 \in \Omega _{[\mathcal L_0]} \mathrm {Pic}^\alpha (X)$ be two loops. As above, choose lifts $\alpha _1$ and $\alpha _2$ starting at $[s_0]$ and ending at $[s_1']$ and $[s_2']$ respectively. Then lift $\mathbb P\alpha _1$ and $\mathbb P\alpha _2$ to paths $\beta _1, \beta _2$ starting at $[s_0]$ and ending at $[s_1]$ and $[s_2]$ respectively. Write $[\varphi _1\cdot s_1] = [s_1']$ and $[\varphi _2 \cdot s_2] = [s_2']$ , for unique $[\varphi _1], [\varphi _2]$ . As we have shown above, the composition of , and the inverse of maps

$$\begin{align*}[\gamma_1] \longmapsto [\varphi_1], \quad [\gamma_2] \longmapsto [\varphi_2]. \end{align*}$$

Let us now consider the concatenation of loops $\gamma _1 \ast \gamma _2$ . First, observe that there exists a unique $[\varphi ]$ such that $[\varphi \cdot s_0] = [s_1']$ . Therefore the loop $\gamma _1 \ast \gamma _2$ can be lifted to the path $\alpha _3 := \alpha _1 \ast (\varphi \cdot \alpha _2)$ ending at $[s_3'] := [\varphi \cdot s_2']$ . Then we may lift $\mathbb P\alpha _3 := \mathbb P(\alpha _1 \ast (\varphi \cdot \alpha _2)) = \mathbb P(\alpha _1 \ast \alpha _2)$ to the path $\beta _1 \ast (\varphi _1^{-1}\cdot \varphi \cdot \beta _2)$ ending at $[s_3]$ . We write $[\varphi _3\cdot s_3] = [s_3']$ for a unique $[\varphi _3]$ . Once again, the composition of , and the inverse of maps

$$\begin{align*}[\gamma_1 \ast \gamma_2] \longmapsto [\varphi_3]. \end{align*}$$

Now observe that, as $[s_3]$ is the end point of $\beta _1 \ast (\varphi _1^{-1}\cdot \varphi \cdot \beta _2)$ , we have

$$\begin{align*}[s_3] = [\varphi_1^{-1}\cdot\varphi\cdot s_2]. \end{align*}$$

Therefore

$$\begin{align*}[\varphi_3\cdot s_3] = [\varphi_3\cdot\varphi_1^{-1}\cdot\varphi\cdot s_2]. \end{align*}$$

But $[\varphi _3\cdot s_3] = [s_3'] = [\varphi \cdot s_2'] = [\varphi \cdot \varphi _2\cdot s_2]$ . By uniqueness

$$\begin{align*}[\varphi_3\cdot\varphi_1^{-1}\cdot\varphi] = [\varphi\cdot\varphi_2], \end{align*}$$

hence $[\varphi _3] = [\varphi _1 \cdot \varphi _2]$ , which achieves the proof.

6.1 Haefliger’s tower of section spaces

Although Theorem 4.6 provides an integral homology isomorphism, we will mainly be interested in the rational cohomology groups for computational reasons. Fibrewise rationalisation (denoted $(-)_{f{\mathbb {Q}}}$ ) yields a fibration

(8) $$ \begin{align} \mathbb P^n_{\mathbb{Q}} \longrightarrow \mathbb P(J^1\mathcal O_X)_{f{\mathbb{Q}}} \longrightarrow X. \end{align} $$

By [Reference Møller26, Theorem 5.3], the natural map $\mathbb P(J^1\mathcal O_X) \to \mathbb P(J^1\mathcal O_X)_{f{\mathbb {Q}}}$ induces a map on section spaces

$$\begin{align*}\Gamma_{\mathcal C^0}(\mathbb P(J^1\mathcal O_X)) \longrightarrow \Gamma_{\mathcal C^0}(\mathbb P(J^1\mathcal O_X)_{f{\mathbb{Q}}}) \end{align*}$$

which is a rationalisation when restricted to a connected component on the source and target. (Beware the connected components of the source are naturally indexed by $H^2(X;\mathbb Z)$ , while those of the target are naturally indexed by $H^2(X;{\mathbb {Q}})$ .) We apply the general strategy described in [Reference Haefliger19, Section 1.3] to compute the rational homotopy type of the section space $\Gamma _{\mathcal C^0}(\mathbb P(J^1\mathcal O_X)_{f{\mathbb {Q}}})$ . The fibration (8) admits a Moore–Postnikov decomposition of the form

where each $p_i \colon Y_i \to Y_{i-1}$ is a principal fibration classified by the k-invariant $k_{i-1}$ . The latter were computed by Møller:

Lemma 6.1 (Compare [Reference Møller27, Lemma 2.1])

The k-invariant $k_0$ is trivial. In particular $Y_1 \simeq X \times K({\mathbb {Q}},2)$ . Writing $z \in H^2(K({\mathbb {Q}},2);{\mathbb {Q}})$ for the generator, $k_1$ corresponds to the cohomology class

$$\begin{align*}\sum_{i=0}^{n+1} (-1)^i c_i(J^1\mathcal O_X) \otimes z^{n+1-i} \in H^*(X;{\mathbb{Q}}) \otimes H^*(K({\mathbb{Q}},2);{\mathbb{Q}}). \end{align*}$$

Let $s \in \Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X)_{f{\mathbb {Q}}})$ with $\alpha \in H^2(X;{\mathbb {Q}})$ . The map $p_2 \circ s$ is a section of $Y_1 \to X$ , and we denote by $\Gamma _1 \subset \Gamma _{\mathcal C^0}(Y_1 \to X)$ its connected component. As $k_0$ is trivial by Lemma 6.1, there is a homotopy equivalence

$$\begin{align*}\Gamma_{\mathcal C^0}(Y_1 \to X) \simeq \mathrm{Map}(X, K({\mathbb{Q}},2)) \simeq K({\mathbb{Q}},2) \times K(H^1(X;{\mathbb{Q}}),1) \times H^2(X;{\mathbb{Q}}). \end{align*}$$

and $\Gamma _1$ corresponds to the connected component indexed by $\alpha $ .

Lemma 6.2 (Compare [Reference Møller27, Lemma 2.2])

Let $\Psi $ be the composite

(9) $$ \begin{align} \Psi \colon K({\mathbb{Q}},2) \times K(H^1(X;{\mathbb{Q}}),1) \times X \simeq \Gamma_1 \times X \overset{\mathrm{ev}}{\longrightarrow} Y_1 \overset{k_1}{\longrightarrow} K({\mathbb{Q}},2n+2). \end{align} $$

Let $z \in H^2(K({\mathbb {Q}},2);{\mathbb {Q}})$ be the generator. Let $\{x_j\}$ be a basis of $H^1(X;\mathbb Z)$ and let $\{x_j'\}$ be the dual basis of $H^1(K(H^1(X;\mathbb Z),1);\mathbb Z) \cong H^1(X;\mathbb Z)^\vee $ . The morphism induced in cohomology $\Psi ^*$ sends the generator $\chi \in H^{2n+2}(K({\mathbb {Q}},2n+2);{\mathbb {Q}})$ to the class:

$$\begin{align*}\Psi^*(\chi) = \sum_{i=0}^{n+1} (-1)^i \bigg(1\otimes 1 \otimes c_i(J^1\mathcal O_X)\bigg) \cup \bigg(z \otimes 1 \otimes 1 + 1 \otimes 1 \otimes \alpha + \sum_j 1 \otimes x_j' \otimes x_j\bigg)^{n+1-i}. \end{align*}$$

Let $\overline {k_1} \colon \Gamma _1 \to K({\mathbb {Q}},2n+2)^X$ be the adjoint of the map (9). There is a homotopy equivalence (see [Reference Haefliger19, Section 1])

(10) $$ \begin{align} K({\mathbb{Q}},2n+2)^X \simeq \prod_{i=2}^{2n+2} K(H^{2n+2-i}(X;{\mathbb{Q}}), i). \end{align} $$

Lemma 6.3 (Compare [Reference Haefliger19])

Let $\varphi _i$ be the map to the i-th factor of the product:

$$\begin{align*}\varphi_i \colon \Gamma_1 \overset{\overline{k_1}}{\longrightarrow} K({\mathbb{Q}},2n+2)^X \longrightarrow K(H^{2n+2-i}(X;{\mathbb{Q}}),i). \end{align*}$$

The morphism induced in cohomology is given explicitly by:

$$ \begin{align*} \varphi_i^* \colon H^{2n+2-i}(X;{\mathbb{Q}})^\vee \cong H^i(K(H^{2n+2-i}(X;{\mathbb{Q}});{\mathbb{Q}}) &\longrightarrow H^i(\Gamma_1;{\mathbb{Q}}) \\ y' &\longmapsto y' \cap \Psi^*(\chi). \end{align*} $$

Here, for $w \otimes y \in H^*(\Gamma _1) \otimes H^*(X)$ and $y' \in H^*(X)^\vee $ , we write $y' \cap (w \otimes y) = y'(y) w$ .

Proposition 6.4 (Compare [Reference Haefliger19])

There is a fibration

$$\begin{align*}K({\mathbb{Q}},2n+1)^X \longrightarrow \Gamma_{\mathcal C^0}^\alpha(\mathbb P(J^1\mathcal O_X)_{f{\mathbb{Q}}}) \longrightarrow \Gamma_1 \end{align*}$$

pulled back from the path space fibration over $K({\mathbb {Q}},2n+2)^X$ via the map $\overline {k_1}$ .

Theorem 6.5. Let z and $\{x_j'\}$ be as in Lemma 6.2. Let $\{y_{ik}'\}$ be a basis of the rational cohomology of (10) where $y_{ik}' \in H^{2n+2-i}(X;{\mathbb {Q}})^\vee $ is in degree i. The rational cohomology of $\Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X)_{f{\mathbb {Q}}})$ is given by the cohomology of the following commutative differential graded algebra:

$$\begin{align*}\Lambda \big(z, \ x_j', \ s^{-1}y_{ik}' \big), \quad d(z) = 0, \ d(x_j') = 0, \ d(s^{-1}y_{ik}') = \varphi_i^*(y_{ik}') \end{align*}$$

where z is in degree $2$ , each $x_j'$ is in degree $1$ , each $s^{-1}y_{ik}'$ is in degree $i-1$ , and $\varphi _i^*$ is given as in Lemma 6.3.

Proof. By Proposition 6.4, there is a homotopy pullback square:

By the Eilenberg–Moore theorem, the cohomology of the pullback is given by the derived tensor product

$$\begin{align*}\Lambda(z,x_j') \otimes^{\mathbb{L}}_{\Lambda(y_{ik}')} {\mathbb{Q}} \end{align*}$$

which can be computed by choosing $\Lambda (y_{ik}') \to \big (\Lambda (s^{-1}y_{ik}', y_{ik}'), \ d(s^{-1}y_{ik}') = y_{ik}' \big ) \simeq {\mathbb {Q}}$ as a cofibrant replacement.

Example 6.6. Let X be a smooth curve ( $n=1$ ) of genus $1$ (i.e., a torus). It is a framed manifold, hence its jet bundle has trivial Chern classes. Write $a, b$ for the standard basis of $H^1(X;\mathbb Z)$ such that $a^2 = b^2 = 0$ and $u = ab$ generates $H^2(X;\mathbb Z)$ . Let $a',b'$ be the dual basis. Let $\alpha = k \cdot u$ for some $k \in \mathbb Z$ . With the notations of Lemma 6.2 we have

$$ \begin{align*} \Psi^*(\chi) &= \big(z \otimes 1 \otimes 1 + 1 \otimes 1 \otimes \alpha + 1 \otimes a' \otimes a + 1 \otimes b \otimes b'\big)^2 \\ &= (2k(z \otimes 1) - 2(1 \otimes a'b')) \otimes u + 2(z \otimes a' \otimes a) + 2(z \otimes b' \otimes b) + z^2 \otimes 1 \otimes 1. \end{align*} $$

The morphisms $\varphi _i^*$ of Lemma 6.3 are given by

$$ \begin{align*} \varphi_2^* \colon u' &\longmapsto u' \cap \Psi^*(\chi) = 2k(z \otimes 1) - 2(1 \otimes a'b') \\ \varphi_3^* \colon a' &\longmapsto a' \cap \Psi^*(\chi) = 2(z \otimes a') \\ b' &\longmapsto b' \cap \Psi^*(\chi) = 2(z \otimes b') \\ \varphi_4^* \colon 1 &\longmapsto 1 \cap \Psi^*(\chi) = z^2 \otimes 1. \end{align*} $$

Therefore the cohomology of $\Gamma _{\mathcal C^0}^\alpha (\mathbb P(J^1\mathcal O_X)) \simeq \mathrm {Map}_\alpha (X, \mathbb P^1)$ is given by the cohomology of the CDGA:

$$\begin{align*}\Lambda(z, a', b', y_1, y_2, y_2', y_3), \quad d(y_1) = 2kz - 2a'b', \ d(y_2) = 2za', \ d(y_2') = 2zb', \ d(y_3) = z^2 \end{align*}$$

where the indices on the last four variables indicate their degrees. (See [Reference Møller27, Section 3] for related computations.)

6.2 Homological stability

Despite the formula given in Theorem 6.5, we do not know in general whether the cohomology varies with $\alpha $ or not. Nonetheless, when X is topologically parallelisable, we can make the following qualitative remark:

Proposition 6.7. Let X be a smooth projective complex variety such that $\Omega ^1_X$ is a topologically trivial vector bundle, and let $\alpha \in H^2(X;{\mathbb {Q}})$ . Then there is a homotopy equivalence

$$\begin{align*}\Gamma_{\mathcal C^0}^{\alpha}(\mathbb P(J^1\mathcal O_X)_{f{\mathbb{Q}}}) \simeq \Gamma_{\mathcal C^0}^{k\alpha}(\mathbb P(J^1\mathcal O_X)_{f{\mathbb{Q}}}) \end{align*}$$

for any nonzero rational number $k \in {\mathbb {Q}}^\times $ .

Proof. As X is topologically parallelisable, the jet bundle $J^1\mathcal O_X$ is topologically trivial. Hence the section space is the mapping space into the fibre:

$$\begin{align*}\Gamma_{\mathcal C^0}^{k\alpha}(\mathbb P(J^1\mathcal O_X)_{f{\mathbb{Q}}}) \simeq \mathrm{Map}_{k\alpha}(X, \mathbb P^n_{\mathbb{Q}}) \end{align*}$$

where the subscript $k\alpha $ on the right-hand side indicates the connected component of the maps which pull back the generator in cohomology to $k\alpha $ . Postcomposing with a self map of $\mathbb P^n_{\mathbb {Q}}$ of degree $1/k$ gives a homotopy equivalence

$$\begin{align*}\mathrm{Map}_{k\alpha}(X, \mathbb P^n_{\mathbb{Q}}) \simeq \mathrm{Map}_{\alpha}(X, \mathbb P^n_{\mathbb{Q}}). \end{align*}$$

Corollary 6.8. Let X be a smooth projective complex variety which is topologically parallelisable. Let $\alpha \in \mathrm {NS}(X)$ be ample, and assume that $d(X,\alpha ) \geq 1$ (see Definition 2.17). Then, for any integer $k \geq 1$ , there is a map

$$\begin{align*}\mathcal{M}_{\mathrm{hyp}}^{k\alpha} \longrightarrow \mathrm{Map}_{\alpha}(X,\mathbb P^n_{\mathbb{Q}}) \end{align*}$$

inducing an isomorphism in rational cohomology in the range of degrees $* < \frac {k \cdot d(X,\alpha ) - 3}{2}$ . In particular, the rational cohomology stabilises as $k \to \infty $ .

Remark 6.9. The rational homotopy type of the mapping space $\mathrm {Map}(X,\mathbb P^n_{\mathbb {Q}})$ can be easily computed without the results of the last section. In [Reference Berglund3, Theorem 1.4], Berglund gives an explicit $L_\infty $ -algebra model whose underlying graded ${\mathbb {Q}}$ -vector space is given by

$$\begin{align*}H^*(X;{\mathbb{Q}}) \otimes {\mathbb{Q}}\cdot\{u,w\} \end{align*}$$

where $H^i(X)$ sits in degree $-i$ , and $u,w$ are respectively in degrees $1$ and $2n$ . (This uses that X is a formal space.) For the connected component corresponding to $\alpha \in H^2(X;{\mathbb {Q}})$ , the associated Maurer–Cartan element is $\tau = \alpha \otimes u$ . In particular, the only possibly nonvanishing brackets are given by

$$\begin{align*}[x_1 \otimes u, \ldots, x_r \otimes u]_\tau = \pm \frac{(n+1)!}{(n+1-r)!} (\alpha^{n+1-r} x_1 x_2\cdots x_r) \otimes w. \end{align*}$$

In fact, in the case of the torus, the Chevalley–Eilenberg complex associated to this $L_\infty $ -algebra is the CDGA given in Example 6.6.

7.1 Scanning

We begin with a brief and intuitive exposition of the general idea behind scanning. Suppose given $M \subset N$ , a d-dimensional submanifold of an n-dimensional manifold. We can try to see what M looks like by looking locally at each point of N. One can imagine looking through a magnifying glass: either we are far from M and see nothing, or close to M and see a first-order approximation of M, that is, a tangent space, together with a small vector from the center of the lens to M. To formalise this intuition, recall the tautological quotient bundle over the Grassmannian of d-dimensional planes in $\mathbb R^n$ :

$$\begin{align*}\mathbb R^{n-d} \longrightarrow \gamma_{d,n}^\perp := \{ (H,v) \mid H \in \mathrm{Gr}(d, \mathbb R^n), \; v \in \mathbb R^n / H \} \longrightarrow \mathrm{Gr}(d, \mathbb R^n). \end{align*}$$

One thinks of a point $(H,v) \in \gamma _{d,n}^\perp $ as a d-dimensional plane together with a normal vector. The Thom space $\mathrm {Gr}(d, \mathbb R^n)^{\gamma _{d,n}^\perp }$ is obtained by one-point compactifying the total space. This construction can be done fibrewise to the tangent bundle $TN$ of N, and we denote by $\mathrm {Gr}(d,TN)^{\gamma _{d,n}^\perp }$ the resulting bundle over N. The submanifold M then gives a section

$$\begin{align*}N \longrightarrow \mathrm{Gr}(d,TN)^{\gamma_{d,n}^\perp} \end{align*}$$

obtained by sending a point far away from M to the point at infinity (in the Thom space), and sending a point $x \in N$ close to a point $y \in M$ to the tangent space $T_yM \subset T_yN$ together with the vector pointing from y to x. Of course this requires to be made precise, for example, by choosing a tubular neighbourhood of M inside N. In many cases, this idea can be implemented in families to obtain a map from a parameter space of submanifolds to a section space.

We are now ready to give an interpretation of the jet map of Theorem 4.6 in the spirit of scanning. We take X and $\alpha $ as in the assumptions of that theorem. We also denote by $n = \dim _{\mathbb {C}} X$ the complex dimension of X. The main observation is the following:

Lemma 7.1. For integers $d, m$ , let $\mathrm {Gr}(d, {\mathbb {C}}^m)$ be the Grassmannian of complex d-dimensional planes in ${\mathbb {C}}^m$ and $\gamma _{d,m}^\perp $ be the tautological quotient bundle. There is a homeomorphism

$$ \begin{align*} \gamma_{d,m}^\perp &\overset{\cong}{\longrightarrow} \mathrm{Gr}(d+1, {\mathbb{C}}^m \oplus {\mathbb{C}}) \setminus \mathrm{Gr}(d+1, {\mathbb{C}}^m) \\ (H,v) &\longmapsto (H,0) \oplus (v,1) \end{align*} $$

where $\mathrm {Gr}(d+1, {\mathbb {C}}^m)$ is embedded inside $\mathrm {Gr}(d+1, {\mathbb {C}}^m \oplus {\mathbb {C}})$ via $P \mapsto (P,0)$ .

When V is an n-dimensional complex vector space, the tautological quotient bundle over $\mathrm {Gr}(n-1,V)$ is homeomorphic to $\mathbb P(V \oplus {\mathbb {C}}) \setminus \{*\}$ . Hence its Thom space is $\mathbb P(V \oplus {\mathbb {C}})$ . Using this observation and the isomorphism $J^1\mathcal O_X \cong \Omega ^1_X \oplus \mathcal O_X$ as smooth complex vector bundles, we see that

$$\begin{align*}\mathbb P(J^1\mathcal O_X) \cong \mathrm{Gr}(n-1, \Omega^1_X)^{\gamma_{n-1,n}^\perp} \cong \mathrm{Gr}(n-1, TX)^{\gamma_{n-1,n}^\perp}. \end{align*}$$

Under these identifications, the jet map

$$\begin{align*}\mathcal{M}_{\mathrm{hyp}}^\alpha \longrightarrow \Gamma_{\mathcal C^0} \big( \mathrm{Gr}(n-1, TX)^{\gamma_{n-1,n}^\perp} \big) \end{align*}$$

is very close to the general idea of scanning described above. Indeed, given a smooth hypersurface $V(s) \subset X$ , $x \mapsto s(x)$ vanishes on the hypersurface and thus records in some sense the distance to the hypersurface, an analogue of the normal vector. Moreover, when $s(x)$ vanishes the derivative $ds(x)$ is nonzero and we can identify the tangent space at $x \in V(s)$ as the kernel $\ker (ds(x))$ .

7.2 Configuration spaces on curves

Let us now describe the case $n=1$ in more detail. The variety X is then a curve, and for $\alpha \in \mathrm {NS}(X) \subset H^2(X;\mathbb Z)$ we write $\deg \alpha \in \mathbb Z$ for its image under the degree isomorphism $\deg \colon H^2(X;\mathbb Z) \overset {\cong }{\to } \mathbb Z$ induced by the complex orientation. A hypersurface of Chern class $\alpha $ is simply an unordered configuration of $\deg \alpha $ points and we have a homeomorphism

$$ \begin{align*} \mathcal{M}_{\mathrm{hyp}}^\alpha &\overset{\cong}{\longrightarrow} \mathrm{UConf}_{\deg \alpha}(X) \\ ([\mathcal L] \in \mathrm{Pic}^\alpha(X), \ [s] \in \Gamma_{\mathrm{ns}}(\mathcal P_{[\mathcal L]})/{\mathbb{C}}^\times) &\longrightarrow V([s]). \end{align*} $$

There is also an identification

$$\begin{align*}\mathbb P(J^1\mathcal O_X) \cong \mathrm{Gr}(0, TX)^{\gamma_{0,1}^\perp} \cong T^+X \end{align*}$$

with the fibrewise one-point compactification of the tangent bundle. In [Reference McDuff25], McDuff studied a scanning map on configuration spaces of points on a manifold, that is, spaces of $0$ -dimensional submanifolds. In the present work, we instead study (complex) codimension $1$ submanifolds. On a curve these agree and we recover a special case of McDuff’s theorem, although our scanning map is now more algebraic in nature:

Theorem 7.2. Let X be a smooth projective complex curve of genus g. Let $\alpha \in H^2(X;\mathbb Z)$ be such that $\deg \alpha> 2g-2$ . The jet map

$$\begin{align*}\mathrm{UConf}_{\deg \alpha}(X) \cong \mathcal{M}_{\mathrm{hyp}}^\alpha \longrightarrow \Gamma_{\mathcal C^0}^\alpha(\mathbb P(J^1\mathcal O_X)) \cong \Gamma_{\mathcal C^0}^\alpha(T^+X) \end{align*}$$

induces an isomorphism in integral homology in the range of degrees $* < \frac {\deg \alpha - 2g - 3}{2}$ .

8.1 Recollections on stable classes

We will shortly recall from [Reference Aumonier1] the geometric interpretation of the stable classes in the rational cohomology of $\Gamma _{\mathrm {ns}}(\mathcal L)$ . As we are here mostly interested in the quotient by the scalars ${\mathbb {C}}^\times $ , we first make two observations.

Lemma 8.1. Let $\mathcal L$ be a $2$ -jet ample line bundle on X. Then there exists a homogeneous polynomial

$$\begin{align*}\Delta \colon \Gamma(\mathcal L) \longrightarrow {\mathbb{C}} \end{align*}$$

such that $\Gamma _{\mathrm {ns}}(\mathcal L) = \Delta ^{-1}({\mathbb {C}}^\times )$ . In other words, the complement of $\Gamma _{\mathrm {ns}} / {\mathbb {C}}^\times \subset \mathbb P(\Gamma (\mathcal L))$ is a hypersurface given by the vanishing of the discriminant polynomial $\Delta $ .

Proof. The general theory of discriminants is explained in [Reference Gelfand, Kapranov and Zelevinsky15], see, for example, page 15 for the definition of the discriminant polynomial. In general, the subspace of singular sections of $\mathcal L$ has codimension at least $1$ , and exactly $1$ in most cases, see, for example, [Reference Gelfand, Kapranov and Zelevinsky15, Corollary 1.2]. We show that when $\mathcal L$ is $2$ -jet ample, we are indeed in the latter case. Consider the incidence variety

$$\begin{align*}R := \left\{ (x,H) \in X \times \mathbb P(\Gamma(\mathcal L)) \mid x \in H \text{ is a singular point}\right\} \end{align*}$$

and its two projections $p_1 \colon R \to X$ and $p_2 \colon R \to \mathbb P(\Gamma (\mathcal L))$ . The set of singular hypersurfaces is $p_2(R)$ . By $1$ -jet ampleness of $\mathcal L$ , $p_1$ is a vector bundle. This implies that R is irreducible of dimension $\dim \mathbb P(\Gamma (\mathcal L)) - 1$ . In [Reference Katz21, Proposition 3.4], Katz shows that the locus of $(x,H) \in R$ where x is a nondegenerate singular point of H is open. By $2$ -jet ampleness, it is nonempty, hence dense as R is irreducible. Now [Reference Katz21, Proposition 3.5] shows that $p_2$ is birational. Therefore $p_2(R)$ has codimension $1$ .

Lemma 8.2. Suppose that $\mathcal L$ is $2$ -jet ample. Then there is an isomorphism of $H^*(\Gamma _{\mathrm {ns}}(\mathcal L); {\mathbb {Q}})$ -modules:

$$\begin{align*}H^*(\Gamma_{\mathrm{ns}}(\mathcal L); {\mathbb{Q}}) \cong H^*(\Gamma_{\mathrm{ns}}(\mathcal L)/ {\mathbb{C}}^\times; {\mathbb{Q}}) \otimes H^*({\mathbb{C}}^\times; {\mathbb{Q}}). \end{align*}$$

Proof. Let $\Delta \colon \Gamma (\mathcal L) \to {\mathbb {C}}$ be the discriminant provided by Lemma 8.1 so that $\Gamma _{\mathrm {ns}}(\mathcal L) = \Gamma (\mathcal L) \setminus \Delta ^{-1}(0)$ . There is a fibre bundle

$$\begin{align*}{\mathbb{C}}^\times \longrightarrow \Gamma_{\mathrm{ns}}(\mathcal L) \longrightarrow \Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times \end{align*}$$

and for any fibre the map ${\mathbb {C}}^\times \hookrightarrow \Gamma _{\mathrm {ns}}(\mathcal L) \overset {\Delta }{\to } {\mathbb {C}}^\times $ is of degree $\deg (\Delta ) \neq 0$ , hence an isomorphism on rational cohomology. The lemma then follows by the Leray–Hirsch theorem.

Consider the universal bundle of hypersurfaces:

$$ \begin{align*} V(s) \longrightarrow U(\mathcal L) := \{ (s, x) \in \Gamma_{\mathrm{ns}}(\mathcal L) \times X \mid s(x) = 0 \} \overset{\pi}{\longrightarrow} \Gamma_{\mathrm{ns}}(\mathcal L). \end{align*} $$

At each point $(s,x) \in U(\mathcal L)$ , the derivative $ds(x)$ is nonzero, thus giving a map

$$ \begin{align*} j \colon U(\mathcal L) \longrightarrow \Omega^1_X \otimes \mathcal L \setminus 0. \end{align*} $$

For $\mathcal L$ ample enough such that the Euler class of $\Omega ^1_X \otimes \mathcal L$ is non zero, the cohomology of the target of j in degrees $* \geq 2n$ is

$$\begin{align*}H^{* \geq 2n}(\Omega^1_X \otimes \mathcal L \setminus 0; \mathbb Z) \cong H^{* \geq 1}(X; \mathbb Z) [2n-1], \end{align*}$$

where $[2n-1]$ indicates a shift of degrees.

Proposition 8.3 (Compare [Reference Aumonier1, Proposition 8.6])

Let $\mathcal L$ be a d-jet ample line bundle on a smooth projective complex variety X. Suppose furthermore that the Euler class of $\Omega ^1_X \otimes \mathcal L$ is nonzero. Then the morphism given by pulling back along j and integrating along the fibres of $\pi $

$$\begin{align*}\pi_! \circ j^* \colon \Lambda(H^{* \geq 2n}(\Omega^1_X \otimes \mathcal L \setminus 0; {\mathbb{Q}})) \longrightarrow H^*(\Gamma_{\mathrm{ns}}(\mathcal L); {\mathbb{Q}}) \longrightarrow H^*(\Gamma_{\mathrm{ns}}(\mathcal L) /{\mathbb{C}}^\times; {\mathbb{Q}}) \end{align*}$$

is an isomorphism in the range of degrees $* < \frac {d-1}{2}$ . (The second arrow is obtained from Lemma 8.2 by projecting away from the second tensor factor.)

Proof. Let us write $g \colon \Gamma _{\mathrm {ns}}(\mathcal L) \times X \to J^1\mathcal L \setminus 0$ for the jet map and $p \colon \Gamma _{\mathrm {ns}}(\mathcal L) \times X \to X$ for the projection. Then [Reference Aumonier1, Proposition 8.6] shows that

$$\begin{align*}p_! \circ g^* \colon \Lambda(H^{* \geq 2n+2}(J^1\mathcal L \setminus 0; {\mathbb{Q}})) \longrightarrow H^*(\Gamma_{\mathrm{ns}}(\mathcal L) /{\mathbb{C}}^\times; {\mathbb{Q}}) \end{align*}$$

is an isomorphism in the range $* < \frac {d-1}{2}$ . There is a pullback square

where $\iota $ and g are transverse, as one sees directly from a local computation using that $\mathcal L$ is globally generated. Thus $i_! \circ j^* = g^* \circ \iota _!$ . Furthermore, $\pi = p \circ i$ so that $\pi _! = p_! \circ i_!$ . The proposition now follows from the fact that $\iota _!$ sends $H^{* \geq 2n}(\Omega ^1_X \otimes \mathcal L \setminus 0; {\mathbb {Q}})$ isomorphically onto $H^{* \geq 2n+2}(J^1\mathcal L \setminus 0; {\mathbb {Q}})$ .

8.2 Comparison with diffeomorphisms

Let us fix a nonsingular section $s \in \Gamma _{\mathrm {ns}}(\mathcal L)$ and write $H := V(s)$ for the associated hypersurface. From the point of view of the Hilbert scheme developed in Section 3, the subspace $\Gamma _{\mathrm {ns}}(\mathcal L) / {\mathbb {C}}^\times $ classifies algebraic bundles with fibres equivalent to H (as divisors) and embedded in X. Seeing H only as a smooth oriented real manifold, one can consider the classifying space $B\mathrm {Diff}^{\mathrm {or}}(H)$ of its group of orientation preserving diffeomorphisms. By definition, this latter space classifies fibre bundles with fibre H and structure group $\mathrm {Diff}^{\mathrm {or}}(H)$ . In particular, there is a map

(11) $$ \begin{align} \Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times \longrightarrow B\mathrm{Diff}^{\mathrm{or}}(H) \end{align} $$

classifying the universal bundle

(12) $$ \begin{align} U(\mathcal L) / {\mathbb{C}}^\times := \{ ([s], x) \in \Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times \times X \mid s(x) = 0 \} \overset{\pi}{\longrightarrow} \Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times. \end{align} $$

One could wonder if the map (11) induces an isomorphism in rational cohomology in a range of degrees. This was shown to be false when $X = \mathbb P^n$ and $\mathcal L = \mathcal O(d)$ by Randal-Williams [Reference Randal-Williams28] (and it will also follow from our results below). On the other hand, one could alter the situation by considering diffeomorphisms preserving other kinds of tangential structures (see Remark 8.4 below, or [Reference Galatius and Randal-Williams13, Definition 2.1, and Section 4.5]): we have picked an orientation, but could have chosen an almost complex structure, or a spin structure in some cases, etc. We construct below a tangential structure $\theta $ such that the map classifying the universal bundle

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L) \longrightarrow B\mathrm{Diff}^\theta(H) \end{align*}$$

is “very close” to being a rational homology isomorphism in a range of degrees.

Choose maps $[TX] \colon X \to BU(n)$ and $[\mathcal L] \colon X \to BU(1)$ classifying respectively the tangent bundle of X and $\mathcal L$ as topological complex vector bundles. Let B be the space defined by the following homotopy pullback square:

(13)

where $"\oplus " \colon BU(n-1) \times BU(1) \to BU(n)$ classifies taking the direct sum of vector bundles. We will adopt the convention that a tangential structure is a space over $BO(2n-2)$ (See [Reference Galatius and Randal-Williams13, Section 4.5] for how it relates to other definitions.) In this language, the tangential structure $\theta $ simply means the map:

$$\begin{align*}\theta \colon B \longrightarrow BU(n-1) \times BU(1) \overset{\mathrm{pr}_1}{\longrightarrow} BU(n-1) \longrightarrow BO(2n-2). \end{align*}$$

Remark 8.4. Up to replacing $\theta $ by a fibration $B \simeq \overline {B} \to BO(2n-2)$ , a $\theta $ -structure on a $(2n-2)$ -manifold M is the datum of a lift of the map classifying the tangent bundle:

By the universal property of the homotopy pullback, this amounts to providing two maps

$$\begin{align*}M \longrightarrow BU(n-1) \times BU(1) \quad \text{ and } \quad M \longrightarrow X \end{align*}$$

which become homotopic after further composing to $BU(n) \times BU(1)$ and such that $M \to BU(n-1)$ classifies the tangent bundles of M. In other words, this is the data of a map $\iota \colon M \to X$ and a complex line bundle $\mathcal L'$ (corresponding to $M \to BU(1)$ ) such that $TH \oplus \mathcal L' \cong \iota ^*TX$ and $\iota ^* \mathcal L \cong \mathcal L'$ . Therefore, a $\theta $ -structure on M should be interpreted as a choice of an immersion $\iota \colon M \to X$ with normal bundle $\iota ^*\mathcal L$ .

We have chosen to construct B via the homotopy pullback (13) as it allowed us to informally understand the geometric meaning of a $\theta $ -structure. But it turns out that we can give more familiar expressions for B and the bundle classified by $\theta $ as explained in the following two lemmas.

Lemma 8.5. There is a homotopy equivalence above X

$$\begin{align*}B \simeq \Omega^1_X \otimes \mathcal L \setminus 0. \end{align*}$$

Proof. We will use the following explicit point set models:

$$ \begin{align*} BU(j) &:= \{ P \subset {\mathbb{C}}^\infty \mid P \text{ is a } j\text{-dimensional plane} \}, \\ \gamma_j &:= \{ (P, v) \mid P \in BU(j), \ v \in P \}, \\ \gamma_j^\vee &:= \{ (P, \varphi) \mid P \in BU(j), \ \varphi \colon P \to {\mathbb{C}} \text{ linear map}\}, \end{align*} $$

for the classifying space, and its tautological vector bundle and the dual of it. Recall that the classical fibration sequence

$$\begin{align*}{\mathbb{C}}^n \setminus 0 \longrightarrow BU(n-1) \longrightarrow BU(n) \end{align*}$$

can be modelled by the sphere bundle of the dual tautological bundle using the homeomorphism

$$\begin{align*}\gamma_n^\vee \setminus 0 \cong BU(n-1), \quad (P, \varphi \colon P \twoheadrightarrow {\mathbb{C}}) \mapsto \ker(\varphi). \end{align*}$$

In fact, we can likewise model the map

$$\begin{align*}BU(n-1) \times BU(1) \overset{("\oplus", \mathrm{pr}_2)}{\longrightarrow} BU(n) \times BU(1) \end{align*}$$

by the sphere bundle $\gamma _n^\vee \boxtimes \gamma _1 \setminus 0$ . Indeed, we may write

$$\begin{align*}\gamma_n^\vee \boxtimes \gamma_1 \setminus 0 \cong \{ (P, L, \varphi \otimes v) \mid P \in BU(n), \ L \in BU(1), \ \varphi \otimes v \in P^\vee \otimes L \setminus 0 \} \end{align*}$$

and use the homeomorphism

$$\begin{align*}\gamma_n^\vee \boxtimes \gamma_1 \setminus 0 \cong BU(n-1) \times BU(1), \quad (P, L, \varphi \otimes v) \mapsto (\ker(\varphi \otimes v), L). \end{align*}$$

Under these identifications, and fixing an isomorphism ${\mathbb {C}}^\infty \oplus {\mathbb {C}}^\infty \cong {\mathbb {C}}^\infty $ , one can check that the map $"\oplus "$ is simply the direct sum $\oplus $ :

$$\begin{align*}(P, L, \varphi \otimes v) \longmapsto \ker(\varphi \otimes v) \oplus L \subset {\mathbb{C}}^\infty \oplus {\mathbb{C}}^\infty \cong {\mathbb{C}}^\infty. \end{align*}$$

Hence, by pulling back along $([TX], [\mathcal L])$ , one sees that $B \simeq \Omega ^1_X \otimes \mathcal L \setminus 0$ .

Proof. We will use the homeomorphism

$$\begin{align*}\Omega^1_X \otimes \mathcal L \setminus 0 \cong \{ (x, \varphi) \mid x \in X, \ \varphi \colon TX|_x \twoheadrightarrow \mathcal L|_x \text{ surjective linear map} \} \end{align*}$$

given by identifying a nonzero vector of $(\Omega ^1_X \otimes \mathcal L)|_x$ with a surjective linear map. As one sees from the point set models described in the proof of Lemma 8.5, the pullback vector bundle $\theta ^*\gamma $ classified by $\theta $ is equivalent to the one whose fibre above a point $(x, \varphi )$ is given by the kernel of $\varphi $ . Writing out the vector bundles

$$ \begin{align*} q^*TX &= \{ (x,\varphi,v) \mid (x,\varphi) \in \Omega^1\otimes \mathcal L \setminus 0, \ v \in TX|_x \}, \\ q^*\mathcal L &= \{ (x,\varphi,v) \mid (x,\varphi) \in \Omega^1\otimes \mathcal L \setminus 0, \ v \in \mathcal L|_x \}, \end{align*} $$

we identify $\theta ^*\gamma $ as the kernel of the morphism of vector bundles

$$\begin{align*}q^*TX \longrightarrow q^*\mathcal L, \ (x,\varphi,v) \longmapsto (x,\varphi,\varphi(v)). \end{align*}$$

We thus obtain the short exact sequence of vector bundles

$$\begin{align*}0 \longrightarrow \theta^*\gamma \longrightarrow q^*TX \longrightarrow q^*\mathcal L \longrightarrow 0 \end{align*}$$

which proves the lemma.

Let $H = V(s)$ be a smooth hypersurface with $s \in \Gamma _{\mathrm {ns}}(\mathcal L)$ . Using nonsingularity, we obtain a map $\ell \colon H \to \Omega ^1_X \otimes \mathcal L \setminus 0$ given by $\ell (x) = ds(x)$ .

Proof. The inclusion $\iota \colon H \hookrightarrow X$ factors as

$$\begin{align*}H \overset{\ell}{\longrightarrow} \Omega^1_X \otimes \mathcal L \setminus 0 \longrightarrow X \end{align*}$$

where the second map is the projection map of the bundle, hence $(2n-1)$ -connected. Therefore it suffices to show that $\iota \colon H \to X$ is $(n-1)$ -connected. But this is precisely the Lefschetz hyperplane theorem.

Recall from [Reference Galatius and Randal-Williams13, Section 4.5] that a $\theta $ -structure on a manifold M is a bundle map (fibrewise linear isomorphism) $TM \to \theta ^*\gamma $ , where $\gamma $ is the universal bundle above $BO(2n-2)$ . In the proposition below, we observe that each $\ell \colon H \to \Omega ^1_X \otimes \mathcal L \setminus 0$ underlies a bundle map $\hat \ell \colon TH \to \theta ^*\gamma $ , and that these naturally assemble when varying H in the universal bundle.

Proof. Let $T_vU(\mathcal L)$ be the vertical tangent bundle of the universal bundle. By definition of a bundle with $\theta $ -structure, we have to provide the horizontal maps in the following diagram to construct a vector bundle map

which restricts to a linear isomorphism in each fibre. Using the notation from the proof of Lemma 8.6, we write

$$\begin{align*}\Omega^1_X \otimes \mathcal L \setminus 0 = \{ (x, \varphi) \mid x \in X, \ \varphi \colon TX|_x \twoheadrightarrow \mathcal L|_x \text{ surjective linear map} \} \end{align*}$$

and

$$\begin{align*}\theta^*\gamma = \{ (x, \varphi, v) \mid (x,\varphi) \in \Omega^1_X \otimes \mathcal L \setminus 0, \ v \in \ker(\varphi) \}. \end{align*}$$

Differentiating a nonsingular section $s \colon X \to \mathcal L$ yields a short exact sequence of vector bundles

$$\begin{align*}0 \longrightarrow TV(s) \longrightarrow TX \overset{ds}{\longrightarrow} \mathcal L \longrightarrow 0 \end{align*}$$

which shows that $\ker (ds(x)) = TV(s)|_x$ . In particular, we have a point set model for the vertical tangent bundle given by

$$\begin{align*}T_vU(\mathcal L) = \left\{ (s,x,v) \mid s \in U(\mathcal L), \ x \in V(s), \ v \in \ker(ds(x)) \right\}. \end{align*}$$

Hence, taking

$$\begin{align*}U(\mathcal L) \longrightarrow \Omega^1_X \otimes \mathcal L \setminus 0, \quad (s,x) \longmapsto (x, ds(x)) \end{align*}$$

and

$$\begin{align*}T_vU(\mathcal L) \longrightarrow \theta^*\gamma, \quad (s,x,v) \longmapsto (x, ds(x), v) \end{align*}$$

gives the wanted square.

Let us now look at hypersurfaces of higher degree. For every integer $d \geq 1$ , we pick a section $s_d \in \Gamma _{\mathrm {ns}}(\mathcal L^{\otimes d})$ and write $H_d = V(s_d) \subset X$ for the associated hypersurface. Replacing $\mathcal L$ by $\mathcal L^{\otimes d}$ in the diagram (13), we obtain spaces $B_d \simeq \Omega ^1_X \otimes \mathcal L^{\otimes d} \setminus 0$ . We write $\theta _d \colon B_d \to BO(2n-2)$ for the tangential structure and $\hat \ell _d \colon TH_d \to \theta _d^*\gamma $ for the tangential structure on $H_d$ induced from the inclusion inside X (as constructed in the proof of Proposition 8.8). Let $\mathrm {Bun}^{\theta _d}(H_d)$ denote the space of bundle maps $TH_d \to \theta _d^*\gamma $ , and write

for the Borel construction. As explained in [Reference Galatius and Randal-Williams13, Section 2.2], this is a moduli space classifying smooth $H_d$ -bundles with $\theta _d$ -structure. We let $\mathcal M^{\theta _d}(H_d, \hat \ell _d) \subset \mathcal M^{\theta _d}(H_d)$ be the connected component of $\hat \ell _d$ . Work of Galatius and Randal-Williams provides the following:

Theorem 8.9 (Compare [Reference Galatius and Randal-Williams13, Theorem 4.5])

Using the notations as above, let $\alpha = c_1(\mathcal L)$ and $N := \int _X \alpha ^{n+1} \neq 0$ . There is a map

$$\begin{align*}\mathcal M^{\theta_d}(H_d, \hat\ell_d) \longrightarrow \Omega^\infty MT\theta_d \simeq \Omega^\infty \big(\Omega^1_X \otimes \mathcal L^{\otimes d} \setminus 0 \big)^{q^*\mathcal L^{\otimes d} - q^*TX} \end{align*}$$

which, when regarded as a map onto the path component that it hits, induces an isomorphism in integral homology in degrees $* \leq \frac {1}{3}Cd^{n+1} + O(d^n)$ , for some constant C depending on n and satisfying $\frac {1}{2}\cdot \frac {13}{15}N \leq C$ .

A major advantage of that theorem resides in the fact that the rational homotopy of the infinite loop space is easily calculated:

Lemma 8.10. Suppose that the Euler class of $\Omega ^1_X \otimes \mathcal L^{\otimes d}$ does not vanish. Then there is an isomorphism of commutative rings

$$ \begin{align*} H^*\big(\mathcal M^{\theta_d}(H_d, \hat\ell_d); {\mathbb{Q}}\big) &\cong \Lambda\big(H^{2n-1}(X)[1] \oplus H^1(X)[2] \oplus H^2(X)[3] \oplus \cdots \oplus H^{2n}(X)[2n+1]\big) \\ &=: \Lambda\big(H^{2n-1}(X)[1]\big) \otimes \Lambda\big(H^{\bullet> 0}(X)[\bullet +1]\big) \end{align*} $$

where $H^i(X)[j]$ denotes the graded ${\mathbb {Q}}$ -vector space $H^i(X;{\mathbb {Q}})$ placed in degree j.

By Proposition 8.8, the universal bundle is pulled back along a map

(14) $$ \begin{align} \Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}) \longrightarrow \mathcal M^{\theta_d}(H_d, \hat\ell_d). \end{align} $$

Our work describes the stable rational cohomology of the domain, whereas Galatius and Randal-Williams compute the one of the codomain. We will shortly reveal the relation between these two rings of characteristic classes. First, as argued as the beginning of this section, it is more geometrically natural to consider the quotient of the domain of (14) by the group ${\mathbb {C}}^\times $ . We have not yet seen a counterpart to this action on the codomain, so we explain how to proceed in the following:

Proposition 8.11. Suppose that $\mathcal L^{\otimes d}$ is $2$ -jet ample, and denote by $\Delta \colon \Gamma (\mathcal L^{\otimes d}) \to {\mathbb {C}}$ the discriminant polynomial (see Lemma 8.1). Let $\mu _{\deg \Delta } \subset {\mathbb {C}}^\times $ be the cyclic subgroup of the ( $\deg \Delta $ )^th roots of unity. Then there is a homeomorphism

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}) / {\mathbb{C}}^\times \cong \Delta^{-1}(1) / \mu_{\deg \Delta}, \end{align*}$$

and an action of $\mu _{\deg \Delta }$ on $\mathcal M^{\theta _d}(H_d, \hat \ell _d)$ such that the map (14) fits into a commutative diagram:

where both vertical arrows are the quotient maps for the actions and the bottom arrow is the induced map

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times \cong \Delta^{-1}(1) / \mu_{\deg \Delta} \longrightarrow \mathcal M^{\theta_d}(H_d, \hat\ell_d) / \mu_{\deg \Delta} \end{align*}$$

on the quotients from $\Delta ^{-1}(1) \subset \Gamma _{\mathrm {ns}}(\mathcal L^{\otimes d}) \to \mathcal M^{\theta _d}(H_d, \hat \ell _d)$ .

Proof. The classical theory of global Milnor fibrations provides a commutative diagram whose top two rows and leftmost two columns are fibre bundles:

The inverse of the homeomorphism

$$\begin{align*}\Delta^{-1}(1) / \mu_{\deg \Delta} \overset{\cong}{\longrightarrow} \Gamma_{\mathrm{ns}}(\mathcal L)/{\mathbb{C}}^\times \end{align*}$$

is explicitly given by $s \mapsto \Delta (s)^{-1/ \deg \Delta } \cdot s$ . This proves the first claim. For the second, recall that we have defined:

where $\mathrm {Bun}^{\theta _d}(H_d, \hat \ell _d)$ denotes the connected component of $\hat \ell _d$ in the space of bundle maps. Recall also from the proof of Proposition 8.8 the point set model:

$$\begin{align*}\theta^*\gamma = \{ (x, \varphi, v) \mid (x,\varphi) \in \Omega^1_X \otimes \mathcal L \setminus 0, \ v \in \ker(\varphi) \}. \end{align*}$$

The group ${\mathbb {C}}^\times $ acts fibrewise on the vector bundle $\Omega ^1_X \otimes \mathcal L$ , thus on $\theta ^*\gamma $ via

$$\begin{align*}\lambda \cdot (x, \varphi, v) = (x, \lambda \cdot \varphi, v) \quad \text{for } \lambda \in {\mathbb{C}}^\times, \end{align*}$$

and therefore acts on $\mathcal M^{\theta _d}(H_d, \hat \ell _d)$ by postcomposition on bundle maps. This action can be restricted to the cyclic subgroup $\mu _{\deg \Delta } \subset {\mathbb {C}}^\times $ . Finally the existence of the claimed commutative square follows by inspection.

We are now ready to summarise the relation between the two rings of characteristic classes in the main result of this section:

Theorem 8.12. Let X be a simply connected smooth projective complex variety. Let $i \geq 0$ be an integer and let $d \gg 0$ be big enough so that

$$\begin{align*}H^*(\mathcal M^{\theta_d}(H_d, \hat\ell_d); {\mathbb{Q}}) \cong \Lambda\big(H^{2n-1}(X)[1]\big) \otimes \Lambda\big(H^{\bullet> 0}(X)[\bullet +1]\big) \end{align*}$$

and

$$\begin{align*}H^*(\Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}) / {\mathbb{C}}^\times; {\mathbb{Q}}) \cong \Lambda\big(H^{\bullet> 0}(X)[\bullet +1]\big) \end{align*}$$

in degrees $* \leq i$ . The map classifying the universal bundle

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}) \longrightarrow \mathcal M^{\theta_d}(H_d, \hat\ell_d) \end{align*}$$

induces a ring morphism in rational cohomology with the following properties in cohomological degrees $* \leq i$ :

1. (i) Its restriction to $\Lambda \big (H^{\bullet> 0}(X)[\bullet +1]\big ) \subset H^*(\mathcal M^{\theta _d}(H_d, \hat \ell _d); {\mathbb {Q}})$ is injective.

2. (ii) Its restriction to $\Lambda \big (H^{2n-1}(X)[1]\big ) \subset H^*(\mathcal M^{\theta _d}(H_d, \hat \ell _d); {\mathbb {Q}})$ is zero.

In particular, its image in degrees $* \leq i$ is the subring $H^*(\Gamma _{\mathrm {ns}}(\mathcal L)/ {\mathbb {C}}^\times; {\mathbb {Q}}) \subset H^*(\Gamma _{\mathrm {ns}}(\mathcal L); {\mathbb {Q}})$ , and the map

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L)/ {\mathbb{C}}^\times \longrightarrow \mathcal M^{\theta_d}(H_d, \hat\ell_d) / \mu_{\deg \Delta} \end{align*}$$

of Proposition 8.11 induces an isomorphism in rational cohomology in degrees $* \leq i$ .

Proof. Recall from [Reference Galatius and Randal-Williams13, Section 3.1] that the characteristic classes of $\theta _d$ -bundles are given by integration along the fibres. From the similar description given in Proposition 8.3, we see that, in degrees $* \leq i$ , a basis of

$$\begin{align*}H^{\bullet> 0}(X)[\bullet +1] \subset H^*(\mathcal M^{\theta_d}(H_d, \hat\ell_d); {\mathbb{Q}}) \end{align*}$$

is sent to a basis of

$$\begin{align*}H^{\bullet> 0}(X)[\bullet +1] \subset H^*(\Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}); {\mathbb{Q}}) \end{align*}$$

under the morphism induced by the map classifying the universal bundle. This proves the first point. To prove the second, we recall that the image of an element $w \in H^{2n-1}(X)$ is the fibre integration

$$\begin{align*}\pi_!(i^*w) \in H^*(\Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}); {\mathbb{Q}}) \end{align*}$$

where $\pi $ is the universal bundle, and $i \colon U(\mathcal L^{\otimes d}) \hookrightarrow \Gamma _{\mathrm {ns}}(\mathcal L^{\otimes d}) \times X \to X$ is the map $(f,x) \mapsto x$ . From the commutative diagram

we compute that

$$\begin{align*}\pi_!(i^*w) = (\mathrm{pr}_1)_!(\iota_!(1) \cup 1 \otimes w) = (\mathrm{pr}_1)_!(1 \otimes c_1(\mathcal L^{\otimes d}) \cup w) = 0. \end{align*}$$

The identification of the subring corresponding to the image follows from Lemma 8.2. Finally, to prove the last claim, it remains to show that the quotient map

$$\begin{align*}\mathcal M^{\theta_d}(H_d, \hat\ell_d) \longrightarrow \mathcal M^{\theta_d}(H_d, \hat\ell_d) / \mu_{\deg \Delta} \end{align*}$$

induces an isomorphism in rational cohomology. As the action is free, the quotient is the homotopy orbit space and we thus have a fibration:

$$\begin{align*}\mathcal M^{\theta_d}(H_d, \hat\ell_d) \longrightarrow \mathcal M^{\theta_d}(H_d, \hat\ell_d) / \mu_{\deg \Delta} \longrightarrow B \mu_{\deg \Delta}. \end{align*}$$

The monodromy action of $\pi _1(B \mu _{\deg \Delta }) = \mu _{\deg \Delta }$ on the cohomology of the fibre is trivial as it can be extended to the connected group ${\mathbb {C}}^\times $ . A finite group has trivial rational cohomology, here $H^*(B \mu _{\deg \Delta }; {\mathbb {Q}}) = {\mathbb {Q}}$ , and the result follows.

Remark 8.13. By general theory, a map from a space T to the homotopy orbit space $\mathcal M^{\theta }(H, \hat \ell ) \mathbin {/\mkern -6mu/} \mu _{\deg \Delta }$ is given by a principal $\mu _{\deg \Delta }$ -bundle $P \to T$ and an equivariant map $P \to \mathcal M^{\theta }(H, \hat \ell )$ . From that point of view, the map

$$\begin{align*}\Gamma_{\mathrm{ns}}(\mathcal L) / {\mathbb{C}}^\times \longrightarrow \mathcal M^{\theta}(H, \hat\ell) / \mu_{\deg \Delta} \end{align*}$$

is given by the datum of a $\theta $ -structure on the pullback of the universal bundle along the étale cover

$$\begin{align*}\Delta^{-1}(1) \longrightarrow \Gamma_{\mathrm{ns}}(\mathcal L)/{\mathbb{C}}^\times \end{align*}$$

with Galois group $\mu _{\deg \Delta }$ .

8.2.1 Removing the quotient

Using the general machinery of [Reference Galatius and Randal-Williams13, Section 4.3], we construct a new tangential structure $\theta _d'$ which takes into account the $\mu _{\deg \Delta }$ -action. The group $\mu _{\deg \Delta }$ acts on $B_d \simeq \Omega ^1_X \otimes \mathcal L^{\otimes d} \setminus 0$ through the scalar action on the bundle $\mathcal L^{\otimes d}$ , and we have:

Lemma 8.14. The map $\theta _d \colon B_d \to BO(2n-2)$ factors through the orbit space of the $\mu _{\deg \Delta }$ -action via a map which we denote

$$\begin{align*}\theta_d' \colon B_d / \mu_{\deg \Delta} \longrightarrow BO(2n-2). \end{align*}$$

Proof. One can argue using the point set models described in Lemma 8.6. Indeed, recall that $B_d \simeq \Omega ^1_X \otimes \mathcal L^{\otimes d} \setminus 0$ can be identified with the space of pairs $(x,\varphi )$ of a point $x \in X$ and a linear surjection $\varphi \colon TX|_x \twoheadrightarrow \mathcal L^{\otimes d}|_x$ , and the map $\theta _d$ to $BO(2n-2)$ sends $(x,\varphi )$ to the kernel $\ker (\varphi )$ . The group $\mu _{\deg \Delta }$ acts on that space by scalar multiplication on $\mathcal L^{\otimes d}$ , and this preserves the kernel of an epimorphism $\varphi $ .

The analogue of Proposition 8.8 also holds for this new tangential structure:

Lemma 8.15. The universal bundle above $\Gamma _{\mathrm {ns}}(\mathcal L^{\otimes d}) / {\mathbb {C}}^\times $ admits the structure of a smooth fibre bundle with $\theta _d'$ -structure.

Proof. A point of the universal bundle is a pair $([f], x)$ of an equivalence class $[f] \in \Gamma _{\mathrm {ns}}(\mathcal L^{\otimes d}) / {\mathbb {C}}^\times $ and a point $x \in V(f)$ . We can choose a representative f of the class $[f]$ with discriminant $\Delta (f) = 1$ and take its derivative at x to define a map:

$$\begin{align*}([f], x) \longmapsto [df(x)] \in \big(\Omega^1_X \otimes \mathcal L^{\otimes d} \setminus 0\big) / \mu_{\deg \Delta}. \end{align*}$$

Although $df(x)$ is only well-defined up to a root of unity, its class is well-defined in the quotient. This map thus construct a $\theta _d'$ -structure on the universal bundle.

As before, if $H_d = V(f)$ is a smooth hypersurface with $f \in \Gamma _{\mathrm {ns}}(\mathcal L^{\otimes d}) / {\mathbb {C}}^\times $ , we write $\hat \ell _d' \colon TH_d \to \theta _d^{\prime *}\gamma $ for the tangential structure constructed in the proof of Lemma 8.15 above. Finally we obtain:

Corollary 8.16. Let X be a simply connected smooth projective complex variety. Let $i,d$ be as in Theorem 8.12. The map classifying the universal bundle

$$\begin{align*}\mathcal{M}_{\mathrm{hyp}}^{c_1(\mathcal L^{\otimes d})} = \Gamma_{\mathrm{ns}}(\mathcal L^{\otimes d}) / {\mathbb{C}}^\times \longrightarrow \mathcal M^{\theta_d'}(H_d, \hat\ell_d') \end{align*}$$

induces an isomorphism in rational cohomology in the range $* \leq i$ .

Proof. The differential map $\ell _d' \colon H_d = V(f) \to (\Omega ^1_X \otimes \mathcal L^{\otimes d} \setminus 0) / \mu _{\deg \Delta }$ is not $(n-1)$ -connected anymore (compare with Lemma 8.7). But the composition

$$\begin{align*}H_d \longrightarrow \Omega^1_X \otimes \mathcal L^{\otimes d} \setminus 0 \longrightarrow (\Omega^1_X \otimes \mathcal L^{\otimes d} \setminus 0) / \mu_{\deg \Delta} \end{align*}$$

is a Moore–Postnikov $(n-1)$ -stage. The group-like topological monoid of weak equivalences of $\Omega ^1_X \otimes \mathcal L^{\otimes d} \setminus 0$ above its quotient by $\mu _{\deg \Delta }$ is directly seen to be equivalent to

$$\begin{align*}\mathrm{Map}(X, \mu_{\deg \Delta}) = \mu_{\deg \Delta}. \end{align*}$$

Hence [Reference Galatius and Randal-Williams13, Theorem 4.5] applies (homotopy orbits and strict orbits are equivalent here as the group action is free) and shows that $\mathcal M^{\theta _d'}(H_d, \hat \ell _d')$ and $\mathcal M^{\theta _d}(H_d, \hat \ell _d) / \mu _{\deg \Delta }$ have the same stable rational cohomology. The result then follows by Theorem 8.12.