Perverse-Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito's decomposition theorem. We study perverse-Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the "Perverse = Hodge" identity of the authors and specializes to Matsushita's theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.
We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory.
Let $S\to C$ be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on $H^*(S^{[n]},\mathbb{Q})$ associated with the natural map $S^{[n]}\to C^{(n)}$ implies that $S\to C$ is an elliptic fibration. The converse is also true when $S\to C$ is a Hitchin-type elliptic fibration.
In this note, we provide a quick introduction to the study of the Milnor fibration via the derived category and perverse sheaves. This is primarily a dictionary for translating from the standard topological setting to the derived category and/or the Abelian category of perverse sheaves.
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variet…
We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.
Using the theory of mixed perverse sheaves, we extend arguments on the Hodge conjecture initiated by Lefschetz and Griffiths to the case of the Tate conjecture, and show that the Tate conjecture for divisors is closely related to the de Rham conjecture for nonproper varieties, finiteness of the Tate-Shafarevich groups, and also to some conjectures in the analytic number theory.
The perverse filtration in cohomology and in cohomology with compact supports is interpreted in terms of kernels of restrictions maps to suitable subvarieties by using the Lefschetz hyperplane theorem and spectral objects. Various mixed-Hodge-theoretic consequences for intersection cohomology and for the decomposition theorem are derived.
This is a report on joint work with T. Hausel and L. Migliorini, where we prove, for each of the groups GL(2,C), PGL(2,C), SL(2,C), that the non-Abelian Hodge theorem identifies the weight filtration on the cohomology of the character variety with the perverse Leray filtration on the cohomology of the domain of the Hitchin map. We review the decomposition theorem, Nĝo's support theorem, the geometric description of the perverse filtration and the sub-additivity of the Leray filtration with respect to the cup product.
We present a Macaulay2 package that produces compactifications of adjoint orbits and of the fibres of symplectic Lefschetz fibrations on them. We use Macaulay2 functions to calculate the corresponding Hodge diamonds, which then reveal topological information about the Lefschetz fibrations.
We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result, one on the topology of the base and fibers of a Lagrangian fibration, the other on the refined Gopakumar-Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration as…
We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.
A homotopical treatment of intersection cohomology recently developed by Chataur-Saralegui-Tanré associates a "perverse algebraic model" to every topological pseudomanifold, extending Sullivan's presentation of rational homotopy to intersection cohomology. In this context, there is a notion of "intersection-formality", measuring the vanishing of Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We endow such invariant with natural mixed Hodge structures. This allows us to prove so…
This work establishes a subtle connection between mirror symmetry for Calabi-Yau threefolds and that of curves of higher genus. The linking structure is what we call a perverse curve. We show how to obtain such from Calabi-Yau threefolds in the Batyrev mirror construction and prove that their Hodge diamonds are related by the mirror duality.
The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variat…
We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for $GL_n$ over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every $n \geq 3$. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare th…
Let $X$ a complex projective variety of complex dimension $n$ with only isolated singularities of simply connected links. We show that we can endow the rational cohomology of the family of the $\overline{p}$-perverse intersection spaces $\{ I^{\overline{p}}X \}$ with compatible mixed Hodge structures.
For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.
For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P=W conjecture in non-abelian Hodge theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge-Tate decomposition for a character variety to each piece of th…
This paper contains two results on Hodge loci in the moduli space of curves. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fibers is contained in a proper Hodge locus. The second result deals with divisors in the moduli space of curves. It is proved that the image of a divisor in the moduli of principally polariz…
We prove that the higher direct images of the dualizing sheaf of a Lagrangian fibration between smooth projective manifolds are isomorphic to the cotangent bundles of base space. As a corollary, we obtain that every Hodge number of the base space of a fibre space of an irreducible symplectic manifold is the same to that of a projective space if the base space is smooth.
We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with $h^{1, 1} \geq 140$ or $h^{2, 1} \geq 140$ that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small $h^{1, 1}$ the fraction of polytopes in the KS database that do not have a genu…
This is an expanded comment on Barannikov's paper math.AG/0006193. A symplectic version of his construction is discussed. It is shown that the duality transformation for mirror torus fibrations over the same Monge-Ampere manifold exchanges semi-infinite A-variations of Hodge structure introduced in this paper with Barannikov's semi-infinite B-variations of Hodge structure.
We introduce the notion of filtered perversity of a filtered differential complex on a complex analytic manifold $X$, without any assumptions of coherence, with the purpose of studying the connection between the pure Hodge modules and the \lt-complexes. We show that if a filtered differential complex $(\cM^\bullet,F_\bullet)$ is filtered perverse then $\aDR(\cM^\bullet,F_\bullet)$ is isomorphic to a filtered $\cD$-module; a coherence assumption on the cohomology of $(\cM^\bullet,F_\bullet)$ implies that, in addition, this $\cD$-module is holonomic. We show the converse: the de Rham complex of …
We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly b…
Mumford constructed a family of abelian fourfolds with special stucture not characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge structure of such a variety decomposes into Hodge substructures via the action of Mumford-Tate group, one of which is of K3 type with Hodge number (1,7,1). We will compute the intersection form of such a Hodge structure and provide a canonical integral Hodge structure on this subspace. Furthermore, we shall show two applications of this invariant. Firstly, using the above formula we shall show that every K3 surface obtained in this way will adm…
Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observ…
The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers d…
We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.
We constructed the analytic Milnor fiber is a non-archimedean model of the classical topological Milnor fibration. In the present paper, we describe the homotopy type of the analytic Milnor fiber in terms of a strictly semi-stable model, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. We give a similar expression for Denef and Loeser's motivic Milnor fiber in terms of a strictly semi-stable model.
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitr…
In its simplest form the Decomposition Theorem asserts that the rational intersection cohomology of a complex projective variety occurs as a summand of the cohomology of any resolution. This deep theorem has found important applications in algebraic geometry, representation theory, number theory and combinatorics. It was originally proved in 1981 by Beilinson, Bernstein, Deligne and Gabber as a consequence of Deligne's proof of the Weil conjectures. A different proof was given by Saito in 1988, as a consequence of his theory of mixed Hodge modules. More recently, de Cataldo and Migliorini foun…
This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito's deep theory of mixed Hodge modules as a "black box", thinking about them as "constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. For the "constant Hodge sheaf", one gets the "motivic characteristic classes" of Brasselet-Schuermann-Yokura, whereas the classes of the "intersection homology Hodge sheaf" wer…
This paper has two aims. The former is to give an introduction to our earlier work on the Hodge theory of algebraic maps and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps. The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y. We provide a few explicit computations of perverse truncations and in…
Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of Kähler manifolds for which each fibre itself is fibred by codimension one Kähler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser-Livné in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the univers…
We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there are at least 36,779 topologically distinct manifolds in our list. We also study the fibration structure of these manifold…
The nonabelian Jacobian $\JA$ of a smooth projective surface $X$ is inspired by the classical theory of Jacobian of curves. It is built as a natural scheme interpolating between the Hilbert scheme $\XD$ of subschemes of length $d$ of $X$ and the stack ${\bf M}_X (2,L,d)$ of torsion free sheaves of rank 2 on $X$ having the determinant $\OO_X (L)$ and the second Chern class (= number) $d$. It relates to such influential ideas as variations of Hodge structures, period maps, nonabelian Hodge theory, Homological mirror symmetry, perverse sheave, geometric Langlands program. These relations manifest…
Fujita's second theorem for Kähler fibre spaces over a curve asserts that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our negative answer (\cite{cd}) to a question by Fujita: is $V$ semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group $(\mathbb Z/…
The aim of this paper is to construct families of Calabi--Yau 3-folds without boundary points with maximal unipotent monodromy and to describe the variation of their Hodge structures. In particular five families are constructed. In all these cases the variation of the Hodge structures of the Calabi--Yau 3-folds is basically the variation of the Hodge structures of a family of curves. This allows us to write explicitly the Picard--Fuchs equation for the 1-dimensional families. These Calabi--Yau 3-folds are desingularizations of quotients of the product of a (fixed) elliptic curve and a K3 surfa…
