Let (S,H) be a rational algebraic surface with an ample divisor. We compute generating functions for the Hodge numbers of the moduli spaces of H-stable rank 2 sheaves on S in terms of certain theta functions for indefinite lattices that were introduced in the paper alg-geom/9612020 written jointly with Don Zagier. If H lies in the closure of the ample cone and has self-intersection 0, it follows that the generating functions are Jacobi forms. In particular the generating functions for the Euler numbers have a similar transformation behaviour under SL(2,Z) as that predicted in Vafa and Witten: A strong coupling test of S-duality. In addition we get that also the generating functions for the signatures can be expressed in terms of modular forms. Finally it turns out that the generating function for the signatures is also (with respect to another developping parameter) the generating function for the Donaldson invariants of S evaluated on all powers of the point class. The paper is related to the recent papers math.AG/9805003 by Yoshioka, math.AG/9805054 and math.AG/9805055 by Qin and Li and hep-th/9802168 by Minahan, Nemeschansky, Vafa and Warner.
In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincare polynomial of the associated cohomology ring.
The purpose of this paper is twofold. First, we survey known results about theta dualities on moduli spaces of sheaves on curves and surfaces. Secondly, we establish new such dualities in the surface case. Among others, the case of elliptic K3 surfaces is studied in detail; we propose further conjectures which are shown to imply strange duality.
We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms …
Summary: The Hodge conjecture asks whether rational Hodge classes on a smooth projective manifolds are generated by the classes of algebraic subsets, or equivalently by Chern classes of coherent sheaves. On a compact Kaehler manifold, Hodge conjecture is known to be false if algebraic subsets are replaced with analytic subsets. Here we show that it is even false that for a Kaehler manifold, Hodge classes are generated by Chern classes of coherent sheaves. We also show that finite free resolution do not in general exist for coherent sheaves on compact Kaehler manifolds.
A constructible sheaf corresponding to Gel'fand Zelevinski hypergeometric functions on a torus is called hypergeometric sheaf. We consider Hodge and Tate conjectrue for hypergeomtric sheaves. Hodge conjecture is formulated in terms of variation of Hodge strucure and Tate conjecture is done for l-adic sheaves on an open set of torus. We prove Hodge and Tate conjecture up to Hodge and Tate cycle of Fermat motifes. We use cohomological Mellin transform to get the main theorem. This is the final revision for preprint.
v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $dim X\le p$, if $k$ has characteristic $p>0$. They show this if $k$ has characteristic 0 and if $n$ is prime to $p$ in characteristic $p>0$. We show the conjecture in characteristic $p>0$ if $n=p$ assuming in addition that $X$ is ordinary (in the sens…
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.
In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If $S$ is a projective K3 surface, $v$ a Mukai vector and $H$ a $v-$generic polarization on $S$, we show that $H^{2}(M_{v},\mathbb{Z})$ is a free $\mathbb{Z}-$module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the Hodge sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are proved for Abelian surfaces.
We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.
We consider arbitrary polarized variations of Hodge structure of weight two and $h^{2,0}=1$ over a non--singular complex algebraic curve $S$ and analyze the boundary behaviour of the associated Kudla--Millson theta series using Schmid's theorems on degenerations of Hodge structure. This allows us to prove that this theta series is always integrable over $S$ and to describe explicitly the non-holomorphic part of the Kudla--Millson generating series in terms of the mixed Hodge structures at infinity.
In this paper we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence, and the existence of proper pushforward. In this way we recover and generalize analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves. We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varie…
We prove that the weight-two Hodge structure of moduli spaces of torsion-free sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we assume the first Chern class is primitive). We prove the moduli space is an irreducible symplectic variety (by Mukai's work it was known to be symplectic). By work of Beauville, this implies that its $H^2$ has a canonical integral non-degenrate quadratic form; Mukai's recepee for $H^2$ includes a description of Beauville's quadratic form. As an application we compute higher-rank Donaldson polynomials of $K3$ surfaces.
This survey is based on my lectures given in last a few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorem, we show, without using vanishing theorem, a new observation that ${\rm dim}\,H^0(\sU_C,Θ_{\sU_C})$ is independent of all of the choices for any smooth curves. The estimate of various codimension and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.
These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing sheaves of Kollár is extended to the sheaves of differential forms of arbitrary degrees in the case of a well prepared birational model. We will also prove the log versions of some of the results, i.e., the case where we allow horizontal boundary components.
We extend the dimension and strong linearity results of generic vanishing theory to bundles of holomorphic forms and rank one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated to irregular varieties. Our main tools are Saito's mixed Hodge modules, the Fourier-Mukai transform for D-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson's harmonic theory for flat bundles. In the process, we discover two natural categories of perverse coherent sheaves, one on the Picard variety, and the other on the moduli space of Higgs line …
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. We survey some of the results known about this primitive cohomology, prove a few general facts and mention some interesting open problems.
We continue our study on the Hodge-Iwasawa theory which is a continuation of our previous work on Hodge-Iwasawa theory, which is aimed at higher dimensional deformation of higher dimensional Hodge structures over general analytic spaces or adic spaces. We still follow closely the approaches of Kedlaya-Liu to study our Frobenius modules over the different kinds of period rings including more generalized perfect period rings and the corresponding imperfect period rings. It is desirable that one can globalize the deformed information in order to organize the deformed period rings into the corresp…
We show that a moduli space of slope-stable sheaves over a K3 surface is an irreducible hyperkähler manifold if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.
These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the "beyond geometric invariant theory" program for analyzing moduli problems in algebraic geometry. We discuss methods for analyzing stability in general moduli problems, focusing on the moduli of coherent sheaves on a smooth projective scheme as a toy example. We describe several applications: a general structure theorem for the derived category of coherent sheaves on an algebraic stack; some results on the topology of moduli stacks; and a "v…
We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semi-positivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semi…
We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering a question raised by Gottsche-Nakajima-Yoshioka.
The primal cohomology $\mathbb{K}_\mathbb{Q}$ of the theta divisor $Θ$ of a principally polarized abelian fivefold (ppav) is the direct sum of its invariant and anti-invariant parts $\mathbb{K}_\mathbb{Q}^{+1}$, resp. $\mathbb{K}_\mathbb{Q}^{-1}$ under the action of $-1$. For smooth $Θ$, these have dimension $6$ and $72$ respectively. We show that $\mathbb{K}_\mathbb{Q}^{+1}$ consists of Hodge classes and, for a very general ppav, $\mathbb{K}_\mathbb{Q}^{-1}$ is a simple Hodge structure of level $2$.
In this paper, we are going to establish a simultaneous generalization of the relative Iwasawa theory proposed by Kedlaya-Pottharst and the relative $p$-adic Hodge theory after Kedlaya-Liu. We call this Hodge-Iwasawa theory in the sense that one could apply the theory to study noncommutative Iwasawa cohomology and noncommutative Iwasawa theories in families and meanwhile one could apply the theory to study the deformation theory of étale local systems or families of representations of fundamental groups or the equivariant constructible $p$-adic sheaves, with more sophisticated point of view co…
We compute the Hodge numbers of the moduli space of semi-stable sheaves on the complex projective plane supported on quintic curves and having Euler characteristic 3. For this purpose we study the fixed-point set for a certain torus action on the moduli space.
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.
We discuss variations of Hodge structures on abelian varieties that arise from intersecting translates of theta divisors with a special focus on the case of abelian varieties of dimension 4
Let (P,X) be Shimura data, M=M(P,X,K) the Shimura variety of level K. To an algebraic representation of P, one can associate a mixed sheaf (variation of Hodge structure, l-adic sheaf) on M. In the paper, we study the degeneration of such sheaves along strata in toroidal compactifications of M. The main result (2.8 in the Hodge setting, 3.9 in the l-adic setting) gives a formula for this degeneration in terms of Hochschild cohomology of certain unipotent subgroups of P. The new version differs from the earlier one in that the proof of 2.8 was rewritten. In particular, the effect of Saito's spec…
We determine the structure of certain moduli spaces of ideal sheaves by generalizing an earlier result of the first author. As applications, we compute the (virtual) Hodge polynomials of these moduli space, and calculate the Donaldson-Thomas invariants of certain 3-folds with trivial canonical classes.
In this note we examine the question of expressing the determinant of the push forward of a symmetric line bundle on an abelian fibration in terms of the pull back of the relative dualizing sheaf via the zero section.
We describe the torus fixed locus of the moduli space of stable sheaves with Hilbert polynomial $4m+1$ on the projective plane. We determine the torus representation of the tangent spaces at the fixed points, which leads to the computation of the Betti and Hodge numbers of the moduli space.
We describe the mod $p^r$ pro $K$-groups $\{K_n(A/I^s)/p^r\}_s$ of a regular local $\mathbb F_p$-algebra $A$ modulo powers of a suitable ideal $I$, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of Geisser-Levine and Bloch-Kato-Gabber. This is achieved by combining the pro Hochschild-Kostant-Rosenberg theorem in topological cyclic homology with the development of the theory of de Rham-Witt complexes and logarithmic Hodge-Witt sheaves on formal schemes in characteristic $p$. Applications include the following: the infinitesimal part of the weak Lefschetz con…
We show that there is a perverse sheaf on a fine moduli space of stable sheaves on a smooth projective Calabi-Yau 3-fold, which is locally the perverse sheaf of vanishing cycles for a local Chern-Simons functional, possibly after taking an etale Galois cover. This perverse sheaf lifts to a mixed Hodge module and gives us a cohomology theory which enables us to define the Gopakumar-Vafa invariants mathematically.
A construction of braid group actions on coherent sheaves using mixed Hodge modules and some well known constructions from geometric representation theory is given.
We prove that given any rational Hodge isometry $ψ:H^2(S_1,\mathbb{Q})\rightarrow H^2(S_2,\mathbb{Q})$ between any two Kähler $K3$ surfaces $S_1$ and $S_2$ the cohomology class of $ψ$ in $H^{2,2}(S_1\times S_2)$ is a polynomial in Chern classes of coherent analytic sheaves over $S_1 \times S_2$. Consequently, the cohomology class of $ψ$ is algebraic whenever $S_1$ and $S_2$ are algebraic.
Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic to Tor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolated singularity, these torsion modules are of finite length for i sufficiently large. The theta invariant of the pair (M,N) is defined by Hochster to be length(Tor_{2i}^R(M,N)) - length(Tor_{2i+1}^R(M,N)) for i sufficiently large. H. Dao has conjectured that the theta invariant is zero for all pairs …
We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is: Theorem: Let $X$ be a K3 surface, $L$ a primitive big and nef line bundle and $H$ a generic polarization. If the moduli space of rank two $H$ semi-stable torsion-free sheaves with determiant $L$ and second Chern class $c_2$ has at least dimension 10 then its Hodge numbers coincide with those of the Hilbert scheme of $l:=2c_2-\frac{L^2}{2}-3$ points on $X$.
We show that the only summands of theta divisors on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the obvious powers of the curve and the Fano surface of lines on the threefold. The proof uses the decomposition theorem for perverse sheaves, a bit of representation theory and a computation of characteristic cycles for Brill-Noether sheaves.
Using the Bialynicki-Birula method, we determine the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (5, 1, 4), (7, 2, 6), respectively, (0, 5, 19). We compute the Hodge numbers of these moduli spaces.
We study the bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors. Furthermore, Fourier-Mukai symmetries of the Verlinde bundles are found, consistently with strange duality. Along the way, a transformation formula for the theta bundles is derived, extending a theorem of Drezet-Narasimhan from curves to abelian surfaces.
Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement.
We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions…
We show how natural functors from the category of coherent sheaves on a projective scheme to categories of Kronecker modules can be used to construct moduli spaces of semistable sheaves. This construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions, about which more complete results can be proved.
In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3 surfaces, and we identify alpha in terms of Mukai's map between the cohomology of X and of M (defined by means of a quasi-universal sheaf). We prove that the derived category of sheaves on X and the derived category of alpha-twisted sheaves on M are equivalent. This suggests a conjecture which describes, in terms of Hodge isometries of lattices, when derived…
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We ap…
M. Hochster defines an invariant namely $Θ(M,N)$ associated to two finitely generated module over a hyper-surface ring $R=P/f$, where $P=k\{x_0,...,x_n\}$ or $k[X_0,...,x_n]$, for $k$ a field and $f$ is a germ of holomorphic function or a polynomial, having isolated singularity at $0$. This invariant can be lifted to the Grothendieck group $G_0(R)_{\mathbb{Q}}$ and is compatible with the chern character and cycle class map, according to the works of W. Moore, G. Piepmeyer, S. Spiroff, M. Walker. They prove that it is semi-definite when $f$ is a homogeneous polynomial, using Hodge theory on Pro…
We study the Albanese image of a compact Kähler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.
We introduce a new arithmetic invariant for hermitian line bundles on an arithmetic variety. We use this invariant to measure the variation of the volume function with respect to the metric. The main result of this paper is a generalized Hodge index theorem on arithmetic toric varieties.
We use superconnections to define and study some natural differential forms on period domains $\mathbb{D}$ that parametrize polarized Hodge structures of given type on a rational quadratic vector space $V$. These forms depend on a choice of vectors $v_1,\ldots,v_r \in V$ and have a Gaussian shape that peaks on the locus where $v_1,\ldots,v_r$ become Hodge classes. We show that they can be rescaled so that one can form theta series by summing over a lattice $L^r \subset V^r$. These series define differential forms on arithmetic quotients $Γ\backslash \mathbb{D}$. We compute their cohomology cla…
In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a su…
