1 Introduction
1.1 Cone conjecture
To understand the geometry of a smooth projective variety X, studying the Mori cone of curves
$\overline {\mathrm {NE}}(X)$
and its dual, the nef cone
$\mathrm {Nef}(X)$
, is central, especially from the viewpoint of the minimal model program (MMP).
An important part of the relationship between the Mori cone and the MMP is captured by the cone theorem, and the contraction theorem. These theorems assert that the
$K_X$
-negative part of the Mori cone of a smooth projective variety X is rational polyhedral away from the
$K_X$
-trivial hyperplane, and the extremal rays of the
$K_X$
-negative part correspond to some morphisms from X involved in the MMP. In particular, when X is a Fano variety (namely,
$-K_X$
is ample), the cone
$\mathrm {Nef}(X)$
is a rational polyhedral cone, and its extremal rays are generated by semiample classes. In general, however, it is difficult to describe the whole Mori cone, or dually the whole nef cone, even under the slightly weaker assumption that
$-K_X$
is semiample. For instance, if X is the blowup of
${\mathbb {P}}^2$
at the base points of a general pencil of cubic curves in
${\mathbb {P}}^2$
, then
$-K_X$
is semiample but
$\mathrm {Nef}(X)$
is not rational polyhedral.
When X is K-trivial, we expect nevertheless that some essential parts of the nef cone of X are rational polyhedral, up to the action of
$\mathrm {Aut}(X)$
. A precise statement, known as the cone conjecture, was first formulated by Morrison [Reference Morrison35] and Kawamata [Reference Kawamata22]. It was later generalized by Totaro [Reference Totaro48] to klt Calabi–Yau pairs
$(X,\Delta )$
(see Section 2.2), thus including many more examples, already in dimension
$2$
. When stated by these authors, the cone conjecture comprises predictions both on the nef cone and on the movable cone of varieties and has both an absolute and a relative version. In what follows, we will only consider the absolute cone conjecture for nef cones of certain Calabi–Yau pairs.
Let us recall the statement formulated by Totaro in [Reference Totaro48, Conjecture 2.1], starting with some notations. For a pair
$(X, \Delta )$
, we define
$$ \begin{align*}\mathrm{Aut}(X,\Delta) \mathrel{:=} \{ f \in \mathrm{Aut}(X) | f(\mathrm{supp}(\Delta)) = \mathrm{supp}(\Delta) \}.\end{align*} $$
We also define the nef effective cone
$\mathrm {Nef}^e(X)$
as
$$\begin{align*}\mathrm{Nef}^e(X) := \mathrm{Nef}(X)\cap \mathrm{Eff}(X), \end{align*}$$
where
$\mathrm {Eff}(X)$
is the effective cone of X.
Conjecture 1.1 (Kawamata–Morrison–Totaro cone conjecture).
Let
$(X, \Delta )$
be a klt Calabi–Yau pair. There exists a rational polyhedral cone
$\Pi $
in
$\mathrm {Nef}^e(X)$
which is a fundamental domain for the action of
$\mathrm {Aut}(X, \Delta )$
on
$\mathrm {Nef}^e(X)$
, in the sense that
$$\begin{align*}\mathrm{Nef}^e(X) =\bigcup_{g\in\mathrm{Aut}(X, \Delta)}g^{\ast}\Pi, \end{align*}$$
and
$\Pi ^{\circ } \cap (g^{\ast }\Pi )^{\circ } = \varnothing $
unless
$g^{\ast } = \mathrm {id}$
.
An important prediction of the cone conjecture for the MMP is that the number of
$\mathrm {Aut}(X, \Delta )$
-equivalence classes of faces of the nef effective cone
$\mathrm {Nef}^e(X)$
corresponding to birational contractions or fiber space structures is finite (see, e.g., [Reference Totaro48, p.243]).
Note that it is standard to replace Conjecture 1.1 by the a priori stronger following conjecture. Let
$\mathrm {Nef}^{+}(X)$
denote the convex hull of
$$\begin{align*}\mathrm{Nef}(X)\cap N^1(X)_{{\mathbb{Q}}},\end{align*}$$
where
$N^1(X)_{{\mathbb {Q}}}$
is the rational Néron–Severi space of X.
Conjecture 1.2. Let
$(X, \Delta )$
be a klt Calabi–Yau pair. Then the following statements hold.
-
1. There exists a rational polyhedral cone in
$\mathrm {Nef}^{+}(X)$
which is a fundamental domain for the action of
$\mathrm {Aut}(X, \Delta )$
on
$\mathrm {Nef}^{+}(X)$
. -
2. We have
$$ \begin{align*}\mathrm{Nef}^{+}(X) = \mathrm{Nef}^e(X).\end{align*} $$
Thanks to the fundamental work of Looijenga [Reference Looijenga32], we prove that the two conjectures are equivalent (see Corollary 2.6).
1.2 Nef cones of fiber products
The starting point of this work is a decomposition theorem for the nef cone of a fiber product over a curve.
It begins with the following general question. Let
$W_1$
and
$W_2$
be projective varieties, and let
$\phi _1 : W_1 \to B$
and
$\phi _2 : W_2 \to B$
be surjective morphisms over a base B. Assume that the fiber product
$W \mathrel {:=} W_1 \times _B W_2$
is irreducible.
Question 1.3. Denote by
$p_i : W \to W_i$
the natural projections. When do we have
$$ \begin{align} p_1^* \mathrm{Nef}(W_1) + p_2^* \mathrm{Nef}(W_2) = \mathrm{Nef}(W)? \end{align} $$
As the nef cone of a projective variety linearly spans the whole space of numerical classes of
${\mathbb {R}}$
-divisors on the variety, the nef cone decomposition (1.1) exists only if
$$ \begin{align} p_1^*N^1(W_1)_{{\mathbb{R}}} + p_2^*N^1(W_2)_{{\mathbb{R}}} = N^1(W)_{{\mathbb{R}}}. \end{align} $$
We may then ask which fiber products satisfying the decomposition (1.2) also have the decomposition (1.1).
When B is a point, it is not hard to see that Equation (1.2) implies Equation (1.1). In this case, indeed, the decomposition (1.2) is a direct sum. Every divisor D decomposes uniquely as
$p_1^{\ast }D_1 + p_2^{\ast }D_2$
, where
$D_1=D_{|W_1\times \{\mathrm {pt}\}}$
and
$D_2=D_{|\{\mathrm {pt}\}\times W_2}$
. If D is nef, then so are its restrictions, and hence Equation (1.1) follows. When B is
${\mathbb {P}}^1$
and the varieties
$W_i$
are certain rational elliptic surfaces, the decomposition (1.1) was proven in [Reference Grassi and Morrison14, Proposition 3.1]. We show that the implication (1.2)
$\Rightarrow $
(1.1) continues to hold for an arbitrary irreducible fiber product over a curve.
Theorem 1.4. For
$i=1,2$
, let
$\phi _i : W_i \to B$
be a surjective morphism from a projective variety to a projective curve B. Assume that
-
1. The fiber product
$W=W_1 \times _B W_2$
is irreducible. -
2. We have
$$ \begin{align*}p_1^*N^1(W_1)_{{\mathbb{R}}}+p_2^*N^1(W_2)_{{\mathbb{R}}}=N^1(W)_{{\mathbb{R}}}.\end{align*} $$
Then
$$ \begin{align*}p_1^*\mathrm{Nef}(W_1) + p_2^*\mathrm{Nef}(W_2)=\mathrm{Nef}(W).\end{align*} $$
As a consequence, we also have
$ p_1^*\mathrm {Amp}(W_1) + p_2^*\mathrm {Amp}(W_2)=\mathrm {Amp}(W)$
.
In Examples 3.5, 3.6 and 3.7, we build examples of fiber products over bases of higher dimension, that fail the implication (1.2)
$\Rightarrow $
(1.1). In Remark 3.8, we recall a classical example emphasizing that a similar decomposition does not hold for the movable cone of divisors of a fiber product over a curve.
We establish the following corollary to this first theorem.
Corollary 1.5. Keep the notations and assumptions of Theorem 1.4. Then the extremal rays of the convex cone
$\mathrm {Nef}(W)$
are exactly the pullbacks of the extremal rays of the two cones
$\mathrm {Nef}(W_1)$
and
$\mathrm {Nef}(W_2)$
. In particular, the cone
$\mathrm {Nef}(W)$
is rational polyhedral if and only if the cones
$\mathrm {Nef}(W_1)$
and
$\mathrm {Nef}(W_2)$
are both rational polyhedral.
This corollary can be seen as a means to construct fiber products over curves, whose nef cones are not rational polyhedral.
1.3 Cone conjecture for Schoen varieties
Among the strict Calabi–Yau manifolds (see Definition 2.2) whose nef cones are known to not be rational polyhedral, to our knowledge, the cone conjecture is only known so far in two special cases. One of them is the desingularized Horrocks–Mumford quintics, studied by Borcea in [Reference Borcea4] (see also [Reference Fryers12]); the other is the fiber product of two general rational elliptic surfaces with sections over
${\mathbb {P}}^1$
, constructed by Schoen in [Reference Schoen44] and investigated by Namikawa and Grassi–Morrison [Reference Namikawa37, Reference Grassi and Morrison14]. Both examples are of dimension three.
The main goal of this paper is to prove the cone conjecture for generalizations of Schoen’s Calabi–Yau threefolds, typically Calabi–Yau pairs, but also higher-dimensional strict Calabi–Yau varieties. In both cases, the underlying varieties, which we call Schoen varieties, are constructed as fiber products over
${\mathbb {P}}^1$
.
Let us first summarize our construction defining Schoen varieties; we refer to Subsections 4.1 and 4.2 for more details. We start with Fano manifolds
$Z_1$
and
$Z_2$
of dimension at least two, which respectively admit an ample and globally generated divisor
$D_i$
(
$i=1,2$
), such that
$-(K_{Z_i}+D_i)$
is globally generated. We take
$W_i \subset {\mathbb {P}}^1\times Z_i$
to be a general member in the linear system
$|{\mathcal {O}}_{{\mathbb {P}}^1}(1)\boxtimes {\mathcal {O}}_{Z_i}(D_i)|$
. There is a fibration
$\phi _i: W_i \to {\mathbb {P}}^1$
. We put another mild condition on the fibrations
$\phi _1$
and
$\phi _2$
to be general with respect to one another (see the second paragraph of Subsection 4.2 for a precise statement). Consider the following fiber product over
${\mathbb {P}}^1$
$$\begin{align*}\phi: X:= W_1 \times_{{\mathbb{P}}^1} W_2 \to {\mathbb{P}}^1. \end{align*}$$
Under our assumptions, the variety X is smooth and projective. All varieties obtained through this procedure are called Schoen varieties.
It follows from the construction that
$-K_X$
is globally generated, so many effective
${\mathbb {Q}}$
-divisors are
${\mathbb {Q}}$
-linearly equivalent to
$-K_X$
. Any such
${\mathbb {Q}}$
-divisor
$\Delta $
yields a Calabi–Yau pair
$(X,\Delta )$
, that we call a Schoen pair.
We prove the following result.
Theorem 1.6. Let
$(X,\Delta )$
be a Schoen pair. Then there exists a rational polyhedral fundamental domain for the action of
$\mathrm {Aut}(X, \Delta )$
on
$\mathrm {Nef}^e(X) =\mathrm {Nef}^{+}(X) = \mathrm {Nef}(X)$
.
Note that, by Corollary 1.5, the cone
$\mathrm {Nef}(X)$
is not rational polyhedral as soon as one of the cones
$\mathrm {Nef}(W_i)$
(
$i=1,2$
) is not, typically if one of the factors
$W_i$
is a rational elliptic surface with
$Z_i\simeq {\mathbb {P}}^2$
and
$D_i={\mathcal {O}}_{{\mathbb {P}}^2}(3)$
. Using this remark, we provide in Example 5.6 the first series of strict Calabi–Yau manifolds (and Calabi-Yau pairs) of arbitrary dimension for which the cone conjecture holds, with nef cones that are not rational polyhedral.
We finally mention two unsurprising consequences of Theorem 1.6 (see Corollary 5.7): The finite presentation of the group of components
$\pi _0\mathrm {Aut}(X)$
and the finiteness of real forms on X, up to isomorphism.
1.4 Relation to other work
1.4.1 Cone conjecture
We refer to [Reference Lazić, Oguiso and Peternell30] and the references therein for a survey of the cone conjecture for varieties (as opposed to pairs). As for the cone conjecture for Calabi–Yau pairs, its two-dimensional case was proven by Totaro [Reference Totaro48]. Kopper [Reference Kopper27] also proved the cone conjecture for Calabi–Yau pairs arising from Hilbert schemes of points on certain rational elliptic surfaces; the underlying varieties in his work may have non rational polyhedral nef cones, but they only appear in even dimensions. The references [Reference Filipazzi, Hacon and Svaldi11, Reference Li and Zhao31] also contain some recent results.
1.4.2 Cone conjectures for varieties with rational polyhedral nef cones
One way of proving the cone conjecture for a smooth projective variety X is to show that
$\mathrm {Nef}(X)$
is a rational polyhedral cone and that
$\mathrm {Nef}(X) = \mathrm {Nef}^e(X)$
(see, e.g., [Reference Lazić29, Proposition 6.5]). This is the case whenever X is a smooth anticanonical hypersurface in a Fano manifold Y of dimension at least
$4$
, by the following theorem, due to Kollár [Reference Borcea and Kollár3, Appendix].
Theorem 1.7. Let D be a smooth anticanonical hypersurface in a smooth Fano variety Y of dimension at least
$4$
. Then the natural restriction map
$\mathrm {Nef}(Y)\to \mathrm {Nef}(D)$
is an isomorphism. In particular,
$\mathrm {Nef}(D)$
is a rational polyhedral cone, generated by classes of semiample divisors.
Other Calabi–Yau pairs
$(X, \Delta )$
for which
$\mathrm {Nef}(X) = \mathrm {Nef}^e(X)$
is rational polyhedral are described in the work of Coskun and Prendergast-Smith [Reference Prendergast-Smith40, Reference Coskun and Prendergast-Smith8, Reference Coskun and Prendergast-Smith9].
1.4.3 Fiber product constructions
Constructing Calabi–Yau threefolds as fiber products of two general rational elliptic surfaces with sections over
${\mathbb {P}}^1$
was first considered and investigated by Schoen [Reference Schoen44]. It recently came back to light as Suzuki considered a certain higher-dimensional generalization of Schoen’s construction and studied its arithmetic properties in [Reference Suzuki46]. Similar ideas are also involved in Sano’s constructions of non-Kähler Calabi–Yau manifolds with arbitrarily large second Betti number in [Reference Sano42].
1.4.4 Cone conjecture for movable cones
We have already mentioned that there is a part of the cone conjecture concerned with movable cones [Reference Totaro48, Conjecture 2.1.(2)]. It predicts that a Calabi–Yau variety should have finitely many minimal models, up to isomorphism [Reference Cascini and Lazić7, Theorem 2.14]). See [Reference Morrison36, Reference Kawamata22, Reference Totaro48, Reference Lazić, Oguiso and Peternell30] for related references. This part of the cone conjecture was verified for some cases. Notably, in [Reference Cantat and Oguiso6], Cantat and Oguiso produced the first series of strict Calabi–Yau manifolds in arbitrary dimension whose movable cones are not rational polyhedral and for which the cone conjecture for movable cones holds. We refer to [Reference Hoff, Stenger and Yáñez17, Reference Ito, Lai and Wang20, Reference Lai and Wang28, Reference Wang50] and references therein for more results.
In [Reference Namikawa37], Namikawa showed that a certain strict Calabi–Yau threefold, constructed as a Schoen variety, has finitely many minimal models, up to isomorphism. Nonetheless, the cone conjecture is still unknown for the movable cone of divisors of this Calabi–Yau threefold. The cone conjecture for the nef cones of each of these minimal models is not known either. Similar questions could be asked for the Schoen varieties of higher dimension constructed here.
2 Preliminaries
We work over the field
${\mathbb {C}}$
of complex numbers throughout this paper. For notions of birational geometry, we refer to [Reference Kollár, Mori, Clemens and Corti26].
2.1 Notations
Let X be a projective variety. We write
$N^1(X)$
for the free abelian group generated by the classes of Cartier divisors modulo numerical equivalence.
Inside the vector space
$N^1(X)_{{\mathbb {R}}}:=N^1(X)\otimes {\mathbb {R}}$
, we denote by
$\mathrm {Nef}(X)$
the nef cone, that is, the closure of the ample cone
$\mathrm {Amp}(X)$
, and by
$\mathrm {Eff}(X)$
the effective cone. The nef effective cone
$\mathrm {Nef}^e(X)$
is defined as
$$\begin{align*}\mathrm{Nef}^e(X) := \mathrm{Nef}(X)\cap \mathrm{Eff}(X). \end{align*}$$
Let
$\mathrm {Nef}^{+}(X)$
denote the convex hull of
$$\begin{align*}\mathrm{Nef}(X)\cap N^1(X)_{{\mathbb{Q}}}, \end{align*}$$
where
$N^1(X)_{{\mathbb {Q}}}:=N^1(X)\otimes {\mathbb {Q}}$
. We denote by
$N_1(X)$
the group of
$1$
-cycles modulo numerical equivalence. The intersection product defines a perfect pairing between the two vector spaces
$N^1(X)_{{\mathbb {R}}}$
and
$N_1(X)_{{\mathbb {R}}}$
. Under this pairing, the nef cone
$\mathrm {Nef}(X)$
is dual to the Mori cone
$\overline {\mathrm {NE}}(X)$
, which is by definition the closure of the convex cone of effective
$1$
-cycles in
$N_1(X)_{{\mathbb {R}}}$
.
The group of automorphisms of X is denoted by
$\mathrm {Aut}(X)$
and acts on
$N^1(X)$
by pullback. This action
$$\begin{align*}\rho: \mathrm{Aut}(X) \to \mathrm{GL}(N^1(X)) \end{align*}$$
linearly extends to
$N^1(X)_{{\mathbb {R}}}$
, preserving the cones
$\mathrm {Nef}^e(X)$
and
$\mathrm {Nef}^{+}(X)$
. The connected component of the identity in
$\mathrm {Aut}(X)$
is a normal subgroup
$\mathrm {Aut}^0(X)$
, which acts trivially on
$N^1(X)$
[Reference Brion5, Lemma 2.8].
2.2 Klt Calabi–Yau pairs
A pair is the data
$(X, \Delta )$
of a normal projective variety X together with an effective
${\mathbb {R}}$
-divisor
$\Delta $
on X such that
$K_X + \Delta $
is
${\mathbb {R}}$
-Cartier.
Definition 2.1. Following [Reference Totaro48], we say that a pair
$(X, \Delta )$
is Calabi–Yau if X is
${\mathbb {Q}}$
-factorial and
$K_X + \Delta $
is numerically trivial.
Let us briefly recall the definition of a Kawamata log terminal (klt) pair. We start with a notation. For any pair
$(X,\Delta )$
and any birational morphism
$\mu : \widetilde {X} \to X$
, there exists a unique
${\mathbb {R}}$
-divisor
$\widetilde {\Delta }$
on
$\widetilde {X}$
such that
$$ \begin{align*}K_{\widetilde{X}} + \widetilde{\Delta} = \mu^*(K_X + \Delta) \ \text{ and }\ \mu_*\widetilde{\Delta} = \Delta.\end{align*} $$
A pair
$(X, \Delta )$
is called klt if, for any birational morphism
$\mu : \widetilde {X} \to X$
, when defining the divisor
$\widetilde {\Delta }$
as above, each irreducible component of
$\widetilde {\Delta }$
has coefficient less than one.
Note that if we can find one resolution of singularities
$\mu : \widetilde {X} \to X$
whose corresponding divisor
$\widetilde {\Delta }$
has simple normal crossings, with irreducible components of coefficients less than one, then
$(X,\Delta )$
is klt.
Definition 2.2. Let X be a smooth projective variety. We say that X is a Calabi–Yau manifold if its canonical line bundle
$K_X$
is trivial and
$h^i(X,{\mathcal {O}}_X)=0$
for any
$0<i<\dim X$
. If in addition, X is simply connected, we call it a strict Calabi–Yau manifold.
2.3 Looijenga’s result
The following result is crucial in this paper.
Proposition 2.3. Let X be a projective variety, and let
$H \le \mathrm {Aut}(X)$
be a subgroup. Assume that there is a rational polyhedral cone
$\Pi \subset \mathrm {Nef}^{+}(X)$
such that
$\mathrm {Amp}(X)\subset H \cdot \Pi .$
Then
-
1.
$H\cdot \Pi = \mathrm {Nef}^{+}(X)$
, and the H-action on
$\mathrm {Nef}^{+}(X)$
has a rational polyhedral fundamental domain. -
2. The group
$\rho (H)$
is finitely presented.
This result should be well known to experts, but we include a proof for the sake of completeness. It relies on the fundamental results due to Looijenga [Reference Looijenga32, Proposition 4.1, Application 4.14, and Corollary 4.15], which we extract and formulate here as Lemma 2.4. Recall that a convex cone
$C \subset N_{{\mathbb {R}}}$
in a finite dimensional
${\mathbb {R}}$
-vector space
$N_{{\mathbb {R}}}$
is called strict if its closure
$\overline {C} \subset N_{{\mathbb {R}}}$
contains no line.
Lemma 2.4. Let N be a finitely generated free
${\mathbb {Z}}$
-module, and let C be a strict convex open cone in the
${\mathbb {R}}$
-vector space
$N_{{\mathbb {R}}}:= N\otimes {\mathbb {R}}$
. Let
$C^{+}$
be the convex hull of
$\overline {C} \cap N_{{\mathbb {Q}}}$
. Let
$(C^\vee )^\circ \subset N_{{\mathbb {R}}}^\vee $
be the interior of the dual cone of C. Let
$\Gamma $
be a subgroup of
$\mathrm {GL}(N)$
which preserves the cone C. Suppose that
-
• there is a rational polyhedral cone
$\Pi \subset C^+$
such that
$C \subset \Gamma \cdot \Pi $
; -
• there exists an element
$\xi \in (C^\vee )^\circ \cap N_{{\mathbb {Q}}}^\vee $
whose stabilizer in
$\Gamma $
(with respect to the dual action
$\Gamma \circlearrowleft N_{{\mathbb {Q}}}^\vee $
) is trivial.
Then
$\Gamma \cdot \Pi = C^+$
and the
$\Gamma $
-action on
$C^{+}$
has a rational polyhedral fundamental domain. Moreover, the group
$\Gamma $
is finitely presented.
To prove Proposition 2.3, it is key to connect abstract convex geometry as in Lemma 2.4 with the specifics of an automorphism group acting on an ample cone. That is the goal of the next lemma.
Lemma 2.5. Let X be a projective variety. Then there exists an ample Cartier divisor on X, whose numerical class
$\eta \in N^1(X)$
satisfies: For every
$g \in \mathrm {Aut}(X)$
, if
$g^*\eta = \eta $
, then
$g^*$
is the identity on
$N^1(X)$
.
Proof. Our proof is inspired by the argument of [Reference Lazić29, Proposition 6.5].
Let
$\Gamma \mathrel {:=} \rho (\mathrm {Aut}(X)) < \mathrm {GL}(N^1(X))$
. For every
$\theta \in N^1(X)_{{\mathbb {Q}}}$
, let
$\Gamma _{\theta }$
denote the subgroup of
$\Gamma $
stabilizing
$\theta $
. We want to find an element
$\eta \in \mathrm {Amp}(X)\cap N^1(X)$
such that
$\Gamma _\eta $
is trivial. By linearity, it is sufficient to find such an element in
$\mathrm {Amp}(X)\cap N^1(X)_{{\mathbb {Q}}}$
.
By Fujiki–Liebermann’s theorem [Reference Brion5, Theorem 2.10], for every element
$\theta $
in
$\mathrm {Amp}(X)\cap N^1(X)_{{\mathbb {Q}}}$
, the stabilizer
$\Gamma _{\theta }$
is finite. Pick an element
$\eta \in \mathrm {Amp}(X)\cap N^1(X)_{{\mathbb {Q}}}$
such that
$\Gamma _{\eta }$
has the smallest possible order. Since the discrete set
$N^1(X)$
is preserved by the action of
$\Gamma $
, there is an open neighborhood
$U \subset \mathrm {Amp}(X)$
of
$\eta $
such that, for every
$\gamma \in \Gamma \setminus \Gamma _{\eta }$
, the intersection
$\gamma U \cap U$
is empty. In particular, for every
$\theta \in U\cap N^1(X)_{{\mathbb {Q}}}$
, we have
$\Gamma _{\theta } \subset \Gamma _{\eta }$
, so
$\Gamma _{\theta } = \Gamma _{\eta }$
by the minimality assumption on
$\eta $
. Hence, we have
$$ \begin{align*}\gamma_{|U\cap N^1(X)_{{\mathbb{Q}}}} = \mathrm{id}_{U\cap N^1(X)_{{\mathbb{Q}}}},\end{align*} $$
which extends by linearity to
$\gamma = \mathrm {id}$
. So the stabilizer
$\Gamma _{\eta }$
is trivial, which concludes the proof.
We can now establish Proposition 2.3.
Proof of Proposition 2.3.
Let us set
$N=N^1(X)$
,
$C=\mathrm {Amp}(X)$
and
$\Gamma =\rho (H)$
. To apply Lemma 2.4 in this setup, it suffices to construct an element
$\xi _0\in (C^{\vee })^{\circ } \cap N^{\vee }_{{\mathbb {Q}}}$
with trivial stabilizer with respect to the dually induced
$\Gamma $
-action. Start by picking any
$\xi \in (C^{\vee })^{\circ } \cap N^{\vee }_{{\mathbb {Q}}}$
.
The idea is to find a minimizer
$\eta $
for the linear functional
$\xi $
on the set
$$ \begin{align*}\Sigma \mathrel{:=} \{\eta \in C \cap N | \Gamma_{\eta} \text{ is trivial}\}\end{align*} $$
and to relate the stabilizer of
$\eta $
(which is then trivial by construction) to the stabilizer of
$\xi $
(which we want to be trivial).
Note that
$\Sigma $
is nonempty by Lemma 2.5 and discrete. By definition, the linear form
$\xi $
takes positive values on the whole convex set
$\overline {C}\setminus \{0\}$
. Picking a large enough positive integer r, the intersection
$$ \begin{align*}\Sigma\cap\{x\in \overline{C}\mid \xi(x)\le r\}\end{align*} $$
is now nonempty and finite. Minimizing
$\xi $
on this finite set is equivalent to minimizing it on
$\Sigma $
, and thus
$\xi $
has finitely many minimizers in
$\Sigma $
.
Since
$C\cap N$
is discrete, we can now perturb
$\xi $
into a new linear form
$\xi _0\in (C^{\vee })^{\circ } \cap N^{\vee }_{{\mathbb {Q}}}$
, which has exactly one minimizer
$\eta $
on
$\Sigma $
. As the set
$\Sigma $
is
$\Gamma $
-invariant and as
$\Gamma _{\eta }$
is trivial, we have, for any nontrivial
$\gamma \in \Gamma $
, that
$\gamma \eta \in \Sigma \setminus \{\eta \}$
, and in particular
$$ \begin{align*}(\gamma \xi_0)(\eta) = \xi_0(\gamma \eta)> \xi_0(\eta).\end{align*} $$
So the stabilizer of
$\xi _0$
in
$\Gamma $
is trivial.
We prove a simple corollary of Proposition 2.3.
Proof. Clearly, Conjecture 1.2 implies Conjecture 1.1. Now, fix a pair
$(X,\Delta )$
for which Conjecture 1.1 holds. Let
$\Pi \subset \mathrm {Nef}^e(X)$
be a rational polyhedral fundamental domain for the action of
$\mathrm {Aut}(X,\Delta )$
on
$\mathrm {Nef}^e(X)$
. Then
$\Pi \subset \mathrm {Nef}^{+}(X)$
by definition of
$\mathrm {Nef}^{+}(X)$
. By Proposition 2.3.(1),
$$ \begin{align*}\mathrm{Nef}^e(X) = \mathrm{Aut}(X,\Delta) \cdot \Pi = \mathrm{Nef}^{+}(X).\end{align*} $$
So Conjecture 1.2 holds.
3 The nef cone of a fiber product over a curve
In this section, we prove Theorem 1.4. Let us recall the notations. For
$i=1,2$
, the map
$\phi _i : W_i \to B$
is a surjective morphism from a projective variety to a projective curve B. We consider the fiber product
and work under the following assumptions:
-
1. The fiber product
$W = W_1 \times _{B} W_2$
is irreducible; -
2. For every
$D \in N^1(W)_{{\mathbb {R}}}$
, there exist
$D_1 \in N^1(W_1)_{{\mathbb {R}}}$
and
$D_2 \in N^1(W_2)_{{\mathbb {R}}}$
such that
$$\begin{align*}D = p_1^*D_1 + p_2^*D_2. \end{align*}$$
Proof of Theorem 1.4.
Let us fix
$D \in \mathrm {Nef}(W)$
and consider a decomposition in real classes
$$ \begin{align*}D = p_1^*D_1 + p_2^*D_2 \in N^1(W)_{{\mathbb{R}}}\end{align*} $$
as in Assumption (2) right above. We prove three lemmas regarding the positivity of these two summands
$D_1$
and
$D_2$
.
Lemma 3.1. Fix
$i=1,2$
. Let
$C_i$
be a curve contained in a fiber of
$\phi _i: W_i\to B$
. Then
$ D_i \cdot C_i \ge 0$
.
Proof. By symmetry, we can focus on
$i = 1$
. Fix any point
$s \in \phi _2^{-1}(\phi _1(C_1))$
and consider the fiber product
$\widetilde {C_1} \mathrel {:=} C_1 \times _{B} \{s\}$
, which can be seen as a curve in
$W_1$
. We have
$$ \begin{align*}0 \le D \cdot \widetilde{C_1} = (p_1^*D_1 + p_2^*D_2) \cdot \widetilde{C_1} = D_1 \cdot p_{1*}\widetilde{C_1} + D_2 \cdot p_{2*}\widetilde{C_1} = D_1 \cdot C_1.\end{align*} $$
This proves the lemma.
Lemma 3.2. Either
$D_1$
or
$D_2$
is nef.
Proof. Assume by contradiction that both
$D_1$
and
$D_2$
are not nef. Then for each i, there exists a curve
$C_i$
in
$W_i$
such that
$D_i \cdot C_i < 0$
. Note that since the fiber product W is assumed to be irreducible, the base B is also irreducible. Hence, and by Lemma 3.1, we have
$\phi _i(C_i) = B$
. So the (possibly reducible) fiber product
$C_1 \times _{B} C_2$
contains a curve
$\widetilde {C}$
dominating B. Let
$\beta _1,\beta _2 \in {\mathbb {Z}}_{>0}$
be such that
$p_{i*}\widetilde {C} = \beta _i C_i$
. Then on one hand,
$$ \begin{align*}\beta_1 D_1 \cdot C_1 + \beta_2 D_2 \cdot C_2 < 0,\end{align*} $$
and on the other hand,
$$ \begin{align*}\beta_1 D_1 \cdot C_1 + \beta_2 D_2 \cdot C_2 = (p_1^*D_1 + p_2^*D_2) \cdot \widetilde{C} = D \cdot \widetilde{C} \ge 0.\end{align*} $$
This is a contradiction.
For the third lemma, we fix a point
$b \in B$
.
Lemma 3.3. Fix
$i=1,2$
. Then there exists
$N_i\in {\mathbb {R}}$
such that for any real number
$n\ge N_i$
, the divisor
$D_i + n \phi _i^*{\mathcal {O}}_B(b)$
is nef.
Proof. By symmetry, we can focus on
$i=2$
. Let
$C_1$
be a curve in
$W_1$
such that
$\phi _1(C_1) = B$
. Set
$$ \begin{align*}N_2 \mathrel{:=} \frac{D_1 \cdot C_1}{ \deg (C_1 \xrightarrow{\phi_1} B)}\end{align*} $$
and consider the following classes
$$ \begin{align*}D_1' \mathrel{:=} D_1 - N_2 \phi_1^*{\mathcal{O}}_B(b) \ \ \text{ and } \ \ D_2' \mathrel{:=} D_2 + N_2 \phi_2^*{\mathcal{O}}_B(b).\end{align*} $$
By construction, we have
$D^{\prime }_1 \cdot C_1 = 0\mbox { and }D = p_1^*D_1' + p_2^*D_2'.$
We want to show that
$D_2'$
is nef. Let
$C_2$
be a curve in
$W_2$
. If it is contained in a fiber of
$\phi _2$
, then
$D^{\prime }_2 \cdot C_2 \ge 0$
by Lemma 3.1. Suppose now that
$\phi _2(C_2) = B$
, let
$\widetilde {C}$
be a curve in the fiber product
$C_1 \times _{B} C_2$
dominating B, and define
$\beta _1,\beta _2 \in {\mathbb {Z}}_{>0}$
such that
$p_{i*}\widetilde {C} = \beta _i C_i$
. We have
$$ \begin{align*} \beta_2 D^{\prime}_2 \cdot C_2 &= \beta_1 D^{\prime}_1 \cdot C_1 + \beta_2 D^{\prime}_2 \cdot C_2 \\ &= (p_1^*D^{\prime}_1 + p_2^*D^{\prime}_2) \cdot \widetilde{C}\\ &= D \cdot \widetilde{C} \ge 0. \end{align*} $$
So
$D^{\prime }_2$
is nef. A fortiori, for
$n\ge N_2$
, the following class
$$ \begin{align*}D_2+n\phi_2^*{\mathcal{O}}_B(b) = D^{\prime}_2+(n - N_2)\phi_2^*{\mathcal{O}}_B(b)\end{align*} $$
is also nef.
Let us resume the proof of Theorem 1.4. For any
$t \in {\mathbb {R}}$
, let
$$ \begin{align*}D_1(t) \mathrel{:=} D_1 - t \phi_1^*{\mathcal{O}}_B(b) \ \ \text{ and } \ \ D_2(t) \mathrel{:=} D_2 + t \phi_2^*{\mathcal{O}}_B(b).\end{align*} $$
By Lemma 3.3, we can define intervals
$$ \begin{align*}I_1 = ]-\infty,-N_{1,\mathrm{min}}] \ \ \text{ and } \ \ I_2 = [N_{2,\mathrm{min}}, +\infty[\end{align*} $$
such that
$D_i(t)$
is nef if and only if
$t \in I_i$
. Since we have for all
$t\in {\mathbb {R}}$
,
$$ \begin{align*} D = p_1^*D_1(t) + p_2^*D_2(t),\end{align*} $$
Lemma 3.2 shows that either
$D_1(t)$
or
$D_2(t)$
is nef, that is,
$I_1 \cup I_2 = {\mathbb {R}}$
. Hence,
$I_1 \cap I_2$
is nonempty and fixing an element t in this intersection, both
$D_1(t)$
and
$D_2(t)$
are now nef, giving a desired decomposition.
The decomposition of the ample cone of W finally follows from the decomposition of the nef cone by [Reference Rockafellar41, Corollary 6.6.2].
Remark 3.4. In the setup of Theorem 1.4, we also have a decomposition of the relative nef cone
$$\begin{align*}\mathrm{Nef}(W/B) = p_1^*\mathrm{Nef}(W_1/B) + p_2^*\mathrm{Nef}(W_2/B) \end{align*}$$
by the projection formula – this is exactly Lemma 3.1.
As a consequence of Theorem 1.4, we prove Corollary 1.5.
Proof of Corollary 1.5.
First, consider
$E\in \mathrm {Nef}(W)$
spanning an extremal ray of
$\mathrm {Nef}(W)$
. Then by Theorem 1.4, there is a decomposition
$E=p_1^*E_1+p_2^*E_2$
, with
$E_i\in \mathrm {Nef}(W_i)$
, and either
$E_1$
or
$E_2$
is nonzero. By extremality, E is thus either in
$p_1^*\mathrm {Nef}(W_1)$
, or in
$p_2^*\mathrm {Nef}(W_2)$
. By symmetry, we can assume that
$E=p_1^*D$
, for some
$D\in \mathrm {Nef}(W_1)$
. Let us show that D spans an extremal ray in
$\mathrm {Nef}(W_1)$
. Let
$D = F + F'$
be any decomposition with
$F, F' \in \mathrm {Nef}(W_1)$
. Then
$E = p_1^*D = p_1^*F + p_1^*F'$
with
$p_1^*F, p_1^*F' \in \mathrm {Nef}(W)$
, and thus by extremality,
$p_1^*F$
and
$p_1^*F'$
are proportional. Since
$p_1^* : N^1(W_1)_{{\mathbb {R}}} \to N^1(W)_{{\mathbb {R}}}$
is injective, F and
$F'$
are proportional as well. This shows that D spans an extremal ray.
We thus know that every extremal ray of
$\mathrm {Nef}(W)$
is obtained by pulling back an extremal ray of either
$\mathrm {Nef}(W_1)$
or
$\mathrm {Nef}(W_2)$
.
Next, assume that
$D \in \mathrm {Nef}(W_1)$
is extremal, and let us prove that
$p_1^*D$
is extremal in
$\mathrm {Nef}(W)$
. Let
$p_1^*D = E + E'$
be a decomposition with
$E,E' \in \mathrm {Nef}(W)$
. Up to adding terms to
$E'$
, we can assume that E spans an extremal ray of
$\mathrm {Nef}(W)$
. By Theorem 1.4, we can write
$$ \begin{align*}E = p_1^*E_1 + p_2^*E_2, \ \text{ and } \ E'= p_1^*E^{\prime}_1 + p_2^*E^{\prime}_2\end{align*} $$
with
$E_i,E^{\prime }_i \in \mathrm {Nef}(W_i)$
. As E is extremal, the divisors E,
$p_1^*E_1$
and
$p_2^* E_2$
are proportional. Moreover
$p_1^*(D-E_1-E_1')=p_2^*(E_2 + E^{\prime }_2) \in \mathrm {Nef}(W)$
. Hence, by the projection formula,
$D-E_1-E_1'$
is nef. But D is extremal in the cone
$\mathrm {Nef}(W_1)$
, so D,
$E_1$
, and
$E_1'$
are proportional. In particular,
$p_1^*D, p_1^*E_1, p_1^*E^{\prime }_1$
and
$p_2^*E_2$
are all proportional, which shows that E and
$E'$
are proportional and thus concludes the proof.
We now construct various fiber products showing that Theorem 1.4 fails in general over bases B of dimension at least 2. Two types of constructions are provided: In Example 3.5, the surjective maps
$\phi _1$
and
$\phi _2$
are birational morphisms; in Example 3.6, they are smooth fibrations. The first construction simply involves
$(-1)$
-curves on blow-ups of
${\mathbb {P}}^2$
; the second construction uses Serre’s construction of vector bundles of rank two.
Example 3.5. Take
$S:={\mathbb {P}}^2$
, and take four points
$P_1$
,
$P_2$
,
$P_3$
,
$P_4$
in S so that no three of them lie on a line. Let
$\ell _1$
be the line through
$P_1$
,
$P_2$
and let
$\ell _2$
be the line through
$P_3$
,
$P_4$
. Take
$$ \begin{align*} W_1:=\mathrm{Bl}_{P_{1}, P_{2}}(S) \ \ \text{ and } \ \ W_2:=\mathrm{Bl}_{P_{3}, P_{4}}(S).\end{align*} $$
We let
$$ \begin{align*}W:=W_1\times_{S} W_2.\end{align*} $$
As the blown-up points are distinct, W is isomorphic to
$\mathrm {Bl}_{P_1,P_2,P_3,P_4}(S)$
, which is smooth. Moreover, the decomposition of the Picard group
$$ \begin{align*}\mathrm{Pic}(W)=p_1^*\mathrm{Pic}(W_1)+p_2^*\mathrm{Pic}(W_2)\end{align*} $$
clearly holds.
Denote by
$\ell _1'$
and
$\ell _2'$
the strict transforms of
$\ell _1$
and
$\ell _2$
in
$W_1$
and
$W_2$
, respectively. Then
$\ell _i'$
is an effective non-nef divisor on
$W_i$
as
$(\ell _i')^2=-1$
. Let
$$ \begin{align*}D:=p_1^*\ell_1'+p_2^*\ell_2'.\end{align*} $$
We show that D is nef; this also shows that Lemma 3.2 fails when
$\dim B \ge 2$
. As D is effective, it is enough to check that its intersections with its components are all nonnegative. By symmetry, it is enough to compute
$$ \begin{align*}D\cdot p_1^*\ell_1'=(\ell_1')^2+\ell_2'\cdot \phi_2^*\ell_1=-1+1=0.\end{align*} $$
So D is nef and has vanishing intersection with the curves
$p_1^*\ell _1'$
and
$p_2^*\ell _2'$
.
Now, assume by contradiction that D has another decomposition
$D=p_1^*D_1+p_2^*D_2$
with
$D_i \in \mathrm {Nef}(W_i)$
. Then we have
$$ \begin{align*}p_1^*(\ell^{\prime}_1 - D_1) = p_2^*(D_2 - \ell^{\prime}_2).\end{align*} $$
As
$p_1^*N^1(W_1)_{{\mathbb {R}}} \cap p_2^*N^1(W_2)_{{\mathbb {R}}}$
clearly has dimension one, it equals
${\mathbb {R}}[p^*{\mathcal {O}}_{{\mathbb {P}}^2}(1)]$
, where p is the natural projection
$W \to S$
. Hence, for some
$c\in {\mathbb {R}}$
, we have
$$ \begin{align*}p_1^*(\ell^{\prime}_1 - D_1) = p_2^*(D_2 - \ell^{\prime}_2) = c p^*{\mathcal{O}}_{{\mathbb{P}}^2}(1).\end{align*} $$
Since
$$ \begin{align*}p_1^*D_1\cdot p_i^*\ell_i'+p_2^*D_2\cdot p_i^*\ell_i'=D\cdot p_i^*\ell^{\prime}_i = 0,\end{align*} $$
and both
$p_1^*D_1$
and
$p_2^*D_2$
are nef, we have
$p_i^*D_i\cdot p_i^*\ell _i'=0$
. Thus,
$$ \begin{align*}-1 = p_1^*\ell_1' \cdot p_1^*(\ell^{\prime}_1 - D_1) = c p_1^*\ell_1' \cdot p^*{\mathcal{O}}_{{\mathbb{P}}^2}(1) = c\end{align*} $$
and similarly,
$$ \begin{align*}1 = p_2^*\ell_2' \cdot p_2^*(D_2 - \ell^{\prime}_2) = c p_2^*\ell_2' \cdot p^*{\mathcal{O}}_{{\mathbb{P}}^2}(1) = c,\end{align*} $$
which is a contradiction.
Example 3.6. Take
$S\mathrel {:=}{\mathbb {P}}^2$
. Let us fix a closed subscheme
$Z_2$
of
${\mathbb {P}}^2$
consisting of two distinct (reduced) points. We fix another closed subscheme
$Z_1$
of
${\mathbb {P}}^2$
consisting of two distinct (reduced) points, chosen generally with respect to
$Z_2$
.
For each
$i = 1,2$
, Serre’s construction (see, e.g., [Reference Huybrechts and Lehn19, Theorem 5.1.1]) produces a locally free sheaf
$E_i$
of rank
$2$
on
${\mathbb {P}}^2$
, which fits into the short exact sequence
$$ \begin{align} 0\to{\mathcal{O}}_{{\mathbb{P}}^2}\to E_i \to {\mathcal I}_{Z_i}(i)\to 0. \end{align} $$
Set
$W_i \mathrel {:=} {\mathbb {P}}(E_i)$
; see [Reference Hartshorne16, Definition in p.162]. Consider
As a projectivized vector bundle, each
$W_i$
is endowed with a tautological line bundle
$\zeta _i$
satisfying
${\phi _i}_*\zeta _i = E_i$
. In particular, this line bundle has a distinguished section given by the inclusion morphism in Equation (3.1), whose zero locus we denote by
$S_i$
. We will describe the geometry of
$S_i$
later.
Note that the Néron–Severi space of W decomposes. Indeed, the smooth fibration
$p_1 : W \to W_1$
identifies with the projectivization of the vector bundle
$\phi _1^*E_2$
over
$W_1$
, which has tautological line bundle
$p_2^*\zeta _2$
, so
$$ \begin{align*} N^1(W)_{{\mathbb{R}}} &= p_1^*N^1(W_1)_{{\mathbb{R}}}+ {\mathbb{R}} \cdot p_2^*[\zeta_2]\\ &= {\mathbb{R}}\cdot p_1^*[\zeta_1] + p^*N^1(S)_{{\mathbb{R}}} + {\mathbb{R}} \cdot p_2^*[\zeta_2]\\ &= p_1^*N^1(W_1)+p_2^*N^1(W_2). \end{align*} $$
Define the line bundle
$$ \begin{align*}D=p_1^*\zeta_1+p_2^*\zeta_2\end{align*} $$
on W. It is effective, as the
$\zeta _i$
both are. To prove that D is nef, let us describe the geometry of the zero loci
$S_i$
.
By [Reference Grothendieck15, Proposition 3.6.2], and since the closed subschemes
$Z_i$
are locally complete intersections, each zero locus
$S_i$
is in fact a (reduced irreducible) surface, isomorphic to
$\mathrm {Bl}_{Z_i}{\mathbb {P}}^2\simeq {\mathbb {P}}({\mathcal I}_{Z_i}(i))$
naturally embedded in
${\mathbb {P}}(E_i)$
through the surjection in Equation (3.1). Through this identification, the restricted line bundle
${\zeta _i}_{|S_i}$
corresponds to the tautological line bundle of
${\mathbb {P}}({\mathcal I}_{Z_i}(i))$
, which in
$\mathrm {Bl}_{Z_i}{\mathbb {P}}^2$
corresponds to the dual of the exceptional line bundle twisted by
$\phi _i^{\ast }{\mathcal {O}}_{{\mathbb {P}}^2}(i)$
. For
$i=1$
, this line bundle corresponds to the divisor obtained by strict transform of the line
$\ell _1$
passing through the two points of
$Z_1$
with the following properties:
-
• It is effective and has a unique section which is irreducible;
-
• It has negative square.
For
$i=2$
, it is the strict transform of any conic through the two points of
$Z_2$
with the following properties:
-
• It is effective and admits an irreducible section;
-
• It has positive square.
Let us summarize: On one hand,
$\zeta _1$
has exactly one negative curve
$\ell _1'$
on
$W_1$
, which is contained in
$S_1$
and has negative square there. On the other hand,
$\zeta _2$
is nef on
$W_2$
.
We now prove that D is nef, arguing by contradiction: Assume that there is a curve C in W such that
$D\cdot C < 0$
. We just proved that
$p_2^*\zeta _2$
is nef, so
$\zeta _1\cdot {p_1}_*C < 0$
, and so there is a positive integer m such that
${p_1}_*C = m\ell _1'$
; moreover, C must lie in
$p_1^{-1}(\ell _1')$
. The restricted map
$\phi _1:\ell _1'\to \ell _1$
is an isomorphism, so its base change
$p_2:p_1^{-1}(\ell _1')\to \phi _2^{-1}(\ell _1)\simeq {\mathbb {P}}({E_2}_{|\ell _1})$
is an isomorphism too. Thus,
${p_2}_*C$
is a reduced curve
$C_2$
in
$W_2$
, and
${\phi _2}_* C_2=m\ell _1$
. By the projection formula,
$$ \begin{align*}D\cdot C=m\zeta_1\cdot\ell^{\prime}_1 + \zeta_2\cdot C_2.\end{align*} $$
We have
${\zeta _1}_{|S_1}=\ell _1'$
, so
$\zeta _1\cdot \ell ^{\prime }_1=-1$
. Moreover, by [Reference Okonek, Schneider and Spindler39, Example 1 in §5.2, Chapter 1], and since we chose
$Z_1$
generally with respect to
$Z_2$
, we have
$$ \begin{align} {E_2}_{|\ell_1} \simeq \mathcal{O}_{{\mathbb{P}}^1}(1) \oplus \mathcal{O}_{{\mathbb{P}}^1}(1), \end{align} $$
so
${E_2}_{|\ell _1}\otimes {\mathcal {O}}_{\ell _1}(-1)$
is nef, in particular
$\zeta _2\cdot C_2\ge \phi _2^*{\mathcal {O}}_{\ell _1}(1)\cdot C_2 = m$
. Hence, we finally have
$0>D\cdot C\ge -m+m = 0$
, contradiction. So D is nef.
We conclude this example by picking a decomposition of D as
$p_1^*D_1+p_2^*D_2$
with
$D_i\in N^1(W_i)_{{\mathbb {R}}}$
, and proving that at least one of the
$D_i$
is not nef. Since the intersection of
$p_1^* N^1(W_1)_{{\mathbb {R}}}$
with
$p_2^* N^1(W_2)_{{\mathbb {R}}}$
is the subspace
${\mathbb {R}}\cdot p^*[{\mathcal {O}}_{{\mathbb {P}}^2}(1)]$
, and since we already have
$D=p_1^*\zeta _1+p_2^*\zeta _2$
, there exists
$a\in {\mathbb {R}}$
such that
$$ \begin{align*}D_1 = \zeta_1 + a \phi_1^*{\mathcal{O}}_{{\mathbb{P}}^2}(1), \ \ D_2 = \zeta_2 - a \phi_2^*{\mathcal{O}}_{{\mathbb{P}}^2}(1).\end{align*} $$
In particular,
$$ \begin{align*}D_1 \cdot \ell_1' = \zeta_1\cdot\ell_1' + a {\mathcal{O}}_{{\mathbb{P}}^2}(1)\cdot{\phi_1}_*\ell_1'= - 1 + a.\end{align*} $$
Moreover, by [Reference Okonek, Schneider and Spindler39, Example 1 in §5.2, Chapter 1] again, there exists a line
$\ell _2$
in
${\mathbb {P}}^2$
such that
$$ \begin{align}{E_2}_{|\ell_2} \simeq \mathcal{O}_{{\mathbb{P}}^1}\oplus \mathcal{O}_{{\mathbb{P}}^1}(2), \end{align} $$
and so there is a section
$\ell _2'$
of the fibration
$\phi _2:{\mathbb {P}}({E_2}_{|\ell _2})\to \ell _2$
such that
$\zeta _2\cdot \ell _2'=0$
. In particular,
$$ \begin{align*}D_2 \cdot \ell_2' = \zeta_2\cdot\ell_2' - a \ell_2\cdot{\phi_2}_*\ell_2' = - a. \end{align*} $$
Since at least one of the two numbers
$a-1$
and
$-a$
is negative,
$D_1$
and
$D_2$
cannot both be nef.
We now use Examples 3.5 and 3.6 to build similar counterexamples over bases of higher dimension.
Example 3.7. Take W,
$W_1$
,
$W_2$
and S as in Example 3.5 or Example 3.6. Note that they all are rationally connected: It is clear in Example 3.5 and follows from [Reference Graber, Harris and Starr13, Corollary 1.3] in Example 3.6. Introduce
$$ \begin{align*}W \times T = (W_1 \times T) \times_{(S \times T)} (W_2 \times T),\end{align*} $$
where T is an arbitrary smooth projective variety. Since W,
$W_1$
and
$W_2$
are rationally connected and smooth, they have trivial irregularity so that
$$ \begin{align*}N^1(Z \times T)_{{\mathbb{R}}} = p_Z^*N^1(Z)_{{\mathbb{R}}}\oplus p_T^*N^1(T)_{{\mathbb{R}}},\end{align*} $$
for
$Z=W$
,
$W_1$
or
$W_2$
. This implies that
$$ \begin{align*}N^1(W \times T)_{{\mathbb{R}}} = (p_1 \times \mathrm{id}_T)^*N^1(W_1 \times T)_{{\mathbb{R}}} + (p_2 \times \mathrm{id}_T)^*N^1(W_2 \times T)_{{\mathbb{R}}}.\end{align*} $$
Note that by the projection formula,
$$ \begin{align*}\mathrm{Nef}(Z \times T) = p_Z^*\mathrm{Nef}(Z)\oplus p_T^*\mathrm{Nef}(T),\end{align*} $$
for
$Z=W$
,
$W_1$
or
$W_2$
. So, if we assume by contradiction that
$$ \begin{align*}\mathrm{Nef}(W\times T)=(p_1\times\mathrm{id}_T)^*\mathrm{Nef}(W_1\times T)+(p_2\times\mathrm{id}_T)^*\mathrm{Nef}(W_2\times T),\end{align*} $$
we get
$\mathrm {Nef}(W)=p_1^*\mathrm {Nef}(W_1)+p_2^*\mathrm {Nef}(W_2)$
, which contradicts Example 3.5 or Example 3.6.
Remark 3.8. We note that Theorem 1.4 also fails if the nef cones are replaced by the movable cones. In general, let X be a smooth projective variety and recall that a divisor D on X is called movable, if there is a positive integer m such that
$mD$
is effective and the base locus of the linear system
$|\mathcal {O}_X(mD)|$
has no component of codimension 1. The closed movable cone
$\overline {\mathrm {Mov}}(X)$
is then defined as the closure of the convex cone in
$N^1(X)_{{\mathbb {R}}}$
generated by the classes of movable divisors. It always holds
$\mathrm {Nef}(X) \subset \overline {\mathrm {Mov}}(X)$
, and, if moreover X is a surface, then
$\mathrm {Nef}(X) = \overline {\mathrm {Mov}}(X)$
.
Take a general fiber product
$W = W_1 \times _{{\mathbb {P}}^1} W_2$
of two very general rational elliptic surfaces
$W_1 \to {\mathbb {P}}^1$
and
$W_2 \to {\mathbb {P}}^1$
with sections. Then W is a strict Calabi–Yau threefold and has nontrivial algebraic flops (see [Reference Namikawa37]). Thus,
$\mathrm {Nef}(W) \subsetneq \overline {\mathrm {Mov}}(W)$
. But since the
$W_i$
are surfaces, we have
$$ \begin{align*}p_1^*\overline{\mathrm{Mov}}(W_1) + p_2^*\overline{\mathrm{Mov}}(W_2) = p_1^*\mathrm{Nef}(W_1) + p_2^*\mathrm{Nef}(W_2) =\mathrm{Nef}(W) \subsetneq \overline{\mathrm{Mov}}(W),\end{align*} $$
where the second equality follows from Theorem 1.4.
Even in this particular case, the version of the cone conjecture stated in [Reference Totaro48, Conjecture 2.1.(2)] is not known.
We conclude this section with a corollary of Theorem 1.4 that will be key in the proof of Theorem 1.6.
For a morphism
$\pi :X\to Y$
, we define
$$ \begin{align*}\mathrm{Aut}(X/Y)=\{g\in\mathrm{Aut}(X)\mid \pi\circ g = \pi\}.\end{align*} $$
Corollary 3.9. For
$i=1,2$
, let
$\phi _i : W_i \to B$
be a surjective morphism from a projective variety to a projective curve B; let
$H_i$
be a subgroup of
$ \mathrm {Aut}(W_i/B)$
. Assume that
-
1. The fiber product
$W=W_1 \times _{B} W_2$
is irreducible; -
2. It holds
where
$$ \begin{align*}p_1^*N^1(W_1)_{{\mathbb{R}}}+p_2^*N^1(W_2)_{{\mathbb{R}}}=N^1(W)_{{\mathbb{R}}},\end{align*} $$
$p_i$
denotes the projection from W onto
$W_i$
;
-
3. For each
$i=1,2$
, there exists a rational polyhedral cone
$\Pi _i$
in
$\mathrm {Nef}^{+}(W_i)$
such that
$\mathrm {Amp}(W_i)\subset H_i\cdot \Pi _i$
.
Then, for any subgroup H of
$\mathrm {Aut}(W)$
containing
$H_1\times H_2$
, there is a rational polyhedral fundamental domain for the H-action on
$\mathrm {Nef}^{+}(W)$
.
Proof. Let
$\Pi $
be the convex hull of
$p_1^*\Pi _1 + p_2^*\Pi _2$
. Then
$\Pi $
is a rational polyhedral cone contained in
$\mathrm {Nef}^+(W)$
. Moreover,
$$\begin{align*}\mathrm{Amp}(W) \subset(H_1 \times H_2) \cdot \Pi \subset H \cdot \Pi \end{align*}$$
as
$p_1^*\mathrm {Amp}(W_1) + p_2^*\mathrm {Amp}(W_2)=\mathrm {Amp}(W)$
by Theorem 1.4. The existence of a rational polyhedral fundamental domain then follows from Proposition 2.3.(1).
4 Construction of Schoen varieties
Schoen varieties are constructed as fiber products of two fibrations over
${\mathbb {P}}^1$
. Let us first construct these fibrations.
4.1 The factor W with a fibration over
${\mathbb {P}}^1$
This construction relies on a pencil of ample hypersurfaces in a Fano manifold.
Let Z be a Fano manifold of dimension at least
$2$
, and let D be an ample divisor in Z such that both
$\mathcal {O}_Z(D)$
and
$\mathcal {O}_Z(-K_Z - D)$
are globally generated. Note that
$\mathcal {O}_Z(-K_Z)$
is then globally generated as well.
Example 4.1. Take any toric Fano manifold Z of dimension at least
$2$
. Since nef line bundles on a projective toric manifold are globally generated, any decomposition
$-K_Z = D + D'$
as the sum of an ample divisor D and a nef divisor
$D'$
yields a pair
$(Z,D)$
satisfying the above condition.
Let
$W \subset {\mathbb {P}}^1\times Z$
be a general member of the ample and basepoint-free linear system
$|{\mathcal {O}}_{{\mathbb {P}}^1}(1)\boxtimes {\mathcal {O}}_{Z}(D)|$
. We have a fibration
$\phi : W \to {\mathbb {P}}^1$
via the first projection, and the second projection
$\varepsilon : W \to Z$
is the blow-up of Z along the smooth subvariety Y of codimension two cut out by the members of the pencil in
$|D|$
defined by W. Since Z is Fano, W is rationally connected. By construction, any point
$y\in Y$
defines a rational curve
$\varepsilon ^{-1}(y)$
which is a section of
$\phi : W \to {\mathbb {P}}^1$
.
By the adjunction formula,
$$ \begin{align} {\mathcal{O}}_W(-K_{W}) = \left({\mathcal{O}}_{{\mathbb{P}}^1}(1)\boxtimes {\mathcal{O}}_{Z}(-K_Z - D)\right)|_{W}, \end{align} $$
so
${\mathcal {O}}_W(-K_W)$
is globally generated, a fortiori nef and effective.
If Z is chosen to be a del Pezzo surface, then the surface W is described by the following lemma. Recall that a smooth projective surface S is called weak del Pezzo if its anticanonical divisor
$-K_S$
is nef and big.
Lemma 4.2. If Z has dimension 2, then either
$D \in |-K_Z|$
and
$W \xrightarrow {\phi } {\mathbb {P}}^1$
is a rational elliptic surface with globally generated anticanonical line bundle, or W is a weak del Pezzo surface.
Proof. Since W is rationally connected and
$\dim W = 2$
, we know that W is rational. If
$D \in |-K_Z|$
, then
$\mathcal {O}_W(-K_{W}) = \phi ^*\mathcal {O}_{{\mathbb {P}}^1}(1)$
, which is globally generated, and which makes W into a rational elliptic surface.
Suppose now that
$D \notin |-K_Z|$
. As
$-K_Z - D$
is effective and nontrivial, and as
$-K_Z$
and D are ample, we have
$-K_Z(-K_Z - D)> 0$
and
$D(-K_Z - D)> 0$
, and thus,
$$ \begin{align*}K_Z^2> -K_Z\cdot D > D^2.\end{align*} $$
As W is the blowup of Z at
$(D^2)$
points, we have
$K_W^2 = K_Z^2 - D^2> 0$
. Since
$-K_W$
is nef, W is a weak del Pezzo surface.
Remark 4.3. Note that, in the case where W is a rational elliptic surface, the fact that it has a section and that it is chosen general in its pencil on
${\mathbb {P}}^1\times Z$
implies that it is isomorphic to
${\mathbb {P}}^2$
blown-up in the base locus of a general pencil of cubics. In particular, W has topological Euler characteristics
$12$
, the canonical fibration
$W\to {\mathbb {P}}^1$
has some singular fibers but no multiple fibers. The fact that the rational elliptic surface W is general implies that the singular fibers of
$W\to {\mathbb {P}}^1$
are exactly 12 nodal rational curves ([Reference Miranda34, p.8]).
Considering the j-invariant in family for the fibration
$W\to {\mathbb {P}}^1$
, we obtain a proper surjective map
$j:{\mathbb {P}}^1\to {\mathbb {P}}^1$
which is finite of degree 12 and has 12 simple poles which occur at the 12 image points of the 12 singular fibers ([Reference Miranda34, Lemma (IV.4.1), Corollary (IV.4.2)]).
In general, the construction of W described above ensures the following properties.
Proposition 4.4. We have
$$ \begin{align*}\mathrm{Nef}^e(W) = \mathrm{Nef}^{+}(W) = \mathrm{Nef}(W).\end{align*} $$
Moreover, if
$\dim W \geq 3$
or if W is a weak del Pezzo surface, then the cone
$\mathrm {Nef}(W)$
is rational polyhedral, spanned by classes of semiample divisors.
Proof. We start with the ‘moreover’ part. It is a corollary of some known results. If W is a weak del Pezzo surface, then W is log Fano (see, e.g., [Reference Massarenti33, Proposition 2.6]). Hence, by the cone theorem [Reference Kollár, Mori, Clemens and Corti26, Theorem 3.7], its nef cone is a rational polyhedral cone spanned by classes of semiample divisors. Assume that
$\dim W \geq 3$
. Since
${\mathbb {P}}^1 \times Z$
is a smooth Fano variety of dimension at least four, and since
$W \subset {\mathbb {P}}^1 \times Z$
is a smooth ample divisor such that
$$ \begin{align*}\mathcal{O}_{{\mathbb{P}}^1 \times Z}(-K_{{\mathbb{P}}^1 \times Z} - W) = \mathcal{O}_Z(-K_Z - D) \boxtimes \mathcal{O}_{{\mathbb{P}}^1}(1)\end{align*} $$
is nef, we can apply [Reference Beltrametti and Ionescu1, Proposition 3.5] (which generalizes [Reference Borcea and Kollár3, Appendix]). It yields an isomorphism
induced by the inclusion
$j : W \hookrightarrow {\mathbb {P}}^1 \times Z$
. Dually, we obtain an isomorphism
As
$\mathrm {Nef}({\mathbb {P}}^1 \times Z)$
is rational polyhedral and spanned by classes of semiample divisors, so is
$\mathrm {Nef}(W)$
.
We now prove the equality of the three cones
$\mathrm {Nef}^e(W)$
,
$\mathrm {Nef}^{+}(W)$
, and
$\mathrm {Nef}(W)$
. If
$\dim W\ge 3$
or if W is a weak del Pezzo surface, the equality clearly follows from the fact that
$\mathrm { Nef}(W)$
is rational polyhedral, spanned by classes of semiample divisors. So by Lemma 4.2, we can focus on the case where W is a rational elliptic surface.
Clearly,
$\mathrm {Nef}^e(W)$
and
$\mathrm {Nef}^{+}(W)$
are subcones of
$\mathrm {Nef}(W)$
. Moreover,
$\mathrm {Nef}^{+}(W)\subset \mathrm {Nef}^e(W)$
by [Reference Totaro48, Lemma 4.2]. We only need to show that
$\mathrm {Nef}(W) = \mathrm {Nef}^{+}(W)$
. By [Reference Nikulin38, Corollary 3.3. (c)], the cone
$\overline {\mathrm {NE}}(W)$
is generated by curve classes, so dually,
$\mathrm {Nef}(W)$
is spanned by Cartier divisors. So
$\mathrm {Nef}(W) = \mathrm {Nef}^{+}(W)$
indeed.
Let us conclude the description of W by describing the general fiber of
$\phi : W\to {\mathbb {P}}^1$
, under the assumption that
$D \in |-K_Z|$
.
Lemma 4.5. Suppose that
$D \in |-K_Z|$
. Then the general fiber F of
$\phi :W\to {\mathbb {P}}^1$
is a Calabi–Yau manifold (as in Definition 2.2).
Proof. Since
$D \in |-K_Z|$
, the general fiber F is linearly equivalent to the anticanonical divisor
$-K_W$
by Equation (4.1). By adjunction, F has trivial canonical bundle. We also have an exact sequence
$$ \begin{align*}0\to {\mathcal{O}}_W(-K_W) \to{\mathcal{O}}_W\to {\mathcal{O}}_F \to 0.\end{align*} $$
Since W is rationally connected, we have
$$ \begin{align*}h^{\dim W - i}(W,-K_W) = h^i(W,{\mathcal{O}}_W)=0\end{align*} $$
for
$i \ge 1$
. Hence,
$h^i(F,{\mathcal {O}}_F)=0$
whenever
$1 \le i \le \dim W - 2 = \dim F - 1 $
.
4.2 The fiber product
$X=W_1{\times }_{{\mathbb {P}}^1} W_2$
We are ready to generalize Schoen’s construction and obtain Calabi–Yau pairs in arbitrary dimension. For
$i = 1, 2$
, let
$Z_i, D_i, W_i$
be as in §4.1. We denote by
$\phi _i:W_i\to {\mathbb {P}}^1$
the associated fibration and recall that it has a section.
We add one assumption, which is automatically satisfied by taking the fibrations
$\phi _i$
for
$i=1,2$
to be general with respect to one another:
$$\begin{align*}\textit{For}\ \textit{every}\ t\in{\mathbb{P}}^1,\ \textit{the fiber of at least one of the}\ \phi_i\ \textit{above}\ t\ \textit{is smooth}. \end{align*}$$
In the case where both
$W_i$
are rational elliptic surfaces, this assumption has an important consequence.
Lemma 4.6. Let
$W_1$
and
$W_2$
be general rational elliptic surfaces, with their canonical fibrations
$\phi _i:W_i\to {\mathbb {P}}^1$
, each admitting a section. Assume that for every
$t\in {\mathbb {P}}^1$
, there is i such that the fiber
$\phi _i^{-1}(t)$
is smooth. Then, for a very general point
$t\in {\mathbb {P}}^1$
, the fibers
$\phi _1^{-1}(t)$
and
$\phi _2^{-1}(t)$
are smooth, nonisogenous elliptic curves.
Proof. For
$i=1,2$
, consider the finite morphism
$j_i:{\mathbb {P}}^1\to {\mathbb {P}}^1$
induced by the j-invariant of the elliptic fibration
$\phi _i$
(see Remark 4.3). Define the morphism
$J\mathrel {:=} (j_1,j_2):{\mathbb {P}}^1\to {\mathbb {P}}^1\times {\mathbb {P}}^1$
. Its image is an irreducible curve in
${\mathbb {P}}^1 \times {\mathbb {P}}^1$
.
For each positive integer n, let
$F_n(x, y) \in {\mathbb {Z}}[x, y]$
be the polynomial as in [Reference Silverman45, Theorem 6.3 in p.146]. Then by [Reference Silverman45, Exercise 2.19. (a) in p.182], we have
$F_n(j(\phi _1^{-1}(t)), j(\phi _2^{-1}(t))) = 0$
if and only if there is an isogeny
$\phi _1^{-1}(t) \to \phi _2^{-1}(t)$
of degree n. By [Reference Silverman45, Exercise 2.18. (e) in p.181], each
$F_n(x, y)$
is a product of some polynomials
$\Phi _m(x, y)$
indexed by positive integers. By the expression in [Reference Silverman45, Exercise 2.18 in p.181], each
$\Phi _m(x, y)$
viewed as a polynomial in the single variable x has leading coefficient
$1$
. Together with [Reference Silverman45, Exercise 2.18.(b) in p.181], we obtain the irreducibility of
$\Phi _m(x, y)$
in
${\mathbb {C}}[x, y]$
.
Let
$\Sigma _m \subset {\mathbb {P}}^1 \times {\mathbb {P}}^1$
be the irreducible curve defined by the homogenization of
$\Phi _m(x, y)$
using
$x = s/t$
and
$y = u/v$
. We claim that
$J({\mathbb {P}}^1)$
intersects with each
$\Sigma _m$
at finitely many points. Indeed, by our assumption, we can take
$t \in {\mathbb {P}}^1$
such that the fiber
$\phi _1^{-1}(t)$
is singular, while the fiber
$\phi _2^{-1}(t)$
is an elliptic curve, so
$J(t)=([1:0],[\alpha :1])$
for some
$\alpha \in {\mathbb {C}}$
. As we mentioned before, each
$\Phi _m(x, y)$
viewed as a polynomial in the single variable x has leading coefficient
$1$
, so
$([1 : 0], [\alpha : 1])\notin \Sigma _m$
. This implies
$J({\mathbb {P}}^1) \neq \Sigma _m$
, and the claim holds because both
$\Sigma _m$
and
$J({\mathbb {P}}^1)$
are irreducible.
Let
$\Sigma \subset {\mathbb {P}}^1 \times {\mathbb {P}}^1$
be the union of the countably many curves
$\Sigma _m$
. Then the set
${\mathbb {P}}^1\setminus (Z_1\cup Z_2\cup J^{-1}(J({\mathbb {P}}^1)\cap \Sigma ))$
is nonempty with the property that each of its elements is a very general point, say t, satisfying that the fibers
$\phi _1^{-1}(t)$
and
$\phi _2^{-1}(t)$
are smooth, nonisogenous elliptic curves.
Now that we better understand the fibrations
$\phi _i$
relatively to one another, we can consider the fiber product over
${\mathbb {P}}^1$
As for every
$t\in {\mathbb {P}}^1$
, the fiber of at least one of the
$\phi _i$
above t is smooth, the variety X is smooth too. We can also view X as a complete intersection of two hypersurfaces in
${\mathbb {P}}^1 \times Z_1 \times Z_2$
, given by general members in the linear systems
$$\begin{align*}|{\mathcal{O}}_{{\mathbb{P}}^1}(1)\boxtimes{\mathcal{O}}_{Z_1}(D_1)\boxtimes {\mathcal{O}}_{Z_2}| \ \ \mathrm{and} \ \ |{\mathcal{O}}_{{\mathbb{P}}^1}(1)\boxtimes{\mathcal{O}}_{Z_1}\boxtimes {\mathcal{O}}_{Z_2}(D_2)|. \end{align*}$$
By adjunction, we obtain that
$$ \begin{align} \mathcal{O}_X(-K_X) = \left({\mathcal{O}}_{{\mathbb{P}}^1}\boxtimes{\mathcal{O}}_{Z_1}(-K_{Z_1} - D_1) \boxtimes {\mathcal{O}}_{Z_2}(-K_{Z_2} - D_2)\right)|_X, \end{align} $$
which is globally generated, hence nef and effective.
Definition 4.7. A smooth projective variety X constructed as above is called a Schoen variety. A pair
$(X, \Delta )$
is called a Schoen pair if X is a Schoen variety, and
$\Delta $
is an effective
${\mathbb {Q}}$
-divisor such that
$K_X + \Delta \sim _{{\mathbb {Q}}} 0$
.
Any Schoen variety X can be associated many Schoen pairs
$(X, \Delta )$
as long as
$-K_X$
is nontrivial. Every Schoen pair is by definition a Calabi–Yau pair (as in Definition 2.1). Moreover, if
$(X,\Delta )$
is a Schoen pair, then there exists a positive integer m such that
$$ \begin{align} \Delta = \frac{1}{m}\Delta_{m,X}, \ \text{ with } \,\Delta_{m,X}\in|-mK_X|. \end{align} $$
If
$m \ge 2$
and
$\Delta _{m,X}\in |-mK_X|$
is general, the Calabi–Yau pair
$(X,\Delta )$
is klt.
To conclude this section, we prove that, if for both
$i=1,2$
, the divisor
$D_i$
chosen when constructing
$W_i$
is in the linear system
$|-K_{Z_i}|$
, then the Schoen variety X is a strict Calabi–Yau manifold.
Lemma 4.8. Any Schoen variety X is simply connected.
Proof. The proof is similar to [Reference Schoen43, Lemma 1] and [Reference Suzuki46, Lemma 2.1].
Let
$U \subset {\mathbb {P}}^1$
be the open subset over which the morphism
$\phi : X \to {\mathbb {P}}^1$
is smooth, and set
$V := \phi ^{-1}(U)$
. Let
$i: V \hookrightarrow X$
and
$j: U \hookrightarrow {\mathbb {P}}^1$
be the natural inclusions. The restriction 
is topologically locally trivial with a fiber, say F. Since both
$\phi _1$
and
$\phi _2$
have sections,
$\phi : X \to {\mathbb {P}}^1$
also admits a section
$\sigma : {\mathbb {P}}^1 \to X$
. Consider the commutative diagram
Here, the first row is exact by the homotopy long exact sequence, and
$i_{\ast } : \pi _1(V) \to \pi _1(X)$
is surjective by [Reference Kollár25, Proposition 2.10.1].
We claim that the image of
$\pi _1(F)$
in
$\pi _1(X)$
equals
$\pi _1(X)$
. Indeed, since
$\pi _1({\mathbb {P}}^1)$
is trivial, the composition
$i_* \circ {\sigma _U}_* = \sigma _* \circ j_*$
is trivial. Using that
$i_*$
is surjective, that this composition is trivial and that
$\pi _1(V)$
is generated by the union of its subgroups
$ \pi _1(F)$
and
${\sigma _U}_*\pi _1(U)$
, we obtain
$$ \begin{align*}\pi_1(X) = i_*\pi_1(V) = i_* \pi_1(F).\end{align*} $$
We are now left to show that the image of
$\pi _1(F)$
in
$\pi _1(X)$
is trivial. Write
$F = F_1 \times F_2$
, where
$F_i$
is a general fiber of
$\phi _i: W_i \to {\mathbb {P}}^1$
for
$i = 1,2$
. Since
$\pi _1(F) = \pi _1(F_1) \times \pi _1(F_2)$
, it is enough to show that the image of
$\pi _1(F_i)$
in
$\pi _1(X)$
is trivial, which we prove for
$i = 1$
.
A section of
$\phi _{2} : W_2 \to {\mathbb {P}}^1$
gives rise to a section s of
$p_1: X \to W_1$
. By construction, the homomorphism
$\pi _1(F_1) \to \pi _1(X)$
is induced by
$F_1 \hookrightarrow W_1 \xrightarrow {s} X$
, thus factors through
$\pi _1(W_1)$
. Since it is rationally connected,
$W_1$
is simply connected and hence the image of
$\pi _1(F_1)$
in
$\pi _1(X)$
is trivial.
Proposition 4.9. Suppose that
$D_i \in |-K_{Z_i}|$
for both
$i = 1,2$
. Then the Schoen variety X is a strict Calabi–Yau manifold (see Definition 2.2).
Proof. By Equation (4.2) and Lemma 4.8, X has trivial canonical bundle and trivial fundamental group. We are left showing that
$h^p(X,{\mathcal {O}}_X)=0$
for every
$0 < p <\dim X$
.
Lemma 4.10. Let
$g: \mathscr {X} \to \mathscr {Y}$
be a surjective morphism between smooth projective varieties. Assume that a general fiber F of g is a Calabi–Yau manifold and that the canonical line bundle
$\omega _{\mathscr {X}}$
is trivial. Then, for every positive integer q, we have
$$ \begin{align*} R^q g_*{\mathcal{O}}_{\mathscr{X}} = \begin{cases} \omega_{\mathscr{Y}}, &\text{if } q = \dim \mathscr{X} - \dim \mathscr{Y}, \\ 0, &\text{otherwise.} \end{cases} \end{align*} $$
Proof. Set
$r \mathrel {:=} \dim \mathscr {X} - \dim \mathscr {Y}$
. By [Reference Kollár23, Theorem 2.1. (i)] and [Reference Kollár24, Corollary 3.9], the sheaf
$R^qg_*\omega _{\mathscr {X}}=R^qg_*\mathcal {O}_{\mathscr {X}}$
is reflexive. Since
$\mathscr {Y}$
is smooth, the invertibility of
$R^qg_*\mathcal {O}_{\mathscr {X}}$
follows provided it has rank one. Its rank is explicitly given by the dimension of
$H^q(F,\mathcal {O}_F)$
, which is one if
$q=0$
or r, and zero otherwise. Hence, we have
$$ \begin{align} R^qg_*\mathcal{O}_{\mathscr{X}} = \begin{cases} \text{a line bundle, if } q = 0 \text{ or } r,\\ 0, \text{otherwise.} \end{cases} \end{align} $$
By Grothendieck–Verdier duality [Reference Huybrechts18, Theorem 3.34], we have
$$ \begin{align*}Rg_* {\mathcal{O}}_{\mathscr{X}} \simeq Rg_* \omega_{\mathscr{X}} \simeq R\mathcal{H}om(Rg_*\mathcal{O}_{\mathscr{X}},\omega_{\mathscr{Y}}[-r]).\end{align*} $$
The Grothendieck spectral sequence gives
$$ \begin{align*}E_2^{p,-q} \mathrel{:=} \mathcal{E}xt^p(R^qg_*\mathcal{O}_{\mathscr{X}}, \omega_{\mathscr{Y}}) \Rightarrow R^{p-q + r}g_* {\mathcal{O}}_{\mathscr{X}}\end{align*} $$
(see, e.g., [Reference Huybrechts18, Example 2.70.ii)]). But by Equation (4.4), the page
$E_2$
has exactly two nonzero entries, namely
$E_2^{0,0} = \omega _{\mathscr {Y}}$
, and
$E_2^{0,-r}$
. So Lemma 4.10 follows.
We return to our Schoen variety X. For
$i=1,2$
, we let
$w_i \mathrel {:=} \dim W_i$
. By Lemma 4.5, and as
$p_2: X \to W_2$
is a base change of
$\phi _1: W_1 \to {\mathbb {P}}^1$
, the general fiber of
$p_2$
is a Calabi–Yau manifold. We can thus apply Lemma 4.10 to
$p_2$
, and obtain that
$$ \begin{align*} R^q{p_2}_*\omega_{X} = R^q{p_2}_*{\mathcal{O}}_{X} = \begin{cases} \mathcal{O}_{W_2}, \text{ if } q = 0, \\ \omega_{W_2}, \,\text{ if } q = \dim w_1 - 1, \\ 0, \ \ \,\quad\text{otherwise.} \end{cases} \end{align*} $$
Together with [Reference Kollár24, Corollary 3.2], this yields
$$ \begin{align*}h^p(X,{\mathcal{O}}_X)=h^p(W_2,{\mathcal{O}}_{W_2})+h^{p-w_1+1}(W_2,\omega_{W_2})\end{align*} $$
for all
$0\le p\le \dim X$
. Since
$W_2$
is rationally connected, this is zero as soon as
$p\ne 0$
and
$p<w_1+w_2-1 = \dim X$
.
5 Application to the cone conjecture
In this section, we prove Theorem 1.6. The setup and the notations were defined in Section 4: We consider a Schoen variety X, fitting in a Schoen pair
$(X,\Delta )$
. Let us recall the Cartesian diagram defining X:
Lemma 5.1. We have
$$ \begin{align*}p_1^*N^1(W_1)_{{\mathbb{R}}}+p_2^*N^1(W_2)_{{\mathbb{R}}}=N^1(X)_{{\mathbb{R}}}.\end{align*} $$
Proof. Let
$p \in {\mathbb {P}}^1$
be a very general point, and let
$F_i \mathrel {:=} \phi _i^{-1}(p) \subset W_i$
.
Claim 5.2. The map
$$ \begin{align*}\Psi : \mathrm{Pic}(F_1) \times \mathrm{Pic}(F_2) \to \mathrm{Pic}(F_1 \times F_2)\end{align*} $$
defined by
$\Psi (L,M) = L \boxtimes M$
is an isomorphism.
Proof. First, suppose that
$W_1$
and
$W_2$
are not both rational elliptic surfaces. If there is i such that
$Z_i$
has dimension at least
$3$
, then
$F_i$
is a smooth ample hypersurface in
$Z_i$
, and so by Lefschetz hyperplane theorem,
$F_i$
has trivial irregularity. If there is i such that
$Z_i$
is a surface and
$D_i\notin |-K_{Z_i}|$
, then
$F_i$
is a smooth curve in
$Z_i$
, and by adjunction, it is in fact a rational curve, which again has trivial irregularity. In any case, Claim 5.2 follows from [Reference Hartshorne16, Exercise III.12.6].
Assume now that both
$W_1$
and
$W_2$
are rational elliptic surfaces. Then, by Lemma 4.6, the fibers
$F_1$
and
$F_2$
are smooth, nonisogenous elliptic curves. We have a short exact sequence of abelian groups [Reference Birkenhake and Lange2, Theorem 11.5.1]
$$ \begin{align*}0 \to \mathrm{Pic}(F_1) \times \mathrm{Pic}(F_2) \xrightarrow{\Psi} \mathrm{Pic}(F_1 \times F_2) \to \mathrm{Hom}(F_1,F_2) \to 0,\end{align*} $$
where
$\mathrm {Hom}(F_1,F_2)$
denotes the group of homomorphisms from
$F_1$
to
$F_2$
preserving both the variety and the group structure. Since
$F_1$
and
$F_2$
are nonisogenous,
$\mathrm {Hom}(F_1,F_2) = 0$
, which proves Claim 5.2.
Let L be a line bundle on X. Claim 5.2 implies that
$$ \begin{align*}L_{|\phi^{-1}(p)} \simeq L_{|F_1 \times \{u\}} \boxtimes L_{|\{v\} \times F_2},\end{align*} $$
for any points
$ u \in F_2$
and
$v \in F_1$
.
For each
$i = 1,2$
, we choose a section
$s_i : {\mathbb {P}}^1 \to W_i$
and let
$\sigma _i : W_i \to X$
be the induced section:
$$ \begin{align*}\sigma_1(w_1) \mathrel{:=} (w_1,s_2(\phi_1(w_1))) \in W_1 \times_{{\mathbb{P}}^1} W_2,\end{align*} $$
and similarly for
$\sigma _2$
. We have
$$ \begin{align*} L_{|\phi^{-1}(p)} & \simeq L_{|F_1 \times \{s_1(p)\}} \boxtimes L_{|\{s_2(p)\} \times F_2} \\ & \simeq (\sigma_1^*L)_{|F_1} \boxtimes (\sigma_2^*L)_{|F_2} \\ & \simeq (p^*_1\sigma_1^*L \otimes p^*_2\sigma_2^*L)_{|\phi^{-1}(p)}. \end{align*} $$
Since
$p \in {\mathbb {P}}^1$
is very general, by applying [Reference Voisin49, Theorem 3.1 and Remark 3.3] to the smooth part of the fibration
$\phi : X \to {\mathbb {P}}^1$
, we obtain
$$ \begin{align*}L \sim_{{\mathbb{Q}}} p^*_1\sigma_1^*L \otimes p^*_2\sigma_2^*L\otimes \mathcal{O}_X(D),\end{align*} $$
for a divisor D whose support is contained in a finite union of fibers of
$\phi : X \to {\mathbb {P}}^1$
. Note that an irreducible component R of a fiber of
$\phi $
embeds in the product
$\phi _1^{-1}(\phi (R))\times \phi _2^{-1}(\phi (R))$
, of which at least one factor
$\phi _i^{-1}(\phi (R))$
is smooth, hence irreducible. It follows that there is an irreducible component
$R'$
of
$\phi _j^{-1}(\phi (R))$
with
$j = \{1, 2\} \setminus \{i\}$
such that
$R=p_j^* R'$
. Applying this to the irreducible components of D, we obtain that
$$ \begin{align*}N^1(W_1)_{{\mathbb{R}}} \times N^1(W_2)_{{\mathbb{R}}} \xrightarrow{p_1^* + p_2^*} N^1(X)_{{\mathbb{R}}}\end{align*} $$
is surjective.
Lemma 5.3. For every
$D \in \mathrm {Nef}(X)$
, one can write
$D = p_1^*D_1 + p_2^*D_2$
, where
$D_i \in \mathrm {Nef}(W_i)$
.
Theorem 5.4 (
$=$
Theorem 1.6).
Let
$(X,\Delta )$
be a Schoen pair. Then
$$ \begin{align*}\mathrm{Nef}(X)=\mathrm{Nef}^{+}(X)=\mathrm{Nef}^e(X),\end{align*} $$
and moreover, there exists a rational polyhedral fundamental domain for the action of
$\mathrm {Aut}(X, \Delta )$
on
$\mathrm {Nef}^e(X)$
.
Proof. Since
$\mathrm {Nef}(W_i)=\mathrm {Nef}^{+}(W_i) = \mathrm {Nef}^e(W_i)$
by Proposition 4.4, we have, by Lemma 5.3,
$\mathrm {Nef}(X)=p_1^*\mathrm {Nef}^{+}(W_1)+p_2^*\mathrm {Nef}^{+}(W_2)\subset \mathrm {Nef}^{+}(X)$
, so
$\mathrm {Nef}(X)=\mathrm {Nef}^{+}(X)$
. Similarly, we have
$\mathrm {Nef}(X)=\mathrm {Nef}^e(X)$
. This proves the first assertion.
Define the subgroups
$H_i \le \mathrm {Aut}(W_i)$
by
$$ \begin{align*}H_i = \begin{cases} \mathrm{Aut}(W_i/{\mathbb{P}}^1), \, \text{ if } W_i \text{ is a rational elliptic surface,} \\ \{\mathrm{id}_{W_i}\}, \, \text{ otherwise. } \end{cases} \end{align*} $$
Then there exists a rational polyhedral cone
$\Pi _i \subset \mathrm {Nef}^{+}(W_i)$
such that
$H_i \cdot \Pi _i$
contains
$\mathrm {Amp}(W_i)$
. Indeed, the case where
$W_i$
is a rational elliptic surface with
$-K_{W_i}$
semiample follows from [Reference Totaro47, Theorem 8.2], and the other cases follow from Proposition 4.4.
We claim that
$H_1\times H_2\le \mathrm {Aut}(X,\Delta )$
. Note that there exists a positive integer m such that
$$ \begin{align*}\Delta = \frac{1}{m}\Delta_{m,X}\end{align*} $$
for some
$\Delta _{m,X}\in |-mK_X|$
. If neither
$W_1$
nor
$W_2$
is a rational elliptic surface, then
$H_1 \times H_2$
is trivial by definition. If both
$W_1$
and
$W_2$
are rational elliptic surfaces, then
$\Delta _{m,X}=0$
and clearly,
$H_1\times H_2\le \mathrm {Aut}(X)$
. Finally, if one of the
$W_i$
, say
$W_1$
, is a rational elliptic surface, and the other, say
$W_2$
, is not, then
$\mathcal {O}_X(-K_X) \simeq p_2^*\mathcal {O}_{W_2}(-K_{Z_2} - D_2)$
. Since
$p_2$
is proper surjective with connected fibers, the pullback
$p_2^*$
induces an isomorphism
$$ \begin{align*}H^0(X, p_2^*{\mathcal{O}}_{W_2}(-m(K_{Z_2}+D_2))) \simeq H^0(W_2, {\mathcal{O}}_{W_2}(-m(K_{Z_2}+D_2))).\end{align*} $$
So
$\Delta _{m,X}=p_2^*\Delta _{m,W_2}$
, for some divisor
$\Delta _{m,W_2}\in |-m(K_{Z_2}+D_2)|$
. Since
$H_2 = \{ \mathrm {id}_{W_2}\}$
in this case, it follows that
$\Delta _{m,X}$
is invariant under
$H_1 \times H_2$
. This proves the claim.
It then follows from Corollary 3.9 that
$\mathrm {Nef}^e(X) = \mathrm {Nef}^{+}(X)$
has a rational polyhedral fundamental domain
$\Pi $
for the
$\mathrm {Aut}(X,\Delta )$
-action.
Remark 5.5. In [Reference Grassi and Morrison14], the authors verified the cone conjecture for a strict Calabi–Yau threefold
$X = W_1 \times _{{\mathbb {P}}^1} W_2$
, where both
$W_i$
are general rational elliptic surfaces with sections. They use the following identification shown by Namikawa [Reference Namikawa37, Proposition 2.2 and Corollary 2.3]
$$\begin{align*}\mathrm{Aut}(X)\cong \mathrm{Aut}(W_1)\times \mathrm{Aut}(W_2), \end{align*}$$
which our proof bypasses, using Looijenga’s result (Lemma 2.4) instead.
Example 5.6. Fix an integer
$n\ge 3$
. Let us explain how to choose
$Z_1,Z_2,D_1,$
and
$D_2$
so that our construction produces a strict Calabi–Yau manifold X of dimension n such that
$\mathrm {Nef}(X)$
admits infinitely many extremal rays and X satisfies the cone conjecture. We take
$Z_1 = {\mathbb {P}}^2$
and
$D_1={\mathcal {O}}_{{\mathbb {P}}^2}(3)$
so that
$W_1$
is a general rational elliptic surface. We take
$Z_2$
to be a Fano variety of dimension
$n-1$
with
$-K_{Z_2}$
globally generated (for example,
$Z_2={\mathbb {P}}^{n-1}$
), and we take
$D_2= -K_{Z_2}$
.
The Schoen variety X obtained from these choices is a strict Calabi–Yau manifold by Proposition 4.9, and
$\mathrm {Nef}(X)$
admits infinitely many extremal rays by Lemma 5.1, by the fact that
$\mathrm { Nef}(W_1)$
admits infinitely extremal rays already, and by Corollary 1.5.
We conclude with an unsurprising corollary of the fact that Schoen varieties satisfy the cone conjecture.
Corollary 5.7. Let X be a Schoen variety. Then the group
$\pi _0 \mathrm {Aut}(X)$
is finitely presented, and there are at most finitely many real forms for X, up to isomorphism.
Proof. The linear action
$\rho : \mathrm {Aut}(X) \to \mathrm {GL}(N^1(X))$
induces and factorizes through an action
$$ \begin{align*}\overline{\rho}: \pi_0\mathrm{Aut}(X) \to \mathrm{GL}(N^1(X)).\end{align*} $$
We let
$\mathrm {Aut}^{\ast }(X) = \rho (\mathrm {Aut}(X)) = \overline {\rho }(\pi _0\mathrm {Aut}(X))$
.
Choose an effective
${\mathbb {Q}}$
-divisor
$\Delta $
on X such that
$(X,\Delta )$
is a Schoen pair. By Theorem 1.6, there exists a rational polyhedral cone
$\Pi \subset \mathrm {Nef}^{+}(X)$
such that
$$ \begin{align*}\mathrm{Amp}(X)\subset \mathrm{Aut}(X,\Delta)\cdot \Pi\subset\mathrm{Aut}^{\ast}(X)\cdot \Pi.\end{align*} $$
It follows from Proposition 2.3 that there is a rational polyhedral fundamental domain for the
$\mathrm {Aut}^{\ast }(X)$
-action on
$\mathrm {Nef}^{+}(X)$
, and that the group
$\mathrm {Aut}^{\ast }(X)$
is finitely presented. By Fujiki–Liebermann’s theorem [Reference Brion5, Corollary 2.11], the kernel
$\mathrm {Ker} (\overline {\rho })$
is finite, and so the first claim follows from [Reference Johnson21, Corollary 10.2].
The second claim follows from Theorem 5.8 below.
Theorem 5.8 [Reference Dinh, Gachet, Lin, Oguiso, Wang and Yu10, Theorem 1.6].
Let V be a smooth complex projective variety. Assume that
$\mathrm {Nef}^{+}(V)$
contains a rational polyhedral cone
$\Pi $
such that
$$ \begin{align*}\mathrm{Amp}(V)\subset \mathrm{Aut}(V)\cdot\Pi.\end{align*} $$
Then V has at most finitely many mutually nonisomorphic real forms.
Acknowledgments
We thank Professors Serge Cantat, Tien-Cuong Dinh, Ching-Jui Lai, Vladimir Lazić, Keiji Oguiso, Burt Totaro, Hokuto Uehara and Claire Voisin for their questions, comments, suggestions and encouragement. We are also grateful to the referee for careful reading and useful comments. The first author would like to thank JSPS Summer Program for providing the opportunity to visit the third author in Tokyo, where this paper was written. The third author would like to thank Department of Mathematics at National University of Singapore, Professor De-Qi Zhang and Doctor Jia Jia for warm hospitality.
Competing interests
The authors have no competing interest to declare.
Financial support
The first author is supported by the ERC Advanced Grant SYZYGY. This project has received funding from the European Research Council (ERC) under the EU Horizon 2020 program (grant agreement No. 834172). The second author is supported by the Ministry of Education Yushan Young Scholar Fellowship (NTU-110VV006) and the National Science and Technology Council (110-2628-M-002-006-, 111-2123-M-002-012-). The third author is supported by JSPS KAKENHI Grant (21J10242), Postdoctoral Fellowship Program of CPSF (GZC20230535) and National Key Research and Development Program of China (#2023YFA1010600).
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