Certain rigid irregular $G_2$-connections constructed by the first-named author are related via pullbacks along a finite covering and Fourier transform to rigid local systems on a punctured projective line. This kind of property was first observed by Katz for hypergeometric connections and used by Sabbah and Yu to compute irregular Hodge filtrations for hypergeometric connections. This strategy can also be applied to the aforementioned $G_2$-connections and we compute jumping indices and dimensions for their irregular Hodge filtrations.
We prove a new inequality for the Hodge number h^1,1 of irregular complex smooth projective surfaces of general type without irregular pencils of genus >1. More specifically we show that if the irregularity q satisfies q=2^k+1 then h^1,1>4q-4.
We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations.
For an algebraic vector bundle $E$ over a smooth algebraic variety $X$, a monodromic $D$-module on $E$ is decomposed into a direct sum of some $O$-modules on $X$. We show that the Hodge filtration of a monodromic mixed Hodge module is decomposed with respect to the decomposition of the underlying $D$-module. By using this result, we endow the Fourier-Laplace transform $M^{\wedge}$ of the underlying $D$-module $M$ of a monodromic mixed Hodge module with a mixed Hodge module structure. Moreover, we describe the irregular Hodge filtration on $M^{\wedge}$ concretely and show that it coincides with…
We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.
Based on work of R. Lazarsfeld and M. Popa, we use the derivative complex associated to the bundle of the holomorphic p-forms to provide inequalities for all the Hodge numbers of a special class of irregular compact Kaehler manifolds. For 3-folds and 4-folds we give an asymptotic bound for all the Hodge numbers in terms of the irregularity. As a byproduct, via the BGG correspondence, we also bound the regularity of the exterior cohomology modules of bundles of holomorphic p-forms.
In this paper we investigate the exactness of the Grassmannian BGG complexes introduced in a previous work (arXiv:1211.2486), and obtain some inequalities between some Hodge numbers of some irregular varieties. In particular, we obtain sharp lower bounds for the Hodge numbers of smooth subvarieties of Abelian varieties, as well as some improvements of results of Lazarsfeld and Popa and Lombardi concerning threefolds and fourfolds.
We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\expφ$ for any meromorphic function $φ$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also…
For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df, extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E_1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of arXiv:0804.4328. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on its proof of the corresponding E_1 degeneration, by reduction to characteris…
Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of arXiv:1302.4537. We show the strictness of the push-forward filtered D-module through any projective morphism, by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in ter…
We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space.
We construct examples of smooth proper rigid-analytic varieties admitting formal model with projective special fiber and violating Hodge symmetry for cohomology in degrees $\geq 3$. This answers negatively a question raised by Hansen and Li.
Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.
N.Katz's middle convolution algorithm provides a description of rigid connections on the projective line with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities.
Using the Katz-Arinkin algorithm we give a classification of irreducible rigid irregular connections on a punctured $\mathbb{P}^1_{\mathbb{C}}$ having differential Galois group $G_2$, the exceptional simple algebraic group, and slopes having numerator 1. In addition to hypergeometric systems and their Kummer pull-backs we construct families of $G_2$-connections which are not of these types.
We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-etale site, which makes all constructions completely functorial.
We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier-Laplace transformation for algebraic integrable twistor $\mathcal{D}$-modules.
Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for lo…
In this article, we generalize the Hodge-Tate decomposition of p-adic étale cohomology to non-smooth rigid spaces. Our strategy is to study pro-étale cohomology of rigid spaces introduced by Scholze, using the resolution of singularities and the simplicial method.
Sen attached to each p-adic Galois representation of a p-adic field a multiset of numbers called generalized Hodge-Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a p-adic local system on a rigid analytic variety over a p-adic field and show that the multiset of generalized Hodge-Tate weights of the local system is constant. The proof uses the p-adic Riemann-Hilbert correspondence by Liu and Zhu, a Sen-Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic propertie…
T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.
It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not true, as already remarked by M. Saito. We explain how one has to modify the latter, show that the Picard group can be expressed by absolute Hodge cohomology, too, and introduce an intermediate object between Picard group and usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne cohomology one can relate line bundles with a regular connec…
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V…
On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that they carry hyperKahler metrics, which are complete when the residue of the connection if semisimple.
The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…
We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be "classical", we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.
We prove the Künneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying chain complexes under a certain elimination of indeterminacy.
In this paper we construct a connection on the trivial G-bundle on the projective line for any simple complex algebraic group G, which is regular outside of the points 0 and infinity, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point infinity, with slope 1/h, the reciprocal of the Coxeter number of G. This connection, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of l-adic representations, which should parametrize a sp…
We give a lower bound on the Hodge number h^{2,0}(X), where X is an irregular compact Kähler (or smooth complex projective) variety, in terms of the minimal rank of an element in the kernel of the wedge product map ψ_2: Λ^2 H^0(X,Ω_X^1) -> H^0(X,Ω_X^2). As a consequence, we obtain a generalization to higher dimensions of the Castelnuovo-de Franchis inequality for surfaces, improving some results of Lazarsfeld and Popa and Lombardi for threefolds and fourfolds.
We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection …
