Clint McCrory, Adam Parusinski
The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).
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We study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of $C^n$ by using the theory of motivic Milnor fibers. The results will be explicitly described by the mixed volumes of the faces of Newton polyhedrons.
Weight of the De Rham-Betti Structures of Abelian Varieties
本文证明了任何定义在有理闭包上的阿贝尔簇的德拉姆-贝蒂群必然包含乘法群作为同源群，从而排除了阿贝尔簇在奇数次上同调群中非零德拉姆-贝蒂类的存在性。
We compute the rational cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci $W^r_d(C)$, parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$.
De Rham-Betti Groups of Type IV Abelian Varieties
本文基于Tannakian观点研究IV型简单阿贝尔簇的de Rham-Betti结构，证明了简单CM阿贝尔四重簇及其自同态代数为4次CM域的简单阿贝尔四重簇的de Rham-Betti群与其Mumford-Tate群一致。
Smooth Calabi-Yau varieties with large index and Betti numbers
本文构造了在每维中具有双指数增长指数的光滑射影Calabi-Yau变种，并探讨了其拓扑不变量的极端性质，认为这些变种在每维中可能是极值的。
Irreducible symplectic varieties with a large second Betti number
本文证明了非紧Lagrangian fibration存在不可约辛紧化，并应用于证明包含固定cubic fourfold的cubic fivefolds的relative Jacobian fibration可被第二Betti数至少为24的不可约辛簇紧化，属于新形变类型。
Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for $GL_n(\mathbb{C})$. A key component of their argument is that the Betti numbers of regular Hessenberg varieties for $GL_n(\mathbb{C})$ are palindromic. In this paper, we extend this result to all reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.
We give two constructions of functorial topological realizations for schemes of finite type over the field $\mathbb{C}(\!(t)\!)$ of formal Laurent series with complex coefficients, with values in the homotopy category of spaces over the circle. The problem of constructing such a realization was stated by D. Treumann, motivated by certain questions in mirror symmetry. The first construction uses spreading out and the usual Betti realization over $\mathbb{C}$. The second uses generalized semistable models and log Betti realization defined by Kato and Nakayama, and applies to smooth rigid analyti…
Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds.
We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers i…
Betti numbers and pseudoeffective cones in 2-Fano varieties
本文提出（弱）k-Fano 簇的定义并猜想其伪有效 k-闭链锥具有多面体性，通过计算一大类 k-Fano 簇的 Betti 数证明了该猜想在指数≥n-2 的 2-Fano 簇等特殊情形下成立，并完成弱 2-Fano 簇的分类。
We present a counterexample to the conjecture of Bihan, Franz, McCrory, and van Hamel concerning the maximality of toric varieties. There exists a six dimensional projective toric variety X with the sum of the mod 2 Betti numbers of X(R) strictly less than the sum of the mod 2 Betti numbers of X(C).
It is well-known that the Eulerian polynomials, which count permutations in $S_n$ by their number of descents, give the $h$-polynomial/$h$-vector of the simple polytopes known as permutohedra, the convex hull of the $S_n$-orbit for a generic weight in the weight lattice of $S_n$. Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this paper we derive recurrences for the $h$-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight…
We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with equivariant homology for compact nonsingular sets, but is different in general. We lay emphasis on the particular case of $Z/2\Z$, and give an application to the study of the singularities of Nash function germs via an analog of the motivic zeta function of Denef & Loeser.
It was noticed in a very recent preprint of T. Eguchi, K. Hori, and Ch.-Sh. Xiong (hep-th/9703086) that a curious identity between Betti numbers and Chern classes holds for many examples of Fano varieties. The goal of this paper is to prove that for varieties with trivial odd cohomology groups this identity is equivalent to having zero Hodge numbers $h^{p,q}$ for $p\neq q$.
On sums of Betti numbers of affine varieties
本文证明由次数至多d的多项式定义的仿射N空间子簇的ℓ-adic贝蒂数之和有界，并回答了Katz的问题。
Betti number bounds for varieties and exponential sums
本文利用周层性质给出仿射簇紧支撑Betti数的上界，并应用于有限域上zeta函数与指数和L函数，改进经典结果，完全交情形渐近最优。
Terminalizations of quotients of Fano varieties of lines on cubic fourfolds
本文研究了立方四维流形上线的Fano簇商的终端化，计算了相关拓扑不变量，并识别出新的四维不可约解析辛流形变形类。
Syzygies of secant varieties of curves of genus 2
本文利用Boij-Söderberg理论，通过计算Betti图的特殊位置确定贡献的纯图，完整计算了亏格2曲线的割线簇的Betti图，解决了权重k+1线性链长度未知的问题。
We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$, $\mathbb{P}^k_{\mathrm{C}}$, as well as products of projective spaces, defined by $r$ polynomials of degrees at most $d$ as a function of $p,d$ and $r$. Unlike previous bounds these bounds are independent of $k$, the dimension of the ambient space. We also prove as consequences of our technique certain homological and representational stability results for …
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.
We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially quadratic polynomials. The bounds we present in the case of partially quadratic polynomials offer a significant improvement over what was previously known. Finally, we extend a result of Barone and Basu on …
Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they have the same Betti numbers.
The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described explicitly. We build on this work by showing that for most of such varieties, the defining ideals are componentwise linear and in particular the componentwise linearity is suitable for classifying the Betti tables of such varieties. As an application, we compute the Betti table of all varieties with almost maximal degree and componentwise…
A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $Δ$ there are families of hypersurfaces with the Newton polytopes $(λΔ)_{λ\in \mathbb{N}}$ that are asymptotically maximal when $λ$ tends to infinity. We also show that these results ge…
Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using …
Density of integral points in the Betti moduli of quasi-projective varieties
本文证明了在参数化拟射影簇上具有固定代数整数迹的 $SL_2$-局部系统的特征簇中整点潜在稠密，通过化简至黎曼曲面情形并构造在映射类群下 Zariski 稠密轨道的整点完成证明。
We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $\mod 2$ Betti numbers for the real form of complex manifolds of complex degree less than $d$.
We construct Eagon--Northcott cycles on Hurwitz space and compare their classes to Kleiman's multiple point loci. Applying this construction towards the classification of Betti tables of canonical curves, we find that the value of the extremal Betti number records the number of minimal pencils. The result holds under transversality hypotheses equivalent to the virtual cycles having a geometric interpretation. We analyse the case of two minimal pencils, showing that the transversality hypotheses hold generically.
Betti bounds for spaces of curves on varieties and Manin's conjecture for quartic del Pezzo surfaces
该研究证明了射影簇上固定亏格曲线态射空间的紧支撑贝蒂数的一致指数界，并将其应用于建立分裂四次德尔佩佐曲面马宁猜想的高亏格函数域版本。
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. Using the exponential tangent cones to these jump loci, we show that each Ω-invariant is contained in the complement of a union of …
Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work…
De Rham-Betti Groups of Type IV Abelian Fourfolds
研究\(\bar{\mathbb{Q}}\)上几类阿贝尔簇的德拉姆 - 贝蒂群，通过采用不同于常规的方法，利用相关结果、分析及约束，确定了简单CM阿贝尔四维簇等的\(\mathrm{G}_{\mathrm{dRB}}(A)=\mathrm{MT}(A)\) 。
The equivalence of several conjectures on independence of $\ell$
本文研究有限域代数闭包上概形平展上同调的ℓ独立性猜想，证明任意概形紧支Betti数的ℓ独立性等价于光滑射影概形闭链同调等价的ℓ独立性，并推出光滑拟射影情形的ℓ独立性可推广到任意分离有限型k-概形。
L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem
本文利用弱一般 Nakano 消没定理，研究 Abel 覆盖塔中规范化 Hodge 数与多重亏格的增长及收敛，并计算相关非正则簇的 L²-Betti 数，推广 Kollár 的可乘性结果。
Betti numbers of the moduli space of Higgs bundles over a real curve
本文基于Mellit、Fedorov-Soibelman-Soibelman和Schiffmann的动机公式，推导出实曲线上的稳定Higgs丛模空间M_r^d的Z_2-Betti数公式，并证明虚拟的Z_2庞加莱多项式是实数上的动机度量。
The comparison theorem for a smooth projective variety $X$ over $\mathbb{C}$ tells us that the Betti numbers are independent of $l$. We aim to understand the $l$ independence of Betti numbers for smooth projective varieties $X$ over $k$, where $k$ is an algebraic extension of $\mathbb{F}_p$.
A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence simple unified proofs are obtained for formulas of Poincare polynomials of toric hyperkahler varieties, Poincare polynomials of Hilbert schemes of points and twisted ADHM spaces of instantons on C^2 and Poincare polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable represent…
We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of delta-vectors of reflexive polytopes.
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。
We define the Z/2Z Hodge spaces H_{pq}(Σ) of a fan Σ. If Σis the normal fan of a reflexive polytope Δthen we use polyhedral duality to compute the Z/2Z Hodge Spaces of Σ. In particular, if the cones of dimension at most e in the face fan Σ^* of Δare smooth then we compute H_{pq}(Σ) for p<e-1. If Σ^* is a smooth fan then we completely determine the spaces H_{pq}(Σ) and we show the toric variety X associated to Σis maximal, meaning that the sum of the Z/2Z Betti numbers of X(R) is equal to the sum of the Z/2Z Betti numbers of X(C).
First Betti number of real Calabi-Yau hypersurfaces: examples
本文通过镜像对称性提出新定理，用于计算实 Calabi-Yau 超曲面的首个贝蒂数，并探讨其拓扑应用。
De Rham-Betti classes with coefficients
本文证明了椭圆曲线积上的L-德拉姆-贝蒂类是L线性组合的代数类，并给出了K3表面的全局德拉姆-贝蒂托里利定理。
Real Toric Varieties: Interactions between their Geometry and their Topology
本文研究实环带簇的拓扑，特别关注环面不可分裂的情形，通过典型纤维化、贝蒂数计算、完全代数同调及可定向性判据的推广，并给出了三维不可分裂环面等变嵌入的拓扑分类。
We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers. Our methods indicate a link between $p$-adic measure and the minimal model theory.
This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of actions of Atkin-Lehner operators and the class group of $K$. Thus we deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties…
We use cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.
We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…
Maximal real varieties from moduli constructions
本文通过研究极大实代数簇上向量丛、Higgs丛、稳定层及Hilbert概型等模空间，构造了一系列新的极大实代数簇，并证明极大性在等变形式性意义下是motivic性质。
We show that for maximal Cohen-Macaulay modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties $X$ computation of Betti numbers can be reduced to computations of dimensions of certain $\operatorname{Hom}$ groups in the bounded derived category $D^b(X)$. In the simplest case of a smooth elliptic curve $E$ imbedded into $\mathbb{P}^2$ as a smooth cubic we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category $D^b(E)$ in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approa…
A complex algebraic variety defined over the reals is maximal when the sum of its Betti numbers for Borel Moore homology with $\zz$ coefficients coincides with the sum of the Betti numbers of its real part. We will show in this paper that toric varieties of dimension 4 are maximal.
We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential $p$-forms on $X$ is bounded by $p(em+1)D$, where $e$, $m$, and $D$ are the maximal codimension, dimension, and degree, respectively, of all irreducible components of $X$. It follows that, for a union $V$ of generi…
We show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology onto the weight complex of McCrory and Parusiński. The group action enriches the information contained in the induced equivariant weight spectral sequence from which we can not extract finite additivity directly if the group is of even order. Still, we manage to construct additive invariants from its elementary bricks, and prove that they are induced from the very equivariant geometry of real algebraic varieties with a…
We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.
We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves.
In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervair resolution, the Betti numbers of this ini…
In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebrai…
Deformations of half-canonical Gorenstein curves in codimension four
本文在同一 Hilbert 概型中构造平坦形变，将若干奇异 Gorenstein 余维四簇实现为此前已知非奇异簇的特殊化，从而连接两类构造。
Hierarchical structure of graded Betti numbers in the quadratic strand
本文研究graded Betti数的分层结构，建立了一般性的层次体系，并证明了广义K_{p,1}定理，揭示了二次链中极值行为的完整描述。
We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.
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