The goal of this paper is to explore the cohomological consequences of the existence of moduli spaces for flat bundles with bounded rank and irregularity at infinity and to prove them unconditionally in the surface case.

Let $U$ be a smooth complex projective variety. From Simpson's work [Reference SimpsonSim94], flat bundles on $U$ with given rank form a complex variety, the de Rham space of $U$. If $U$ is quasi-projective, flat bundles acquire singularities at infinity making the situation more involved. However, regular singular flat bundles again give rise to moduli spaces at the cost of rigidifying the situation by a choice of logarithmic lattice [Reference NitsureNit93]. Let $X$ be a smooth compactification of $U$ such that $D:=X\setminus U$ has simple normal crossings and let $j : U\to X$ be the inclusion. A flat bundle $\mathcal {E}=(E,\nabla )$ on $U$ is regular singular [Reference DeligneDel70] if $E$ extends to a vector bundle $F$ on $X$ with $\nabla (F)\subset F\otimes _{\mathcal {O}_X}\Omega ^1_X(\log D)$. Regular singularity is independent of a choice of compactification. When an extra condition on the residues is imposed, $F$ is called a Deligne lattice. A natural question to ask is what lies beyond the regular singular case. In general, the connection $\nabla$ no longer has simple poles at infinity, but Sabbah observed in the 1990s [Reference SabbahSab00] that $\nabla$ still has a simple expression, the so-called good formal decomposition at the cost of working formally along $D$ away from a codimension-$1$ subset of $D$. The locus of $D$ where this simple expression does not hold is the turning locus of $\mathcal {E}$. Away from the turning locus, Deligne's lattices admit straightforward generalizations and Malgrange proved in [Reference MalgrangeMal96] that they extend canonically to $X$. Armed with Deligne–Malgrange lattices and conjectural bounds for their Chern classes, it is an expectation of Esnault and Langer dating back from 2014 that there should exist a moduli of finite type for flat bundles on $U$ with bounded rank and irregularity at infinity. See [Reference EsnaultEsn14] and [Reference KedlayaKed21, 4.3.4]. Bounded irregularity means very roughly that we bound the poles of the connection restricted to a Deligne–Malgrange lattice. See Definition 2.1 for a rigorous definition. The bound on irregularity is thus embodied by an effective divisor $R$ supported on $D$. Let us denote by $\mathcal {M}_r(X,D,R)$ Esnault and Langer's expected moduli of rank-$r$ flat bundles on $U$ with irregularity bounded by $R$ along $D$. As pointed out to the authors by Langer, there is a mismatch between Nitsure's construction and $\mathcal {M}_r(X,D,R)$ for $R=0$. Indeed, rigidifying the moduli problem with choices of logarithmic lattices makes Nitsure's construction depend on a choice of compactification for $X$. On the other hand, having regular singularity at infinity is independent of a choice of compactification. Furthermore, different points of Nitsure's construction may underlie the same flat bundle.

The existence of $\mathcal {M}_r(X,D,R)$ has deep consequences for flat bundles which can be stated and studied independently. The first consequence is cohomological. For a flat bundle $\mathcal {E}$ on $U$, let $\operatorname {DR}\mathcal {E}$ be the algebraic de Rham complex of $\mathcal {E}$. Following the construction of the jumping loci for character varieties, the following subsets

\[ \mathcal{V}^{j}:=\{\mathcal{E}\in \mathcal{M}_r(X,D,R) \text{ such that } \dim H^\ast(U, \operatorname{DR} \mathcal{E} )\geq j\} \]

should form a decreasing sequence of closed subsets of $\mathcal {M}_r(X,D,R)$ when $j$ increases. Since $\mathcal {M}_r(X,D,R)$ is expected to be of finite type, its underlying topological space should be noetherian, so the sequence of $\mathcal {V}^{j}$ should stabilize for $j$ big enough. On the other hand, de Rham cohomology is always finite dimensional. Hence, there should exist an integer $j_0$ such that $\mathcal {V}^{j_0}$ is empty. Put otherwise, there should exist a universal bound for the algebraic de Rham cohomology of rank $r$ flat bundles on $U$ with bounded irregularity at infinity. Furthermore, this bound should depend only on $X$, $D$, $R$ and $r$. See Conjecture 2.5 for a statement making precise the dependency in $r$ and $R$. One of the main results of this paper is an unconditional proof of the existence of this bound in the surface case (Theorem 6.6). To state it, we let $\operatorname {Div}(X,D)$ be the group of divisors of $X$ supported on $D$ and denote by $k$ a field of characteristic $0$. For every effective divisor $R$ of $X$ supported on $D$, for every integer $r\geq 0$, we let $\operatorname {MIC}_r(X,D,R)$ be the category of flat bundles on $U$ with rank smaller than $r$ and irregularity bounded by $R$ along $D$.

Theorem 0.1 Let $X$ be a smooth projective surface over $k$. Let $D$ be a normal crossing divisor of $X$. Then, there exists a quadratic polynomial $C : \operatorname {Div}(X,D)\oplus \mathbb {Z}\to \mathbb {Z}$ affine in the last variable such that for every effective divisor $R$ of $X$ supported on $D$, for every integer $r\geq 0$ and every object $\mathcal {E}$ of $\operatorname {MIC}_r(X,D,R)$, we have

\[ \dim H^{*}(U,\operatorname{DR} \mathcal{E} )\leq C(R,r). \]

A second consequence of the existence of $\mathcal {M}_r(X,D,R)$ is the Lefschetz recognition principle. It relies on the heuristic principle that the formation of $\mathcal {M}_r(X,D,R)$ should come packaged with functorialities. If $f : Y\to X$ is a morphism of smooth projective varieties over $k$ such that $f^*R$ makes sense and such that $f^{-1}(D)$ is a normal crossing divisor, one expects the existence of a morphism of varieties $\mathcal {M}_r(X,D,R)\to \mathcal {M}_r(Y,f^{-1}(D),f^*R)$ induced by the pull-back along $f: Y\to X$. Functorialities would make the study of $\mathcal {M}_r(X,D,R)$ tractable through the curve case, where they should be much easier to construct since flat bundles on curves have no turning points. If $\mathcal {E}_0$ and $\mathcal {E}_1$ are non-isomorphic flat bundles on $U$, one can find a hyperplane $H$ such that the restrictions of $\mathcal {E}_0$ and $\mathcal {E}_1$ to $X\cap H$ are again non-isomorphic. When viewed as points of $\mathcal {M}_r(X,D,R)$, this means geometrically that $\mathcal {E}_0$ and $\mathcal {E}_1$ do not lie in the same fibre of $\mathcal {M}_r(X,D,R)\to \mathcal {M}_r(X\cap H,f_H^{-1}(D), f_H^*R)$ where $f_H : X\cap H \to X$ is the inclusion. One thus expects the induced morphism of schemes

\[ \mathcal{M}_r(X,D,R)\longrightarrow \prod_H\mathcal{M}_r(X\cap H,f_H^{-1}(D), f_H^*R) \]

to be a closed immersion. Since $\mathcal {M}_r(X,D,R)$ should be of finite type, this implies the existence of hyperplanes $H_1,\ldots, H_N$ depending only on $X,D,R$ and $r$ such that if we put $f: \bigsqcup X\cap H_i \to X$, then $\mathcal {M}_r(X,D,R)\to \mathcal {M}_r(Y,f^{-1}(D), f^*R)$ is a closed immersion. Put in the language of flat bundles, two objects of $\operatorname {MIC}_r(X,D,R)$ are isomorphic if and only their pull-back to $\bigsqcup X\cap H_i$ are isomorphic. This is the Lefschetz recognition principle. We are led to introduce the following.

Our second main result is a proof of the Lefschetz recognition principle for $\mathcal {C} = \operatorname {MIC}_r(X,D,R)$ in any dimension.

See Theorem 7.5 for explicit conditions defining the dense open set $\Omega (R,r)$.

Although the statements of cohomological boundedness and the Lefschetz recognition principle were derived separately from the existence of $\mathcal {M}_r(X,D,R)$, it turns out that for surfaces, the latter is a consequence of the former. Let us explain how. Let $j : U\to X$ be the open immersion and let $\mathcal {E}$ be a flat bundle on $U$. The $\mathcal {D}_X$-module $j_*\mathcal {E}$ is coherent and as such, a canonical $\dim U$-cycle $CC( j_*\mathcal {E})$ of the cotangent bundle $T^*X$ was attached to it by Kashiwara and Schapira [Reference Kashiwara and SchapiraKS90]. This is the characteristic cycle of $j_* \mathcal {E}$. The cycle $CC( j_*\mathcal {E})$ tells how far the $\mathcal {D}_X$-module $j_*\mathcal {E}$ is from being a flat connection on $X$ and thus provides a geometric measure of the complexity of the differential system underlying $j_*\mathcal {E}$. The cycle $CC( j_*\mathcal {E})$ is Lagrangian. If $X$ is a surface, $CC( j_*\mathcal {E})$ is thus a combination of the zero section of $T^*X$, the conormal bundles of the components of $D$ and the conormal bundles of some points of $D$. If $\mathcal {E}'$ is another flat bundle on $U$, then any smooth curve transverse to $D$ and avoiding the points whose conormal bundle contributes to $CC(j_*{\mathcal {H}}\! \operatorname {om}(\mathcal {E}_1,\mathcal {E}_2))$ where $\mathcal {E}_1,\mathcal {E}_2\in \{\mathcal {E},\mathcal {E}'\}$ distinguishes $\mathcal {E}$ and $\mathcal {E}'$ after restriction. See Corollary 7.4. This observation translates the statement of Theorem 0.3 into the problem of finding a universal bound for the number of points of $D$ whose conormal bundle contributes to $CC( j_*\mathcal {E})$ for $\mathcal {E}$ in $\operatorname {MIC}_r(X,D,R)$. This question is cohomological since by Kashiwara and Dubson's formula [Reference DubsonDub84], every such point contributes to de Rham cohomology. This is how Theorem 0.3 follows from Theorem 0.1 for surfaces.

The main result of [Reference TeyssierTey23] implies that the points of the smooth locus of $D$ whose conormal bundle contributes to $CC( j_*\mathcal {E})$ and $CC( j_*{\mathcal {E}} \operatorname {nd}\mathcal {E})$ are exactly the turning points of $\mathcal {E}$ along $D$. From the above discussion, cohomological boundedness thus yields a universal bound on the number of turning points of objects in $\operatorname {MIC}_r(X,D,R)$. We prove in any dimension a stronger universal bound (Theorem 6.9).

For a flat bundle $\mathcal {E}$, Kedlaya [Reference KedlayaKed11] and Mochizuki [Reference MochizukiMoc11] proved that the turning locus of $\mathcal {E}$ along $D$ can be eliminated by enough blow-up. Theorem 0.4 is thus consistent with Esnault and Langer's expectation that there should exist a universal bound on the number of blow-up needed to achieve good formal decomposition for objects in $\operatorname {MIC}_r(X,D,R)$. Their insight is that controlling the number of blow-ups should give the required bound on the Chern classes of Deligne–Malgrange lattices to construct the moduli $\mathcal {M}_r(X,D,R)$. By purity of turning loci proved by André for $D$ smooth [Reference AndréAnd07] and by Kedlaya in general [Reference KedlayaKed21], we deduce the following (Theorem 6.10).

We summarize the interplay (at least for surfaces) of the above results and heuristic in the following diagram, to be understood with bounded rank and irregularity.

We finally introduce the main tool of this paper and describe an extra application. Let $X$ be a smooth connected variety over $k$. Let $D$ be a normal crossing divisor of $X$. Let $\mathcal {E}$ be a flat bundle on $j :U:=X\setminus D\to X$. For every proper birational morphism $p : Y\to X$ where $Y$ is smooth and $E:=p^{-1}(D)$ is a normal crossing divisor, Kedlaya attached to $\mathcal {E}$ an effective divisor $\operatorname {Irr}(Y,p^+\mathcal {E})$ supported on $E$ whose coefficient along a component $Z$ is the generic irregularity number of $\mathcal {E}$ along $Z$ or, equivalently, the coefficient of $T^*_ZY$ in $CC(\jmath _{*}\mathcal {E})$, where $\jmath : U\to Y$ is the inclusion. This collection of divisors organizes into an element $\operatorname {Irr} \mathcal {E}$ of the group $\textbf {Div}(X):=\varprojlim _{Y\rightarrow X} \operatorname {Div}(Y)$ of $b$-divisors, where $p : Y\to X$ runs over the poset of morphisms as above and where the transition maps are push-forward. In particular, an element of $\textbf {Div}(X)$ is a $\mathbb {Z}$-valued function on the set of divisorial valuations on $X$. The $b$-divisor $\operatorname {Irr}\mathcal {E}$ is the irregularity $b$-divisor of $\mathcal{E}$ along $D$. In the group $\textbf {Div}(X)$ lies the subgroup of Cartier $b$-divisor defined as $\varinjlim _{Y\rightarrow X}\operatorname {Div}(Y)$ where the transition maps are pull-backs. Fundamental to this paper is Kedlaya's theorem [Reference KedlayaKed21] that $\operatorname {Irr}\mathcal {E}$ is a nef Cartier $b$-divisor with the property that $\operatorname {Irr}\mathcal {E}$ and $\operatorname {Irr}{\mathcal {E}}\!\operatorname{nd}\mathcal {E}$ lie in the subgroup $\operatorname {Div}(X)$ of $\textbf {Div}(X)$ if and only if $\mathcal {E}$ has good formal decomposition along $D$. We show in § 5.3 that $\operatorname {Irr}\mathcal {E}$ and $\operatorname {rk} \mathcal {E}$ determine $CC(j_*\mathcal {E})$, and it is an intriguing question to understand how $CC(j_*\mathcal {E})$ relates to $\operatorname {Irr}\mathcal {E}$. If $X$ is a surface, we answer this question using a new operation on $b$-divisors, the partial discrepancy $\delta : \textbf {Div}(X)\to \textbf {Div}(X)$. The partial discrepancy measures the failure of a $b$-divisor $Z$ to lie in the image of $\operatorname {Div}(X)\to \textbf {Div}(X)$ in the sense that if it does, then $\delta Z=0$. We prove that if $Z$ is a nef Cartier $b$-divisor, then $\delta Z$ is an effective $b$-divisor with finite support when viewed as a function on the set of divisorial valuations on $X$. Hence, the sum of its values, denoted by $\int _X \delta Z$ is a well-defined positive integer. For a subset $A$ of $X$, we denote by $\int _A \delta Z$ the sum of the values of $\delta Z$ over the set of divisorial valuations whose centres on $X$ lie in $A$. Using the partial discrepancy, we prove the following formula for the characteristic cycle of flat bundles on surfaces (Theorem 5.18).

Theorem 0.6 Let $X$ be a smooth projective surface over an algebraically closed field of characteristic $0$. Let $D$ be a normal crossing divisor of $X$. Let $\mathcal {E}$ be a flat bundle on $j :U:=X\setminus D\to X$. Then

\[ CC(j_*\mathcal{E}) = \operatorname{rk} \mathcal{E} \cdot CC(\mathcal{O}_{X}(\ast D)) + LC(\operatorname{Irr}(X,\mathcal{E})) + \sum_{P\in D} \bigg( \int_{P} \delta\operatorname{Irr}\mathcal{E}\bigg) \cdot T_{P}^*X. \]

In the above formula, $LC(\operatorname {Irr}(X,\mathcal {E}))$ is a Lagrangian cycle depending only on the divisor $\operatorname {Irr}(X,\mathcal {E})$ supported on $D$ attached to $\mathcal {E}$. In particular, it depends on $\mathcal {E}$ only via generic data along the components of $D$. A nice feature of the above formula is to make explicit the means by which the lack of good formal decomposition reflects in the characteristic cycle. By definition of the partial discrepancy, it shows, in particular, that the characteristic cycle is only sensitive to the turning points lying in the smooth locus of the successive inverse images of $D$ by the successive blow-up needed to achieve good formal decomposition. From the Kashiwara and Dubson formula [Reference DubsonDub84], we deduce the following Grothendieck–Ogg Shafarevich-type formula for flat bundles on surfaces (Theorem 5.20).

Theorem 0.7 Let $X$ be a smooth projective surface over $\mathbb {C}$. Let $D$ be a normal crossing divisor of $X$. Let $\mathcal {E}$ be a flat bundle on $j :U:=X\setminus D\to X$. Then

\[ \chi(U,\operatorname{DR}\mathcal{E}) = \operatorname{rk} \mathcal{E} \cdot \chi(U(\mathbb{C})) + (LC(\operatorname{Irr}(X,\mathcal{E})),T^*_X X)_{T^* X} + \int_D \delta\operatorname{Irr}\mathcal{E}. \]

We now give a linear overview of the paper. Section 1 gathers some general material on $\mathcal {D}$-modules as well as Kedlaya's results on irregularity $b$-divisors. Section 2 introduces the cohomological boundedness conjecture for flat bundles with bounded rank and irregularity at infinity. The main upshot of § 2 is Corollary 2.17 stating that cohomological boundedness is equivalent to an a priori weaker conjecture, the $\chi$-boundedness conjecture asking for a universal bound on the global Euler–Poincaré characteristic of the de Rham cohomology. Section 3 is the main technical core of this paper. In a relative situation, it provides a mechanism for deducing $\chi$-boundedness out of $\chi$-boundedness for the generic fibre provided the turning locus is contained in a fibre. See Proposition 3.23. Section 4 is devoted to the construction of the partial discrepancy $b$-divisor attached to a $b$-divisor on a smooth surface. Its upshot is Proposition 4.15 ensuring that the partial discrepancy of a nef Cartier $b$-divisor is a $b$-divisor with finite support when viewed as a function on the set of divisorial valuations. Section 5 is an application of § 4 to the proof of Theorems 0.6 and 0.7. Section 6 gives the proof of cohomological boundedness for surfaces. Section 7 provides the proof of the Lefschetz recognition principle. In § 8, the techniques of this paper are used to obtain a Lefschetz theorem for the differential Galois group of flat bundles under some uncountability assumption of the base field.

1. Geometric and $\mathcal {D}$-module preparations

1.3 Transversality

We recall the transversality conditions from [Reference BeilinsonBei16].

Definition 1.1 Let $f:Y\to X$ be a morphism between smooth varieties over $k$. Let $C$ be a closed conical subset of $T^*X$. Let $\overline y$ be a geometric point above a point $y$ of $Y$. We say that $f:Y\to X$ is $C$-transversal at $y$ if

\[ \operatorname{Ker} df_{\overline y}\cap C_{f(\overline y)}\subset \{0\}\subset T^*_{f(\overline y)}X, \]

where $df_{\overline y}: T^*_{f(\overline y)}X \to T^*_{\overline y}Y$ is the cotangent map of $f$ at $\overline y$. We say that $f:Y\to X$ is $C$-transversal if it is $C$-transversal at every point of $Y$.

If $f:Y\to X$ is $C$-transversal, let $f^\circ C$ be the scheme-theoretic image of $Y\times _XC$ in $T^*Y$ by the canonical map $df: Y\times _XT^*X\to T^*Y$. As proved in [Reference BeilinsonBei16, Lemma 1.2], the map $df:Y\times _XC\to f^\circ C$ is finite and $f^\circ C$ is a closed conical subset of $T^*Y$.

In the next transversality criterion, we use an extra operation on closed conical subsets of cotangent bundles. Let $f:X\to Y$ be a proper morphism between smooth varieties over $k$. Let $C$ be a closed conical subset of $T^*X$. We denote by $f_{\circ }C$ the closed conical subset of $T^*Y$ defined as the image of $df^{-1}(C)\subset X \times _Y T^*Y$ by the projection $X \times _Y T^*Y\to T^*Y$, where $df : X\times _Y T^*Y\to T^*X$ is the canonical map.

Lemma 1.2 Let

(1.3)

be a cartesian diagram of smooth varieties over $k$ whose arrows are closed immersions. Let $C$ be a closed conical subset of $T^*X$. Let $x\in Y$ and let us abuse notation by viewing $x$ as a point in $X$, $H$ and $P$. Then, the following statements hold.

1. (i) If $g$ is $i_{\circ }C$-transversal at $x$, then $f$ is $C$-transversal at $x$.

2. (ii) Assume that $X$ and $H$ are transverse at $x$. If $f$ is $C$-transversal at $x$, then $g$ is $i_{\circ }C$-transversal at $x$.

Proof. Assume that $f$ is not $C$-transversal at $x$. This means that there exists a non-zero form $\omega$ in $C_x$ vanishing on $T_xY \subset T_xX$. Since (1.3) is cartesian, so is the following induced diagram.

Via the choice of a supplementary for $T_x Y$ in $T_xH$ and a supplementary for $T_xX + T_xH$ in $T_xP$, we can extend $\omega$ to a non-zero form $\eta$ on $T_xP$ vanishing on $T_xH$. In particular, $\eta$ lies in $i_\circ C$ and $g$ is not $i_\circ C$-transversal at $x$ and part (i) is proven.

We now prove part (ii). Assume that $T_xX$ and $T_x H$ generate $T_xP$ and that $f$ is $C$-transversal at $x$. We argue by contradiction. Let $\eta$ be a non-zero form of $i_\circ C$ above $x$ such that $\eta$ vanishes on $T_x H$. If $i^*\eta =0$, the fact that $T_xX$ and $T_x H$ generate $T_xP$ yields $\eta =0$, which is not possible. Thus, $i^*\eta$ is a non-zero form of $C$ above $x$. By the transversality assumption, $i^*\eta$ does not vanish on $T_xY$, so $\eta$ does not vanish on $T_x H$. Contradiction. This concludes the proof of Lemma 1.2.

We gather in Lemma 1.4 standard facts on transversality. See [Reference BeilinsonBei16, 1.2,2.2] and [Reference SaitoSai17, 3.4].

Lemma 1.4 Let $f : Y \to X$ be a morphism of smooth varieties over $k$. Let $C,C' \subset T^*X$ be closed conical subsets. Let $x\in X$. Then the following hold.

1. (i) Assume $C_x\subset C'_x$. If $f$ is $C'$-transversal at $x$, then $f$ is $C$-transversal at $x$.

2. (ii) If $f$ is $C'$-transversal at $x$ and $C$-transversal at $x$, then $f$ is $C\cup C'$-transversal at $x$.

3. (iii) Let $g : Z \to Y$ be a morphism of smooth varieties over $k$. The following conditions are equivalent:

   1. (a) $f$ is $C$-transversal on a neighbourhood of $g(Z)$ and $g$ is $f^{\circ }(C)$-transversal;

   2. (b) $f\circ g$ is $C$-transversal.

4. (iv) If $C'' \subset T^*Y$ is a closed conical subset, the set of points $y\in Y$ at which $f$ is $C''$-transversal is open in $Y$.

1.4 Universal hyperplane

Let $E$ be a finite-dimensional vector space over $k$. Let $E^{\ast }$ be its dual. Let $\mathbb {P}=\mathrm {Proj}(\operatorname {Sym} E^{\ast })$ and $\mathbb {P}^{\vee }=\mathrm {Proj}(\operatorname {Sym} E)$ be the associated projective spaces. The closed subscheme of $\mathbb {P}\times _k\mathbb {P}^{\vee }$ defined by

\begin{align*} Q:=\{(x,H)\in \mathbb{P}\times_k\mathbb{P}^{\vee} \;|\; x\in H\} \end{align*}

is the universal family of hyperplanes of $\mathbb {P}$. Let $(X,D)$ be a quasi-projective normal crossing pair over $k$. Let $i:X\to \mathbb P$ be an immersion. Put $X_Q=X\times _{\mathbb {P}}Q$. Denote by $p^{\vee }_X:X_Q\to \mathbb {P}^{\vee }$ the composition of $X_Q\to Q$ with the canonical projection $p^{\vee }:Q\to \mathbb {P}^{\vee }$. Then, Bertini's theorem ensures the existence of a dense open subset $V\subset \mathbb {P}^{\vee }$ of hyperplanes $H$ transverse to $X$ such that $D\cap H$ has normal crossings. In particular, if $\eta$ denotes the generic point of $\mathbb {P}^{\vee }$, we have the following commutative diagram with cartesian squares:

where $(X_\eta,D_\eta )$ is a quasi-projective pair over $\eta$ of dimension $\dim X-1$ and where $D_\eta$ has normal crossings. The pair $(X_\eta,D_\eta )$ is the generic hyperplane section of $(X,D)$.

The following lemma provides a slight generalization of [Reference SaitoSai21, 1.3.7 (2)].

Lemma 1.5 Let $\mathbb {P}$ be a projective space over $k$. Let $X$ be a smooth subvariety of $\mathbb {P}$. Let $C\subset T^*X$ be a closed conical subset of pure dimension $\dim X$. Then there exists a dense open subset of hyperplanes $H$ transverse to $X$ such that $X\cap H \to X$ is $C$-transversal.

Proof. By Bertini's theorem, there exists a dense open subset $V_1\subset \mathbb {P}^{\vee }$ of hyperplanes transverse to $X$. Observe that $i_\circ C$ has pure dimension $\dim \mathbb {P}$. Thus, [Reference SaitoSai21, 1.3.7 (2)] ensures that there exists a dense open subset $V_2 \subset \mathbb {P}^{\vee }$ of hyperplanes $H$ such that $H\to \mathbb {P}$ is $i_\circ C$-transversal. From Lemma 1.2(i), $V_1\cap V_2$ thus meets our requirements.

1.5 Characteristic cycle for coherent $\mathcal {D}$-modules

Let $k$ be a field of characteristic $0$. Let $X$ be a smooth variety over $k$. We endow $\mathcal {D}_X$ with its filtration $F$ by the order of differential operators. In particular, there is a canonical isomorphism $\operatorname {gr}^F \mathcal {D}_X\simeq \pi _*\mathcal {O}_{T^*X}$ where $\pi : T^*X \to X$ is the canonical projection. Every coherent $\mathcal {D}_X$-module $\mathcal {M}$ admits a filtration $F_{\mathcal {M}}$ compatible with $F$ such that $\operatorname {gr}^{F_{\mathcal {M}}}\mathcal {M}$ is a coherent $\pi _*\mathcal {O}_{T^*X}$-module. Then,

\[ \widetilde{\operatorname{gr}^{F_{\mathcal{M}}}}\mathcal{M}:=\mathcal{O}_{T^*X} \otimes_{\pi^{-1}\pi_*\mathcal{O}_{T^*X}}\operatorname{gr}^{F_{\mathcal{M}}}\mathcal{M} \]

is a coherent $\mathcal {O}_{T^*X}$-module. Its characteristic cycle is a cycle of $T^*X$ which does not depend on a choice of filtration as above. It is the characteristic cycle of $\mathcal {M}$. We denote it by $CC(\mathcal {M})$. The support of $CC(\mathcal {M})$ is a closed conical subset of $T^*X$ called the singular support of $\mathcal {M}$. We denote it by $SS(\mathcal {M})$.

Remark 1.6 The formation of the characteristic cycle commutes with flat base change. In particular, it commutes with base field extension and analytification.

As a consequence of [Reference GabberGab81], we have the following.

Theorem 1.7 For any coherent ${D}_X$-module $\mathcal {M}$, the irreducible components of $SS(\mathcal {M})$ have dimension $\geq \dim X$.

Definition 1.8 We say that a coherent ${D}_X$-module $\mathcal {M}$ is holonomic if $SS(\mathcal {M})$ has pure dimension $\dim X$.

Holonomy is the correct finiteness condition for ${D}_X$-modules, as will be clear from Proposition 1.10 below.

1.6 De Rham cohomology

Let $k$ be a field of characteristic $0$. Let $X$ be a smooth algebraic $k$-variety of dimension $d$. Let $\mathcal {M}$ be a $\mathcal {D}_X$-module. We denote by

\[ \operatorname{DR} \mathcal{M} : \mathcal{M}\longrightarrow \Omega^1_X \otimes_{\mathcal{O}_X}\mathcal{M} \longrightarrow \cdots \longrightarrow \Omega^d_X \otimes_{\mathcal{O}_X}\mathcal{M} \]

the algebraic de Rham complex of $\mathcal {M}$, where $\mathcal {M}$ lies in degree $0$. The algebraic de Rham cohomology of $\mathcal {M}$ is the cohomology of $\operatorname {DR} \mathcal {M}$.

If $k=\mathbb {C}$, let $\mathcal {M}^{\operatorname {an}}$ be the analytification of $\mathcal {M}$ and define similarly the analytic de Rham complex of $\mathcal {M}$. The analytic de Rham cohomology of $\mathcal {M}$ is the cohomology of $\operatorname {DR} \mathcal {M}^{\operatorname {an}}$.

In the proper complex setting, the GAGA theorem for quasi-coherent cohomology identifies algebraic and analytic de Rham cohomology. See [Reference DeligneDel70, 6.6.1]. This is the following.

Proposition 1.9 Let $X$ be a smooth projective variety over $\mathbb {C}$. Let $\mathcal {M}$ be a quasi-coherent $\mathcal {D}_X$-module. Then, the canonical comparison morphism

\begin{align*} H^{\ast}(X,\operatorname{DR} \mathcal{M})\longrightarrow H^{\ast}(X(\mathbb{C}),\operatorname{DR} \mathcal{M}^{\operatorname{an}}) \end{align*}

is an isomorphism.

When holonomy is imposed, algebraic de Rham cohomology is finite dimensional even when the ambient variety is not proper.

Proposition 1.10 Let $X$ be a smooth variety over $k$. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Then, for every integer $n$, the space $H^{n}(X,\operatorname {DR} \mathcal {M})$ is finite dimensional over $k$ and vanishes if $n\neq 0,\ldots, 2\dim X$.

Proof. Since the formation of the de Rham complex commutes with push-forward [Reference Hotta, Takeuchi and TanisakiHTT00, 4.2.5] and since holonomy is preserved under push-forward [Reference Hotta, Takeuchi and TanisakiHTT00, 3.2.3], we can at the cost of replacing $X$ by a smooth compactification suppose that $X$ is proper over $k$. From Remark 1.11, we reduce to the case where $k=\mathbb {C}$. From Proposition 1.9, we are left to prove a variant of Proposition 1.10 where now $X$ is a smooth compact complex manifold and where $\mathcal {M}$ is a holonomic $\mathcal {D}_X$-module. From Kashiwara's perversity theorem [Reference KashiwaraKas75], we are left to prove that for every perverse complex $\mathcal {F}$ on $X$, the $\mathbb {C}$-vector space $H^{n}(X,\mathcal {F})$ is finite dimensional and vanishes if $n\neq -\dim X,\ldots,\dim X$. This is a standard fact from the theory of perverse complexes [Reference Beilinson, Bernstein, Deligne and GabberBBDG18, 4.2.4].

Remark 1.11 As a consequence of the invariance of quasi-coherent cohomology under flat base change, de Rham cohomology is invariant under base field extensions.

1.7 The solution and the irregularity complexes

Let $X$ be a complex manifold of dimension $d$. Let $\mathcal {M}$ be a $\mathcal {D}_X$-module. The solution complex of $\mathcal {M}$ is defined as

\[ \operatorname{Sol} \mathcal{M} := \operatorname{R}{\mathcal{H}}\! \operatorname{om}_{\mathcal{D}_X}(\mathcal{M},\mathcal{O}_X). \]

The following theorem is due to Kashiwara [Reference KashiwaraKas75].

Theorem 1.12 Let $X$ be a smooth complex manifold. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Then the complexes $\operatorname {Sol} \mathcal {M}[\dim X]$ and $\operatorname {DR} \mathcal {M}[\dim X]$ are perverse complexes.

Let $Z$ be a hypersurface in $X$. Let $i: Z\to X$ be the inclusion. Let $\mathcal {M}(\ast Z)$ be the localization of $\mathcal {M}$ along $Z$. Define the irregularity complex of $\mathcal {M}$ along $Z$ by

\[ \operatorname{Irr}_Z^*\mathcal{M} := i_* i^*\operatorname{Sol} \mathcal{M}(\ast Z) \]

The following theorem is due to Mebkhout [Reference MebkhoutMeb90].

Theorem 1.13 Let $X$ be a complex manifold. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Let $Z$ be a hypersurface in $X$. Then $\operatorname {Irr}_Z^*\mathcal {M}[\dim X]$ is a perverse complex supported on $Z$.

The de Rham and solution complexes are related under sheaf duality; see [Reference MebkhoutMeb82].

Theorem 1.14 Let $X$ be a complex manifold. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. There is a canonical isomorphism of complexes

\[ \operatorname{DR} \mathcal{M} \longrightarrow \operatorname{R}\!{\mathcal{H}}\!\operatorname{om}_{\mathbb{C}}(\operatorname{Sol} \mathcal{M}, \mathbb{C}) \]

in the derived category of complexes of sheaves.

1.8 Characteristic cycle and the de Rham and solution complexes

For a complex of sheaves $\mathcal {F}$ with bounded and constructible cohomology, Kashiwara and Schapira defined the characteristic cycle $CC(\mathcal {F})$ by means of microlocal analysis [Reference Kashiwara and SchapiraKS90]. The following theorem identifies the characteristic cycle of a holonomic $\mathcal {D}_X$-module with that of its solution complex. See [Reference Kashiwara and SchapiraKS90, 11.3.3] and [Reference DubsonDub84, Theorem 4].

Theorem 1.15 Let $X$ be a complex manifold. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Then

\[ SS(\mathcal{M}) = SS(\operatorname{Sol} \mathcal{M}) \quad \text{and} \quad CC(\mathcal{M}) = CC(\operatorname{Sol} \mathcal{M}). \]

In particular, the computation of $CC(\mathcal {M})$ can be reduced to a sheaf-theoretic question. If the singular support of $\mathcal {M}$ is already known, the following theorem [Reference KashiwaraKas75, Theorem 3.5] tells us how to chop off $X$ in order to compute $CC(\operatorname {Sol} \mathcal {M})$.

Theorem 1.16 Let $X$ be a complex manifold. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Let $X_1,\ldots,X_n$ be a Whitney stratification of $X$ such that $SS(\mathcal {M})$ lies in $\bigcup _{i=1}^n T^*_{X_i}X$. Then, the cohomology sheaves of $(\operatorname {Sol} \mathcal {M})|_{X_i}$ are local systems on $X_i$, $i=1,\ldots,n$.

From a cohomological perspective, computing the characteristic cycle is useful because of Kashiwara and Dubson's formula [Reference DubsonDub84]. See also [Reference LaumonLau83].

Theorem 1.17 Let $X$ be a proper complex manifold. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Then, we have

\[ \chi(X,\operatorname{DR}\mathcal{M})=(CC(\mathcal{M}),T^*_XX)_{T^*X}. \]

where $\chi (X,\operatorname {DR}\mathcal {M})$ denotes the Euler–Poincaré characteristic of $\operatorname {DR}\mathcal {M}$ and where $(-,-)_{T^*X}$ denotes the intersection number of cycles in $T^*X$.

1.9 Characteristic cycle and functorialities

The characteristic cycle for $\mathcal {D}$-modules commutes with proper push-forward. This is the following.

Lemma 1.18 Let $f: X\to Y$ be a proper morphism between smooth varieties over $k$. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Then, $CC(f_+\mathcal {M})=f_*CC(\mathcal {M})$.

Proof. From Remark 1.6, we can suppose $k=\mathbb {C}$. We have the following chain of equalities:

\begin{align*} CC(f_+\mathcal{M})&= CC(f_+^{\operatorname{an}}\mathcal{M}^{\operatorname{an}}) = CC(\operatorname{Sol}(f_+^{\operatorname{an}}\mathcal{M}^{\operatorname{an}}))\\ & = CC(Rf_*\operatorname{Sol}\mathcal{M}^{\operatorname{an}}) =f_*CC(\operatorname{Sol}\mathcal{M}^{\operatorname{an}}) =f_*CC(\mathcal{M}). \end{align*}

The first equality follows from Remark 1.6 applied to $f_+\mathcal {M}$. The second equality follows from Theorem 1.15. The third equality follows from the compatibility of the formation of $\operatorname {Sol}$ with proper push-forward. The fourth equality follows from [Reference Kashiwara and SchapiraKS90, 9.4.2]. The last equality follows from the above arguments applied to $\mathcal {M}$.

Definition 1.19 Let $f: Y\to X$ be a morphism between smooth varieties over $k$ or between complex manifolds. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. We say that $f: Y\to X$ is non-characteristic for $\mathcal {M}$ if it is $SS(\mathcal {M})$-transversal.

The following results are due to Kashiwara [Reference KashiwaraKas95].

Theorem 1.20 Let $f: Y\to X$ be a morphism between smooth varieties over $k$. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module such that $f: Y\to X$ is non-characteristic for $\mathcal {M}$. Then the following statements hold.

1. (i) The $\mathcal {D}_Y$-module pull-back $f^+\mathcal {M}$ is concentrated in degree $0$. That is, $f^+\mathcal {M}\simeq f^* \mathcal {M}$.

2. (ii) We have $SS(f^+\mathcal {M} )=f^\circ SS(\mathcal {M})$.

3. (iii) If $k=\mathbb {C}$, the comparison morphisms

   \[ f^* \operatorname{Sol} \mathcal{M}^{\operatorname{an}} \longrightarrow \operatorname{Sol} f^+\mathcal{M}^{\operatorname{an}} \quad \text{and} \quad \operatorname{DR} f^+\mathcal{M}^{\operatorname{an}} \longrightarrow f^* \operatorname{DR}\mathcal{M}^{\operatorname{an}} \]
   are isomorphisms in the derived category of sheaves.

1.10 Meromorphic flat connections

Let $X$ be a smooth algebraic variety over $k$. A flat connection on $X$ or module with integrable connection on $X$ is a $\mathcal {D}_X$-module $\mathcal {E}:=(E,\nabla )$ whose underlying $\mathcal {O}_X$-module $E$ is a vector bundle of finite rank on $X$. We denote by $\operatorname {MIC}(X)$ the category of flat connections on $X$.

If $D$ is a divisor in $X$, a meromorphic flat connection on $X$ with poles along $D$ is a $\mathcal {D}_X$-module $\mathcal {M}:=(M,\nabla )$ whose underlying $\mathcal {O}_X$-module $M$ is a locally free sheaf of $\mathcal {O}_X(\ast D)$-modules of finite rank. We denote by $\operatorname {MIC}(X,D)$ the category of meromorphic flat connections on $X$ with poles along $D$.

Remark 1.21 The above definitions make sense in the analytic setting, where the same notation will be used.

Remark 1.22 Meromorphic flat connections are holonomic $\mathcal {D}$-modules.

In the algebraic setting, there is no difference between flat connections and meromorphic flat connections. This is expressed by the following proposition [Reference Hotta, Takeuchi and TanisakiHTT00, 5.3.1].

Proposition 1.23 Let $(X,D)$ be a pair over $k$. Let $j: U=X\setminus D\to X$ be the inclusion. Then $(j_*, j^*)$ induce an equivalence of categories between $\operatorname {MIC}(U)$ and $\operatorname {MIC}(X,D)$.

Since all the action in this paper happens at infinity, we will use the meromorphic viewpoint. The next lemma says that this does not make any difference for cohomology.

Lemma 1.24 Let $(X,D)$ be a pair over $k$. Put $U:=X\setminus D$ and let $j: U\to X$ be the inclusion. Let $\mathcal {E}$ be an object in $\operatorname {MIC}(U)$. Then the canonical restriction morphism

\[ R\Gamma(X,\operatorname{DR} j_* \mathcal{E})\longrightarrow R\Gamma(U,\operatorname{DR} \mathcal{E}) \]

is an isomorphism in the derived category of vector spaces over $k$.

Proof. We have

\[ R\Gamma(U,\operatorname{DR} \mathcal{E}) \simeq R\Gamma(X,Rj_*\operatorname{DR} \mathcal{E}) \simeq R\Gamma(X,j_*\operatorname{DR} \mathcal{E}) \simeq R\Gamma(X,\operatorname{DR} j_*\mathcal{E}), \]

where the second equality comes from the fact that $j$ is an affine morphism, and where the last equality follows from $j_*\operatorname {DR} \mathcal {E}\simeq \operatorname {DR} j_*\mathcal {E}$.

Proposition 1.25 Let $(X,D)$ be a pair over $k$. Let $\mathcal {M}_1$ and $\mathcal {M}_2$ be objects of $\operatorname {MIC}(X,D)$. Then, there is a canonical isomorphism

\[ \operatorname{RHom}_{\mathcal{D}_X}(\mathcal{M}_1,\mathcal{M}_2)\longrightarrow R\Gamma(X,\operatorname{DR}{\mathcal{H}}\! \operatorname{om}(\mathcal{M}_1,\mathcal{M}_2)) \]

in the derived category of $k$-vector spaces.

Proof. Combining [Reference BorelBor87, VII 9.8] and [Reference BorelBor87, VI 5.3.2] yields a canonical isomorphism

\[ \operatorname{RHom}_{\mathcal{D}_X}(\mathcal{M}_1,\mathcal{M}_2)\simeq R\Gamma(X,\operatorname{DR}\mathcal{M}_1^{\vee}\otimes_{\mathcal{O}_X}^{L}\mathcal{M}_2), \]

where $\mathcal {M}_1^{\vee }$ denotes the $\mathcal {D}_X$-module dual to $\mathcal {M}_1$. Since $\mathcal {M}_2$ is a locally free $\mathcal {O}_X$-module localized along $D$, we have

\[ \mathcal{M}_1^{\vee}\otimes_{\mathcal{O}_X}^{L}\mathcal{M}_2\simeq \mathcal{M}_1^{\vee} \otimes_{\mathcal{O}_X} \mathcal{M}_2\simeq\mathcal{M}_1^{\vee}(\ast D)\otimes_{\mathcal{O}_X} \mathcal{M}_2. \]

Observe that $\mathcal {M}_1^{\vee }(\ast D)$ is an object of $\operatorname {MIC}(X,D)$. Let $\mathcal {M}_1^*$ be the meromorphic connection dual to $\mathcal {M}_1$. Then [Reference Hotta, Takeuchi and TanisakiHTT00, 2.6.10] ensures that the restriction of $\mathcal {M}_1^{\vee }(\ast D)$ and $\mathcal {M}_1^*$ to $U:=X\setminus D$ are canonically isomorphic. From Proposition 1.23, we deduce that $\mathcal {M}_1^{\vee }(\ast D)$ and $\mathcal {M}_1^*$ are canonically isomorphic. We thus conclude the proof of Proposition 1.25 using the identification $\mathcal {M}_1^{\ast }\otimes _{\mathcal {O}_X} \mathcal {M}_2\simeq {\mathcal {H}}\! \operatorname {om}(\mathcal {M}_1,\mathcal {M}_2)$.

1.11 Good formal structure for connections

In the next definition, we follow [Reference KedlayaKed21, Definition 2.1.3] and fix a field $k$ of characteristic $0$.

Definition 1.26 Let $(X,D)$ be a normal crossing pair over $k$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. We say that $\mathcal {M}$ has good formal structure at $x\in D$ if there is a decomposition

(1.27)\begin{equation} \mathcal{M}_x\otimes_{\mathcal{O}_{X,x}}\mathcal{S}\cong \bigoplus_{\alpha \in I} \mathcal{E}^{\varphi_\alpha}\otimes_{\mathcal{S}} \mathcal{R}_{\alpha}, \end{equation}

where

1. (1) the ring $\widehat {\mathcal {O}}_{X,x}(\ast D)$ is the localization along $D$ of the completion of ${\mathcal {O}}_{X,x}$ along the intersection of the irreducible components of $D$ containing $x$ and $\mathcal {S}$ is a finite étale $\widehat {\mathcal {O}}_{X,x}(\ast D)$-algebra;

2. (2) $I$ is a finite set;

3. (3) each $\mathcal {R}_{\alpha }$ is a regular differential module over $\mathcal {S}$;

4. (4) $\varphi _\alpha$ $(\alpha \in I)$ are elements in $\mathcal {S}$ satisfying that, if $\varphi _\alpha$ does not lie in the integral closure $\mathcal {S}_0$ of $\mathcal {O}_{X,x}$ in $\mathcal {S}$, then $\varphi _\alpha$ is a unit of $\mathcal {S}$ and $\varphi _\alpha ^{-1}\in \mathcal {S}_0$; furthermore, if $\varphi _\alpha -\varphi _\beta$ does not lie in $\mathcal {S}_0$, then $\varphi _\alpha -\varphi _\beta$ is a unit of $\mathcal {S}$ and $(\varphi _\alpha -\varphi _\beta )^{-1}\in \mathcal {S}_0$.

Definition 1.28 Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. We say that a point $x$ of $D$ is a turning point of $\mathcal {M}$ if $\mathcal {M}$ does not admit a good formal structure at $x$. The set of turning points of $\mathcal {M}$ is called the turning locus of $\mathcal {M}$. We denote it by $\operatorname {TL}(\mathcal {M})$. If $\operatorname {TL}(\mathcal {M})$ is empty, we say that $\mathcal {M}$ admits good formal structure along $D$.

Remark 1.29 The formation of the turning locus commutes with regular base change. See [Reference KedlayaKed21, 4.3.2] for a proof.

The compatibility of good formal structure and pull-back follows from Definition 1.26:

Proposition 1.30 Let $f: (Y,E)\to (X,D)$ be a morphism of normal crossing pairs over $k$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. Let $x$ be a point in $E$ such that $\mathcal {M}$ has good formal structure at $f(x)$. Then, $f^+\mathcal {M}$ has good formal structure at $x$. In particular, if $\mathcal {M}$ has good formal structure, so does $f^+\mathcal {M}$.

Purity of the turning locus was proved by André when $D$ is smooth [Reference AndréAnd07, 3.4.3] and by Kedlaya when $D$ has normal crossing [Reference KedlayaKed21, 2.3.1].

Theorem 1.31 Let $(X,D)$ be a normal crossing pair over $k$. Then, for any object $\mathcal {M}$ of $\operatorname {MIC}(X,D)$, the turning locus of $\mathcal {M}$ is a closed subscheme of $D$ of pure codimension $1$.

The following fundamental theorem was proved by Kedlaya [Reference KedlayaKed11, Theorem 8.1.3] and Mochizuki [Reference MochizukiMoc11, Theorem 19.5] with complement from [Reference KedlayaKed21, Corollary 3.2.5].

Theorem 1.32 Let $(X,D)$ be a pair over $k$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. Then, there exists a morphism of smooth algebraic varieties $f:Y\to X$ obtained as a composition of blow-up with smooth centres above $\operatorname {TL}(\mathcal {M})$ such that $f^{-1}(D)$ is a normal crossing divisor of $Y$ and $f^+\mathcal {M}$ admits good formal structure along $f^{-1}(D)$.

We thank the anonymous referee for pointing out the following.

Remark 1.33 In Theorem 1.32, it is not obvious from [Reference KedlayaKed11, Theorem 8.1.3] and [Reference MochizukiMoc11, Theorem 19.5] that $f:Y\to X$ can be taken as a composition of blow-up with smooth centres above $\operatorname {TL}(\mathcal {M})$. Such control on $f$ is guaranteed by the subsequent results from [Reference KedlayaKed21].

1.12 Irregularity number

Let $(X,D)$ be a normal crossing pair over $k$ of dimension $d+1\geq 1$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. In the setting of Definition 1.26, good formal structure always holds at the generic point $\eta$ of an irreducible component $Z$ of $D$. In that case, $\widehat {\mathcal {O}}_{X,\eta }(\ast D)$ has the form $k[\![ x_1,\ldots,x_d,y]\!] [y^{-1}]$ and $S$ has the form $K [\![ x_1,\ldots, x_d,y^{1/m}]\!] [(y^{1/m})^{-1}]$ where $m\geq 1$ and where $K/k$ is a finite extension.

Definition 1.34 For an object $\mathcal {M}$ in $\operatorname {MIC}(X,D)$, the generic slopes of $\mathcal {M}$ along $Z$ are the poles orders in the $y$-variable of the $\varphi _\alpha$ contributing to (1.27). Let us denote them by $\operatorname {ord}_y \varphi _\alpha$ where $\alpha \in I$ and put

\[ r(Z,\mathcal{M}):=\max_{\alpha \in I} \operatorname{ord}_y \varphi_\alpha. \]

The generic irregularity number of $\mathcal {M}$ along $Z$ is defined as

\[ \operatorname{irr}(Z,\mathcal{M}):=\sum_{\alpha \in I} (\operatorname{ord}_y \varphi_\alpha )\cdot \operatorname{rk} \mathcal{R}_\alpha. \]

Remark 1.35 In the setting of Definition 1.34, the generic irregularity number of $\mathcal {M}$ along $Z$ is an integer independent of the choice of the finite étale $\widehat {\mathcal {O}}_{X,\eta }(\ast D)$-algebra $S$.

In general, irregularity numbers and irregularity complexes are related through a theorem of Malgrange [Reference MalgrangeMal71, Theorem 1.4]. We record it in the following form useful for us.

Lemma 1.36 Let $(X,D)$ be an analytic pair. Let $Z$ be an irreducible component of $D$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. Then, the generic irregularity number of $\mathcal {M}$ along $Z$ is the generic rank along $Z$ of the perverse complex $\operatorname {Irr}^*_{D} \mathcal {M}$.

1.13 $b$-divisors

Let $X$ be a smooth variety over $k$. A modification of $X$ is the datum of a smooth variety $Y$ over $k$ and a map $Y\to X$ which is proper, dominant and an isomorphism away from a nowhere-dense closed subset of $X$. Following [Reference KedlayaKed21], we introduce the following definition.

Definition 1.37 The group of integral $b$-divisors $\textbf {Div}(X)$ of $X$ is the limit

\[ \varprojlim_{Y\rightarrow X} \operatorname{Div}(Y), \]

where $Y\to X$ runs over the category of modifications of $X$ and where the transition maps are push-forward.

For $D$ in $\textbf {Div}(X)$, we denote by $D(Y)$ the component of $D$ along a modification $Y\to X$. For every irreducible divisor $E$ of $Y$, we denote by $m(E,D)$ the multiplicity of $D(Y)$ along $E$.

Definition 1.38 The group of integral Cartier $b$-divisors $\textbf {CDiv}(X)$ on $X$ is the colimit

\[ \varinjlim_{Y\rightarrow X}\operatorname{CDiv}(Y), \]

where $Y\to X$ runs over the category of modifications of $X$ and where the transition maps are pull-back.

Let $\operatorname {ZR}^{\operatorname {divis}}(X)$ be the subset of the Zariski–Riemann space of $X$ consisting of divisorial valuations centred on $X$. As explained in [Reference KedlayaKed21], the group of integral $b$-divisors identifies with the set of functions $m : \operatorname {ZR}^{\operatorname {divis}}(X)\to \mathbb {Z}$ such that for every modification $Y\to X$, there is only a finite number of divisorial valuations $v\in \operatorname {ZR}^{\operatorname {divis}}(X)$ centred at an irreducible divisor of $Y$ such that $m(v)\neq 0$. Thus, the order on $\mathbb {Z}$ induces an order $\leq$ on $\textbf {Div}(X)$. Furthermore, there is a canonical injective map

\[ \textbf{CDiv}(X) \longrightarrow \textbf{Div}(X). \]

One of the main players of this paper is a Cartier $b$-divisor with an extra property called nef in [Reference KedlayaKed21]. In view of Lemmas 1.4.9 and 1.4.10 from [Reference KedlayaKed21], we can define nef Cartier $b$-divisor as follows.

Definition 1.39 Let $X$ be a smooth variety over $k$. Let $D$ be a Cartier $b$-divisor on $X$. We say that $D$ is nef if for every modification $Y\to X$, we have $D\leq D(Y)$ in $\textbf {Div}(X)$ where $D(Y)$ is viewed as a $b$-divisor via $\textbf {CDiv}(X) \to \textbf {Div}(X)$.

1.14 The irregularity $b$-divisor

Let $(X,D)$ be a pair over $k$. We recall the following definition from [Reference KedlayaKed21, 3.1.1].

Definition 1.40 Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. We denote by $\operatorname {Irr} \mathcal {M}$ the unique $b$-divisor on $X$ such that for every modification $p: Y\to X$, the multiplicity of $\operatorname {Irr} \mathcal {M}$ along an irreducible divisor $E$ of $Y$ is the generic irregularity $\operatorname {irr}(E,p^+\mathcal {M})$ of $p^{+}\mathcal {M}$ along $E$. We put

\[ \operatorname{Irr}(Y,p^+\mathcal{M}):=(\operatorname{Irr} \mathcal{M})(Y) \text{ in } \operatorname{Div}(Y). \]

The following lemma is obvious.

Lemma 1.41 Let $(X,D)$ be a pair over $k$. Let $k\subset K$ be a field extension. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. Then, $\operatorname {Irr}(X_K,\mathcal {M}_K)=\operatorname {Irr}(X,\mathcal {M})_K$.

Remark 1.42 In the setting of Definition 1.40, one can also define $R(\mathcal {M})$ the unique rational $b$-divisor on $X$ such that for every modification $p: Y\to X$, the multiplicity of $R(\mathcal {M})$ along an irreducible divisor $E$ of $Y$ is the highest generic slope $r(E,p^+\mathcal {M})\in \mathbb {Q}_{\geq 0}$ of $p^{+}\mathcal {M}$ along $E$, and put $R(Y,p^+\mathcal {M}):=(R(\mathcal {M}))(Y) \text { in } \operatorname {Div}(Y)_{\mathbb {Q}}$.

Remark 1.43 Equivalently, the $b$-divisors $R(\mathcal {M})$ and $\operatorname {Irr} \mathcal {M}$ will be viewed as $\mathbb {Q}$-valued functions on the set of divisorial valuations of $X$.

The following theorem is due to Kedlaya [Reference KedlayaKed21, 3.2.3].

Theorem 1.44 Let $(X,D)$ be a normal crossing pair over $k$. For any object $\mathcal {M}$ of $\operatorname {MIC}(X,D)$, the $b$-divisor $\operatorname {Irr} \mathcal {M}$ is a nef Cartier $b$-divisor. In particular, we have

\[ \operatorname{Irr} \mathcal{M} \leq \operatorname{Irr}(X,\mathcal{M}) \]

in $\textbf {Div}(X)$, where $\operatorname {Irr}(X,\mathcal {M})$ is viewed as a $b$-divisor via $\textbf {CDiv}(X) \to \textbf {Div}(X)$.

Corollary 1.45 Let $(X,D)$ be a normal crossing pair over $k$. Let $R$ be an effective divisor of $X$ supported on $D$. Let $\mathcal {M}$ be an object in $\operatorname {MIC}(X,D)$. Then the following conditions are equivalent:

1. (i) $\operatorname {Irr}(X,\mathcal {M})\leq R$ in $\operatorname {Div}(X)$;

2. (ii) for every point $0$ in $D$, for every locally closed smooth curve $C \to X$ in $X$ meeting $D$ at $0$ only, we have

   \[ \operatorname{irr}(0,\mathcal{M}|_{C})\leq (C,R)_0 \]
   where $(C,R)_0$ denotes the intersection number of $C$ with $R$ at $0$.

Proof. If condition (ii) holds, we get condition (i) by using generically enough smooth curves transverse to the irreducible components of $D$. We now show that condition (i) implies condition (ii). To do this, we can suppose that $R=\operatorname {Irr}(X,\mathcal {M})$. Let $0$ be a point in $D$ and let $C \to X$ be a locally closed smooth curve meeting $D$ at $0$ only. Let $p : Y\to X$ be a modification of $X$ which is an isomorphism above $X\setminus D$. By valuative criterion for properness, the immersion $C\to X$ factors uniquely through an immersion $C\to Y$ followed by $p$. In particular, we have

\[ \operatorname{irr}(0,\mathcal{M}|_{C})=\operatorname{irr}(0,(p^+\mathcal{M})|_{C}). \]

On the other hand, the projection formula yields

\[ (C,\operatorname{Irr}(X,\mathcal{M}))_0 = ( C,p^* \operatorname{Irr}(X,\mathcal{M}))_{0}. \]

From Theorem 1.44, we further have

\[ \operatorname{Irr}(Y,p^+\mathcal{M})\leq p^* \operatorname{Irr}(X,\mathcal{M}) \]

in $\operatorname {Div}(Y)$. Hence, the sought-after inequality for $X, \mathcal {M}, C\to X$ and $R=\operatorname {Irr}(X,\mathcal {M})$ follows from the analogous inequality for $Y, p^+\mathcal {M}, C\to Y$ and $R=\operatorname {Irr}(Y,p^+\mathcal {M})$. From Theorem 1.32, we are thus left to suppose that $\mathcal {M}$ has good formal structure and that $C$ is transverse to $D$ at $0$. In that case, the sought-after inequality is an equality.

Remark 1.46 For surfaces, Corollary 1.45 was proved by Sabbah [Reference SabbahSab00, 3.2.3].

The property of having good formal structure can be read from the irregularity $b$-divisor. This is due to Kedlaya [Reference KedlayaKed21, 3.1.2] as a consequence of Theorem 1.44.

Theorem 1.47 Let $(X,D)$ be a normal crossing pair over $k$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. Then, $\mathcal {M}$ has good formal structure if and only if

\[ \operatorname{Irr} \mathcal{M} = \operatorname{Irr}(X,\mathcal{M}) \quad \text{and} \quad \operatorname{Irr} \operatorname{End}\mathcal{M} = \operatorname{Irr}(X,\operatorname{End}\mathcal{M}) \]

in $\textbf {Div}(X)$.

The proof of Corollary 1.45 combined with Theorem 1.47 yields the following curve-like criterion for testing good formal structure.

Corollary 1.48 Let $(X,D)$ be a normal crossing pair over $k$. Then, an object $\mathcal {M}$ in $\operatorname {MIC}(X,D)$ has good formal structure if and only if for every point $0$ in $D$, for every locally closed smooth curve $C \to X$ in $X$ meeting $D$ at $0$ only, we have

\[ \operatorname{irr}(0,\mathcal{M}|_{C})= (C,\operatorname{Irr}(X,\mathcal{M}))_0 \quad \text{and} \quad \operatorname{irr}(0,(\operatorname{End}\mathcal{M})|_{C})= (C,\operatorname{Irr}(X,\operatorname{End}\mathcal{M}))_0. \]

Putting Corollaries 1.45 and 1.48 together yields the following differential analogue of a result [Reference HuHu19] of the first named author.

Corollary 1.49 Let $(X,D)$ be a pair over $k$ where $D$ is smooth and irreducible. Let $\mathcal {M}$ be an object in $\operatorname {MIC}(X,D)$. Then,

\[ \operatorname{irr}(D,\mathcal{M})=\sup_{(0,C)} \frac{\operatorname{irr}(0,\mathcal{M}|_{C})}{(C,D)_0} \]

where $0$ runs over the points of $D$ and where $C$ runs over all smooth locally closed curves of $X$ meeting $D$ at $0$ only.

2. Cohomological and $\chi$-boundedness conjectures

2.2 Bounded irregularity and ${\mathcal {H}}\! \operatorname {om}$

The ${\mathcal {H}}\! \operatorname {om}$ construction for connections preserves boundedness of irregularity in the following sense.

Proposition 2.2 Let $(X,D)$ be a normal crossing pair over $k$. Let $R$ be an effective divisor of $X$ supported on $D$. Let $r\geq 0$ be an integer. Let $\mathcal {M}_1$ and $\mathcal {M}_2$ be objects of $\operatorname {MIC}_r(X,D,R)$. Then, ${\mathcal {H}}\! \operatorname {om}(\mathcal {M}_1,\mathcal {M}_2)$ is an object of $\operatorname {MIC}_{r^2}(X,D,2r^2\cdot R)$.

Proof. We can suppose that $D$ is irreducible with generic point $\eta$. Let $(\varphi _\alpha,\mathcal {R}_\alpha ), \alpha \in I$ and $(\psi _\beta,\mathcal {S}_{\beta }), \beta \in J$ be the constituents of the good formal structures of $\mathcal {M}_1$ and $\mathcal {M}_2$ at $\eta$ as in (1.27). With the notation from Definition 1.34, we have

\[ \operatorname{irr}(D,{\mathcal{H}}\! \operatorname{om}(\mathcal{M}_1,\mathcal{M}_2))=\sum_{I\times J} (\operatorname{ord}_y \psi_\beta-\varphi_\alpha )\cdot (\operatorname{rk} \mathcal{R}_\alpha)\cdot (\operatorname{rk} \mathcal{S}_\beta). \]

Since $(\operatorname {ord}_y \psi _\beta -\varphi _\alpha )\leq (\operatorname {ord}_y \varphi _\alpha )+(\operatorname {ord}_y \psi _\beta )$, we deduce

\begin{align*} \operatorname{irr}(D,{\mathcal{H}}\! \operatorname{om}(\mathcal{M}_1,\mathcal{M}_2)) & \leq |J|\operatorname{rk} \mathcal{M}_2\sum_{I } (\operatorname{ord}_y \varphi_\alpha )\cdot (\operatorname{rk} \mathcal{R}_\alpha)+|I|\operatorname{rk} \mathcal{M}_1 \sum_{ J} (\operatorname{ord}_y \psi_\beta )\cdot (\operatorname{rk} \mathcal{S}_\beta)\\ & \leq (\operatorname{rk}\mathcal{M}_2)^2 \cdot \operatorname{irr}(D,\mathcal{M}_1) +(\operatorname{rk}\mathcal{M}_1)^2 \cdot \operatorname{irr}(D,\mathcal{M}_2). \end{align*}

The proof of Proposition 2.2 thus follows.

2.3 Bounded irregularity and pull-back

Bounded irregularity behaves well with respect to inverse image.

Proposition 2.3 Let $f: (Y,E)\to (X,D)$ be a morphism of normal crossing pairs over $k$. Let $R$ be an effective divisor of $X$ supported on $D$. Then, for every object $\mathcal {M}$ in $\operatorname {MIC}(X,D,R)$, the connection $f^+\mathcal {M}$ has irregularity bounded by $f^*R$.

Proof. By writing $f$ as a composition of a closed immersion followed by a smooth morphism, we are left to suppose that $f$ is either a closed immersion or a smooth morphism. In both cases, the sought-after bound is local around the maximal points of $E$, so we can suppose in each case that $E$ is smooth and irreducible.

Let us assume that $f$ is a smooth morphism. Then, the pull-back $E\to D$ is smooth. Since $E$ is smooth, so is the open set $f(E)$ of $D$. Thus, $f(E)$ lies in a unique irreducible component $T$ of $D$. Since $f$ is smooth, so is the pull-back $E\to T$. Hence, the generic point $\eta _E$ of $E$ lies above the generic point $\eta _T$ of $T$ and a uniformizer of $\mathcal {O}_{X,\eta _T}$ pulls back to a uniformizer of $\mathcal {O}_{Y,\eta _E}$. Thus, the irregularity numbers of $\mathcal {M}|_{\eta _T}$ and $(f^+\mathcal {M})|_{\eta _E}$ are equal. Hence, $\operatorname {Irr}(Y,f^+\mathcal {M})=f^*\operatorname {Irr}(X,\mathcal {M})$ and Proposition 2.3 is proved if $f$ is smooth.

We now assume that $f$ is a closed immersion with $E$ smooth and irreducible. Let $0$ be a point in $E$ and let $C\to Y$ be a locally closed smooth curve of $Y$ meeting $E$ at $0$ only. Corollary 1.45 applied to $f(C)$ and $\mathcal {M}$ on $X$ gives

\[ \operatorname{irr}(0,(f^+\mathcal{M})|_{C})=\operatorname{irr}(f(0),\mathcal{M}|_{f(C)})\leq (f(C),R)_{f(0)}= (C,f^*R)_0, \]

where the last equality follows from the projection formula. Hence, condition (ii) of Corollary 1.45 is satisfied and Proposition 2.3 follows.

2.4 Bounded irregularity and change of compactification I

Bounded irregularity behaves well with respect to compactification by normal crossing pairs.

Proposition 2.4 Let $U$ be a smooth variety over $k$. Let $(X_1,D_1)$ and $(X_2,D_2)$ be proper normal crossing pairs over $k$ compactifying $U$, that is $U=X_1\setminus D_1=X_2\setminus D_2$. Let $j_1 : U\to X_1$ and $j_2 : U\to X_2$ be the inclusions. Let $R_1$ be an effective divisor of $X_1$ supported on $D_1$. Then, there exists an effective divisor $R_2$ of $X_2$ supported on $D_2$ depending only on $(X_1,D_1)$, $(X_2,D_2)$ and linearly on $R_1$ such that for every object $\mathcal {M}_1$ of $\operatorname {MIC}(X_1,D_1,R_1)$, the connection $j_{2*}j_1^*\mathcal {M}_1$ lies in $\operatorname {MIC}(X_2,D_2,R_2)$.

Proof. Using resolution of singularities, we construct a pair $(X_3,D_3)$ dominating $(X_1,D_1)$ and $(X_2,D_2)$ such that $D_3$ has normal crossing. That is, there is a commutative diagram

with modifications as horizontal arrows where $D_3$ lies over $D_1$ and $D_2$. Let $\mathcal {M}_1\in \operatorname {MIC}(X_1,D_1,R_1)$. Put $\mathcal {M}_2=j_{2*}j_1^*\mathcal {M}_1$ and $\mathcal {M}_3=j_{3*}j_1^*\mathcal {M}_1$. From Proposition 1.23, we have

\[ p^+_1\mathcal{M}_1=\mathcal{M}_3=p^+_2\mathcal{M}_2. \]

Let $E$ be an irreducible component of $D_2$. We have to bound $\operatorname {irr}(E,\mathcal {M}_2)$ by means of $(X_i,D_i)$, $i=1,2$ and $R_1$ only. Let $E'$ be the strict transform of $E$ in $X_3$. Then,

\[ \operatorname{irr}(E,\mathcal{M}_2)=\operatorname{irr}(E',p_2^+\mathcal{M}_2)=\operatorname{irr}(E',\mathcal{M}_3)=\operatorname{irr}(E',p_1^+\mathcal{M}_1). \]

From Theorem 1.44, we further have

\[ \operatorname{Irr}(X_3,p_1^+\mathcal{M}_1)\leq p_1^*\operatorname{Irr}(X_1,\mathcal{M}_1) \leq p_1^* R_1. \]

We deduce

\[ \operatorname{irr}(E,\mathcal{M}_2)\leq m(E',p_1^* R_1)=m(E,p_{2 \ast} p_1^* R_1). \]

Hence, $R_2=p_{2 \ast } p_1^* R_1$ depends linearly on $R_1$ and satisfies the condition of Proposition 2.4.

2.5 Cohomological boundedness conjecture

Recall that if $G$ is an abelian group and $d\geq 0$ an integer, a polynomial $C : G\to \mathbb {Z}$ of degree at most $d$ is an element of $\operatorname {Sym}^{\bullet }_{\mathbb {Z}}(G^{\vee })$ of degree at most $d$, where $G^{\vee }$ is the abelian group dual to $G$.

Conjecture 2.5 Let $(X,D)$ be a projective normal crossing pair of dimension $d$ over $k$. There exists a polynomial $C : \operatorname {Div}(X,D)\oplus \mathbb {Z}\to \mathbb {Z}$ of degree at most $d$, affine in the last variable such that for every effective divisor $R$ of $X$ supported on $D$, for every integer $r\geq 0$ and every object $\mathcal {M}$ of $\operatorname {MIC}_r(X,D,R)$, we have

\[ \dim H^{*}(X,\operatorname{DR} \mathcal{M})\leq C(R,r). \]

Remark 2.6 If a polynomial $C : \operatorname {Div}(X,D)\oplus \mathbb {Z}\to \mathbb {Z}$ as above exists, we say that cohomological boundedness holds for $(X,D)$ with bound $C$.

Remark 2.7 Let $d\geq 0$ be an integer. Let $k$ be a field of characteristic $0$. We say that cohomological boundedness holds in dimension $d$ over $k$ if it holds for every choice of $(X,D)$ as above with $(X,D)$ defined over $k$ and $\dim X=d$.

Remark 2.8 Let $d\geq 0$ be an integer. We say that cohomological boundedness holds in dimension $d$ if it holds in dimension $d$ over every field of characteristic $0$.

Lemma 2.9 Cohomological boundedness holds in dimension $1$.

Proof. From Lemma A.1, it is enough to treat the case of a smooth connected projective curve $X$ of genus $g$ over $\mathbb {C}$. Let $D$ be a reduced divisor of $X$. Let $\mathcal {M}$ be an object of $\operatorname {MIC}(X,D)$. From Proposition 1.9 and Theorem 1.14, we are left to bound the cohomology of $\operatorname {Sol}(\mathcal {M}^{\operatorname {an}})$. Let $j : U=X\setminus D \to X$ be the open immersion and put $\mathcal {L} :=\operatorname {Sol}(\mathcal {M}^{\operatorname {an}})|_{U(\mathbb {C})}$. Observe that $\mathcal {L}$ is a local system of rank $\operatorname {rk} \mathcal {M}$ on $U(\mathbb {C})$. From Lemma 1.36, there is the following distinguished triangle.

We have $H^{0}(X(\mathbb {C}), j_!\mathcal {L} )\simeq 0$. If $\mathcal {L}^*$ denotes the dual of $\mathcal {L}$, Poincaré duality further gives

\[ \dim H^{2}(X(\mathbb{C}), j_!\mathcal{L} )=\dim H^{0}(U(\mathbb{C}), \mathcal{L}^{\ast} ) \leq \operatorname{rk} \mathcal{M}. \]

On the other hand, $\chi _c(U(\mathbb {C}), \mathcal {L} )=\operatorname {rk} \mathcal {M} \cdot \chi (U(\mathbb {C}), \mathbb {C} )=\operatorname {rk} \mathcal {M} \cdot (2-2g-|D|)$. Then, Lemma 2.9 follows from the long exact sequence in cohomology induced by the above distinguished triangle.

2.6 The $\chi$-boundedness conjecture

We formulate an a priori weaker version of the cohomological boundedness conjecture.

Conjecture 2.10 Let $(X,D)$ be a projective normal crossing pair of dimension $d$ over $k$. There exists a polynomial $C : \operatorname {Div}(X,D)\oplus \mathbb {Z}\to \mathbb {Z}$ of degree at most $d$, affine in the last variable such that for every effective divisor $R$ of $X$ supported on $D$, for every integer $r\geq 0$ and every object $\mathcal {M}$ of $\operatorname {MIC}_r(X,D,R)$, we have

\[ |\chi(X,\operatorname{DR} \mathcal{M}) |\leq C(R,r). \]

Remark 2.11 We adopt for the $\chi$-boundedness conjecture the terminology from Remarks 2.6–2.8 for the cohomological boundedness conjecture.

Remark 2.12 Let $d\geq 0$ be an integer. Similarly as in Lemma A.1, the $\chi$-boundedness conjecture holds in dimension $d$ if it holds in dimension $d$ over $\mathbb {C}$.

Cohomological boundedness trivially implies $\chi$-boundedness. The goal of the next subsection is to show that they are equivalent.

2.7 The cohomological and $\chi$-boundedness conjectures are equivalent

Proposition 2.13 Let $\mathbb {P}$ be a projective space over $k$. Let $X$ be a smooth subvariety of $\mathbb {P}$. Let $\mathcal {M}$ be a holonomic $\mathcal {D}_X$-module. Let $H\in \mathbb {P}^{\vee }(k)$ be a hyperplane such that $X\cap H$ is smooth and $X\cap H \to X$ is non-characteristic for $\mathcal {M}$. Then the canonical comparison morphism

\[ H^n(X,\operatorname{DR} \mathcal{M})\longrightarrow H^n(X\cap H,\operatorname{DR} (\mathcal{M}|_{X\cap H})) \]

is an isomorphism if $n<\dim X-1$ and is injective if $n=\dim X-1$.

Proof. From Remark 1.11, we reduce to the case where $k=\mathbb {C}$. From Proposition 1.9, we are left to prove a variant of Proposition 2.13 where now $X$ is the analytification of a smooth complex algebraic variety and where $\mathcal {M}$ is the analytification of a holonomic $\mathcal {D}$-module. Then Theorem 1.12 ensures that $\operatorname {DR} \mathcal {M}[\dim X]$ is a perverse complex. From the Lefschetz hyperplane theorem for perverse complexes [Reference de CataldodCat17, 2.4.2], the canonical comparison morphism

\[ H^n(X,\operatorname{DR} \mathcal{M})\longrightarrow H^n(X\cap H,(\operatorname{DR} \mathcal{M})|_{X\cap H}) \]

is an isomorphism if $n<\dim X-1$ and is injective if $n=\dim X-1$. Since $X\cap H \to X$ is non-characteristic for $\mathcal {M}$, we conclude the proof of Proposition 2.13 using Theorem 1.20.

Remark 2.14 When $X$ is a projective complex manifold, Proposition 2.13 is also valid via the same reasoning with either the analytic de Rham complex or the solution complex.

Corollary 2.15 In the setting of Proposition 2.13, assume the existence of $C\geq 0$ with

\[ \dim H^{\ast}(X\cap H,\operatorname{DR} (\mathcal{M}|_{X\cap H})) \leq C. \]

Then, if $d$ denotes the dimension of $X$, we have

\[ \dim H^{\ast}(X,\operatorname{DR} \mathcal{M})\leq |\chi(X,\operatorname{DR} \mathcal{M})| + 4d \cdot C. \]

Proof. We can suppose that $k=\mathbb {C}$. From Proposition 2.13, we have for $n< d$,

\[ \dim H^{n}(X,\operatorname{DR} \mathcal{M})\leq C. \]

Assume that $n>d$. Then,

\begin{align*} \dim H^{n}(X,\operatorname{DR} \mathcal{M}) &= \dim H^{n}(X(\mathbb{C}),\operatorname{DR} \mathcal{M}^{\operatorname{an}}) \\ & =\dim H^{2d-n}(X(\mathbb{C}),\operatorname{Sol} \mathcal{M}^{\operatorname{an}})\\ & \leq \dim H^{2d-n}((X\cap H)(\mathbb{C}),\operatorname{Sol} (\mathcal{M}^{\operatorname{an}}|_{(X\cap H)(\mathbb{C})}))\\ & \leq \dim H^{n-2}((X\cap H)(\mathbb{C}),\operatorname{DR}(\mathcal{M}^{\operatorname{an}}|_{(X\cap H)(\mathbb{C})})) \\ & \leq \dim H^{n-2}(X\cap H,\operatorname{DR}(\mathcal{M}|_{X\cap H})) \leq C, \end{align*}

where the first and fifth inequality follow from Proposition 1.9, the second follows from Theorem 1.14 and Poincaré–Verdier duality on $X(\mathbb {C})$, the third follows from Remark 2.14 since $2d-n< d$ and the fourth inequality follows from Theorem 1.14 and Poincaré–Verdier duality on $(X\cap H)(\mathbb {C})$. Furthermore, Proposition 1.10 yields

\begin{align*} \dim H^{d}(X,\operatorname{DR} \mathcal{M})& \leq |\chi(X,\operatorname{DR} \mathcal{M})| + \displaystyle{\sum_{n\neq d}} \dim H^{n}(X,\operatorname{DR} \mathcal{M})\\ & \leq |\chi(X,\operatorname{DR} \mathcal{M})| + 2d \cdot C. \end{align*}

Putting all the above inequalities together finishes the proof of Corollary 2.15.

Proposition 2.16 Let $(X,D)$ be a projective normal crossing pair of dimension $d$ over $k$. Let $X\to \mathbb {P}$ be a closed immersion in some projective space. Assume that $\chi$-boundedness holds for $(X,D)$ with bound $K$ and that cohomological boundedness holds with bound $C$ for the generic hyperplane section $(X_\eta,D_\eta )$ of $(X,D)$. Then, cohomological boundedness holds for $(X,D)$ with bound $K+4d\cdot C'$ where $C'$ is the composition of $C$ with the linear map $\operatorname {Div}(X,D)\oplus \mathbb {Z}\to \operatorname {Div}(X_\eta,D_\eta )\oplus \mathbb {Z}$ deduced from $X_\eta \to X$.

Proof. Let $R$ be an effective divisor of $X$ supported on $D$. Let $r$ be an integer. By Bertini's theorem, there exists a dense open subset $V$ in $\mathbb {P}^{\vee }$ such that for every hyperplane $H\in V(k)$, the pair $(X\cap H, D\cap H)$ is smooth of dimension $d-1$ and $D\cap H$ has normal crossing. Let $\eta$ be the generic point of $\mathbb {P}^{\vee }$. Following the notation from § 1.4, consider the following commutative diagram with cartesian squares.

Let $\pi : X_Q\to \mathbb {P}^{\vee }$ be the composition morphism and let $\pi _V : X_V\to V$ and $\pi _{\eta } : X_{\eta }\to \eta$ be its pull-back above $V$ and $\eta$, respectively. Let $D_V$ and $R_V$ be the pull-back of $D$ and $R$ to $X_V$ and let $D_\eta$ and $R_\eta$ be their pull-back to $X_\eta$. Observe that $(X_\eta,D_\eta )$ is a pair of dimension $d-1$ over $\eta$ where $D_{\eta }$ has normal crossing. Since cohomological boundedness holds for $(X_\eta,D_\eta )$ with bound $C$, we have for every object $\mathcal {N}_\eta$ in $\operatorname {MIC}_r(X_\eta,D_\eta,R_\eta )$,

\[ \dim H^{\ast}(X_\eta,\operatorname{DR} \mathcal{N}_\eta)\leq C(R_\eta,r). \]

On the other hand, for every object $\mathcal {N}$ in $\operatorname {MIC}_r(X_V,D_V ,R_V)$ we have

\[ (\pi _{V+}\mathcal{N})_\eta \simeq \pi _{\eta+}\mathcal{N}_\eta\simeq R\Gamma(X_\eta,\operatorname{DR} \mathcal{N}_\eta) [d-1]. \]

Thus, the space $H^n(X_\eta,\operatorname {DR} \mathcal {N}_\eta )$ is the generic fibre of the $\mathcal {D}_V$-module

\[ \mathcal{H}^{n-d+1} \pi _{V+}\mathcal{N} \]

for every $n\geq 0$. Let $\mathcal {M}$ be an object in $\operatorname {MIC}_r(X,D,R)$. From Proposition 2.3, the pull-back $\mathcal {M}_V$ of $\mathcal {M}$ to $X_V$ lies in $\operatorname {MIC}_r(X_V,D_V ,R_V)$. Since $k$ is infinite, Lemma 1.5 ensures the existence of a hyperplane $H\in V(k)$ such that $X\cap H\to X$ is non-characteristic for $\mathcal {M}$ and such that the cohomology modules of $\pi _{V+}\mathcal {M}_V$ are flat connections in a neighbourhood of $H\in V(k)$. In particular, the inclusion $i : \{H\} \to V$ is non-characteristic for the cohomology modules of $\pi _{V+}\mathcal {M}_V$. Hence, Theorem 1.20 combined with base change for $\mathcal {D}$-modules [Reference Hotta, Takeuchi and TanisakiHTT00, 1.7.3] yields

\begin{align*} i^*\mathcal{H}^{n-d+1} \pi _{V+}\mathcal{M}_V & \simeq \mathcal{H}^{n-d+1} i^+ \pi _{V+}\mathcal{M}_V \\ & \simeq \mathcal{H}^n \pi _{H+}(\mathcal{M}_V|_{X\cap H}) \\ & \simeq H^n(X\cap H, \operatorname{DR}\mathcal{M}|_{X\cap H} ), \end{align*}

where $\pi _{H} : X\cap H \to \{H\}$ is the pull-back of $\pi$ above $H\in V(k)$. Since $\mathcal {H}^{n-d+1} \pi _{V+}\mathcal {M}_V$ is a flat connection in a neighbourhood of $H$, we also have

\[ \dim H^{n}(X_\eta,\operatorname{DR} (\mathcal{M}_V)_\eta) = \dim i^*\mathcal{H}^{n-d+1} \pi _{V+}\mathcal{M}_V. \]

Thus,

\[ \dim H^\ast(X\cap H, \operatorname{DR}\mathcal{M}|_{X\cap H} ) \leq C(R_\eta,r). \]

Then, Corollary 2.15 yields

\[ \dim H^{\ast}(X,\operatorname{DR} \mathcal{M})\leq |\chi(X,\operatorname{DR} \mathcal{M})| + 4d \cdot C(R_\eta,r)\leq K(R,r)+ 4d \cdot C(R_\eta,r), \]

which concludes the proof of Proposition 2.16.

Corollary 2.17 Let $d\geq 2$ be an integer. Then cohomological boundedness holds in dimension $d$ if it holds in dimension $d-1$ and if $\chi$-boundedness holds in dimension $d$.

From Lemma 2.9 and Corollary 2.17, we deduce the following.

Corollary 2.18 The cohomological and $\chi$-boundedness conjectures are equivalent.