The primitive cohomology lattice of a complete intersection

二次超曲面完全交的上同调

Cohomology of complete intersections of quadrics

本文证明了一般二次超曲面完全交的变上同调可等同于双重覆盖的交上同调，并得出奇数维射影空间中四个二次超曲面完全交的中上同调同构于双重体奇点解消的中上同调。

The primitive cohomology of the theta divisor of an abelian fivefold

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.

Intersection Cohomology, Monodromy, and the Milnor Fiber

We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen to be equivalent to $X$ being a homology manifold. Given an analytic function $f$ on an intersection cohomology manifold, we describe a simple relation between $V(f)$ being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of $f$. We then describe how the Sebastiani-Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Fin…

Intersection Cohomology and Perverse Eigenspaces of the Monodromy

We describe the relationship between intersection cohomology with twisted coefficients and the perverse sheaves which play the role of the eigenspaces for the Milnor monodromy of an affine hypersurface.

Intersection cohomology of character varieties for punctured Riemann surfaces

We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier.

From moment graphs to intersection cohomology

We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.

The semi-infinite intersection cohomology sheaf

We introduce the semi-infinite category of sheaves on the affine Grassmannian, and construct a particular object in it, which we call the the semi-infinite intersection cohomology sheaf. We relate it to several other entities naturally appearing in the geometric Langlands theory.

Non-affine Landau-Ginzburg models and intersection cohomology

We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.

Eigenvalues of Frobenius acting on the $\ell$-adic cohomology of complete intersections of low degree

The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos.

The intersection cohomology of the Satake compactification of ${\mathcal A}_g$ for $g\le 4$

We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 4. We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.

The Decomposition Theorem and the Intersection Cohomology of Quotients in Algebraic Geometry

This paper applies the decomposition theorem in intersection cohomology to geometric invariant theory quotients, relating the intersection cohomology of the quotient to that of the semistable points for the action. Suppose a connected reductive complex algebraic group $G$ acts linearly on a complex projective variety $X$. We prove that if $1 \to N \to G \to H \to 1$ is a short exact sequence of connected reductive groups, and $X^{ss}$ the set of semistable points for the action of $N$ on $X$, then the $H$-equivariant intersection cohomology of the geometric invariant theory quotient $X^{ss}//N…

Hodge type of the exotic cohomology of complete intersections

We improve the known Hodge type bound for the exotic cohomology of complete intersections. In the revised version, we included a simplification of our original argument due to Pierre Deligne. The note appears in the C. R. de l'Aca. des Sc. de Paris.

Formality of equivariant intersection cohomology

We present several approaches to equivariant intersection cohomology. We show that for a complete algebraic variety acted by a connected algebraic group $G$ it is a free module over $H^*(BG)$. The result follows from the decomposition theorem or from a weight argument.

Hyperbolic localization of intersection cohomology

For a normal variety X defined over an algebraically closed field with an action of the multiplicative group G_m, we consider the ``hyperbolic localization'' functor from D^b(X) to D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.

Intersection cohomology and Severi's varieties

Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}π{_*}\mathbb{Q}$, where $π$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(\mathbb{P} ^N)}^{\vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}π{_*}\mathbb{Q})$ over the points of a Severi's variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $\mathbb{P} ^N$.

关于射影曲面本原上同调格的一个注记

A note on the primitive cohomology lattice of a projective surface

本文研究射影曲面的本原上同调格，证明在不定情形下它可由相交格和本原丰沛类的自交数确定，并给出 Godeaux 曲面、Kunev 曲面及 Horikawa 曲面等例子，同时讨论了一般型典范极化曲面的负定本原格结果。

高阶 Teichmüller 分量的相交上同调

Intersection cohomology of higher rank Teichmüller components

该研究结合非 Abel Hodge 对应与 Cayley 对应，给出紧 Riemann 曲面基本群特征簇中奇异高阶 Teichmüller 分量的相交上同调计算。

自同构与上同调 II：完全交集

Automorphism and Cohomology II: Complete intersections

本文证明了射影空间中一般完全交集的自同构群平凡且忠实作用于上同调，仅存在少数已知例外。

Homology and cohomology intersection numbers of GKZ systems

We describe the homology intersection form associated to regular holonomic GKZ systems in terms of the combinatorics of regular triangulations. Combining this result with the twisted period relation, we obtain a formula of cohomology intersection numbers in terms of a Laurent series. We show that the cohomology intersection number depends rationally on the parameters. We also prove a conjecture of F. Beukers and C. Verschoor on the signature of the monodromy invariant hermitian form. This is a continuation of the previous work arXiv:1904.00565.

On intersection cohomology with torus actions of complexity one

The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus $\mathbb{T}$, one of our result determines the intersection cohomology Betti numbers of any normal projective $\mathbb{T}$-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.

Pfaffian of Appell's hypergeometric system $F_4$ in terms of the intersection form of twisted cohomology groups

We study a Pfaffian of the system of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$ by twisted cohomology groups associated with integrals representing solutions to this system. We simplify its connection matrix by the pull-back under a double cover of the complement of the singular locus. We express the simplified connection matrix in terms of the intersection form of the twisted cohomology groups.

On the mixed Hodge structures of the intersection cohomology stalks of complex hypersurfaces

We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.

Shimura 簇的 L_2-上同调与交上同调的 Hodge 分解

Hodge decomposition of L_2-cohomology and intersection cohomology of a Shimura variety

本文证明了 Shimura 簇的平方可积上同调与 Baily-Borel 紧化的交上同调之间的 Hodge 分解在已有的 Zucker 猜想证明下是一致的。

Equivariant cohomology and equivariant intersection theory

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications. As another appplication of equivariant intersection theory, we obtain simple versions of criteria for smoothness or rati…

Parity and symmetry in intersection and ordinary cohomology

In this paper, we show that the Galois representations provided by $\ell$-adic cohomology of proper smooth varieties, and more generally by $\ell$-adic intersection cohomology of proper varieties, over any field, are orthogonal or symplectic according to the degree. We deduce this from a preservation result of orthogonal and symplectic pure perverse sheaves by proper direct image. We show moreover that the subgroup of the Grothendieck group generated by orthogonal pure perverse sheaves of even weights and symplectic pure perverse sheaves of odd weights are preserved by Grothendieck's six opera…

Towards a combinatorial Intersection Cohomology for Fans

The real intersection cohomology of a toric variety is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' on the fan as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of Stanley's generalized f- and g-vector of an arbitrary polytope or complete fan under a natural vanishing condition. -- The results presented in this note originate from joint work with G.Barthel, J.-P.Brasselet and L.Kaup, continui…

Intersection cohomology of Drinfeld's compactifications

Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification $\widetilde{\on{Bun}}_P$ of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\on{Bun}}_P$. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.

The semi-infinite intersection cohomology sheaf-II: the Ran space version

This paper is a sequel to [Ga1]. We study the semi-infinite category on the Ran version of the affine Grassmannian, and study a particular object in it that we call the semi-infinite intersection cohomology sheaf. Unlike the situation of [Ga1], this version of the semi-infinite intersection IC sheaf is defined as the middle of extension of the constant (more precisely, dualizing) sheaf on the basic stratum, in a certain t-structure. We give several explicit description and characterizations of our semi-infinite intersection IC sheaf: we describe its !- and *- stalks; we present it explicitly a…

Intersection cohomology of pure sheaf spaces using Kirwan's desingularization

Let $\mathbf{M}_n$ be the Simpson compactification of twisted ideal sheaves $\mathcal{I}_{L,Q}(1)$ where $Q$ is a rank $4$ quardric hypersurface in $\mathbb{P}^n$ and $L$ is a linear subspace of dimension $n-2$. This paper calculates the intersection Poincaré polynomial of $\mathbf{M}_n$ using Kirwan's desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree $\geq 8$ by considering wall-crossings of stable pairs and complexes.

$L^\infty$ cohomology is intersection cohomology

Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.

Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties

We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted projective spaces and singular toric varieties. We generalize the Riemann-Roch equations to weighted projective spaces. We compute counting matrices of smooth Fano threefolds with Picard group Z and anticanonical degrees 2, 8, and 16.

A型环面簇的交点同调

Intersection cohomology of type-A toric varieties

本文研究A型环面簇的交点同调，通过GIT商构造，提出组合模型计算Betti数，并证明交点同调群具有典范乘积结构。

Rational Integrals of the second kind on a complex projective manifold and its primitive cohomology

Let X be a complex algebraic manifold of dimension n+1 embedded in a sufficiently higher dimensional complex projective space, and Y a generic hyperplane section of X. We describe the mixed Hodge structure on H^p(X-Y,C) and the Hodge filtration of the middle primitive cohomology group H^n(Y,C)_0 of Y in terms of rational integrals on X. (Key words: Primitive cohomology, Rational integral of the 2nd kind, Generalized Poincare residue map, Hodge filtration, Mixed Hodge structure)

Cohomology of twisted symmetric powers of cotangent bundles of smooth complete intersections

In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symme…

曲线上向量丛模空间的交叉上同调

Intersection cohomology of moduli spaces of vector bundles over curves

本文通过将交叉上同调的Hodge-Deligne多项式与Donaldson-Thomas不变量关联，计算了曲线上半稳定向量丛模空间的交叉上同调，并推广至任意遗传范畴中半稳定对象的模空间。

Weight filtration on the cohomology of algebraic varieties

We show that the etale cohomology (with compact supports) of an algebraic variety $X$ over an algebraically closed field has the canonical weight filtration $W$, and prove that the middle weight part of the cohomology with compact supports of $X$ is a subspace of the intersection cohomology of a compactification $X'$ of X, or equivalently, the middle weight part of the (so-called) Borel-Moore homology of $X$ is a quotient of the intersection cohomology of $X'$. We are informed that this has been shown by A. Weber in the case $X$ is proper (and $k=\bC$) using a theorem of G. Barthel, J.-P. Bras…

Counting ramified coverings and intersection theory on spaces of rational functions I (Cohomology of Hurwitz spaces)

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations …

Eichler-Shimura Relation on Intersection Cohomology

We prove the Eichler-Shimura relation on intersection cohomolgoy of minimal compactifications of Shimura Varieties of Hodge type.

曲线上$\mathrm{SL}_n$-希格斯丛模空间的交截上同调

On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

本文利用Ngô型支撑不等式研究曲线上任意次数SL_n-希格斯丛模空间的交截上同调，证明了奇异情形下的支撑定理及Hausel-Thaddeus拓扑镜像对称猜想的一个版本，并推广了Harder-Narasimhan定理。

带权上同调、Hodge理论与Shimura簇的交上同调

Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties

本文证明了复Shimura簇的Baily-Borel紧化交上同调与重合Hodge结构的顶权商相关，引入了与特殊循环相关的规范循环类，将非紧Shimura簇的分析几何体积与拓扑项联系起来。

Intersection cohomology of quotients of nonsingular varieties

The purpose of this paper is to provide a way to compute the intersection cohomology of the GIT quotient of a nonsingular projective variety. We show that the middle perversity intersection cohomology of the GIT quotient $M//G$ is naturally embedded into the $G$ equivariant cohomology of $M^{ss}$ under an assumption satisfied by many interesting spaces. This embedding enables us to compute the intersection pairing in terms of the cup product structure of the equivariant cohomology ring. We also show that the image is a subspace defined by truncation. Overall, we can compute the intersection co…

Localization formulas of cohomology intersection numbers

We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of …

Intersection cohomology of a rank one local system on the complement of a hyperplane-like divisor

Under a certain condition A we give a construction to calculate the intersection cohomology of a rank one local system on the complement to a hyperplane-like divisor

Intersection cohomology of representation spaces of surface groups

We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(π_1(Σ),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$ equivariant cohomology of $Hom(π_1(Σ),G)$ where $Σ$ is a closed Riemann surface. This enables us to compute the intersection cohomology as a graded vector space with intersection pairing, in terms of the equivariant cohomology ring. The case where $G=SU(2)$ -- the moduli space of rank 2 holomorphic vector bundles of even degree -- is discussed in detail.

The intersection form in the cohomology of the moduli space of genus 0 $n$-pointed curves $H^*(\overline{M}_{0,n})$ and the explicit Künneth formula in quantum cohomology

We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of $\overline {M}_{0,n}$. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology and formal Frobenius manifolds. In the context of quantum cohomology this establishes the explicit Künneth formula.

Cohomology of complete intersections in toric varieties

This paper explicitly describes Hodge structures of complete intersections of ample hypersurfaces in compact simplicial toric varieties.

Intersection Cohomology of reductive varieties

We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.

抛物向量丛与曲线上向量丛模空间的交点上同调

Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves

本文提出纯几何框架，利用分解定理分析抛物模空间遗忘映射，几何证明递归公式，将0次模空间交点Betti数计算归约到1次情形。

代数拓扑中托里安品种的交集上同调Hodge模

The intersection cohomology Hodge module of toric varieties

本文研究了托里安品种的交集上同调Hodge模的Hodge滤波，通过Ishida复形计算反射微分 sheaf 的更高直接像，并给出了与交集上同调留数的精确公式。

Intersection cohomology of moduli spaces of sheaves on surfaces

We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.

Prym-Tjurin Constructions on Cubic Hypersurfaces

In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups for a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on both of the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi homomorphism induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients.

行列式平凡曲线上秩2 Higgs丛模空间交集上同调的爆破公式

A blowing-up formula for the intersection cohomology of the moduli of rank 2 Higgs bundles over a curve with trivial determinant

本文证明了行列式平凡曲线上秩2 Higgs丛模空间交集上同调的爆破公式，并据此在技术假设下推导出其庞加莱多项式。

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

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…

Complete Intersections of Two Quadrics and Galois Cohomology

For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.