On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbb{P}^2$

Perverse coherent sheaves on blow-up. II. wall-crossing and Betti numbers formula

This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli…

Spectral action in Betti Geometric Langlands

Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bun_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bun_G(X)$. We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of $X$. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli $Loc_{G^\vee}(X)$ of dual group local systems. In particular, we establish the "automorphic to Galois" direction in the Betti Geometric Langlands corr…

On the geometry of the moduli space of sheaves supported on curves of genus two in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 2, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers and we give a classification of the stable sheaves involving locally free resolutions.

Motivic measures of moduli spaces of 1-dimensional sheaves on rational surfaces

We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.

Torus action on the moduli spaces of plane sheaves

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.

曲面上一维层模空间的渐近行为

Asymptotic Behaviors of Moduli of One-dimensional Sheaves on Surfaces

研究曲面上曲线类一维层模空间的 Betti 数与 Picard 数渐近行为，在 del Pezzo 曲面及充分正定曲线类情形确定相交上同调 Betti 数，并提出局部 del Pezzo 曲面精细 BPS 不变量的 $P=C$ 猜想。

The first two Betti numbers of the moduli spaces of vector bundles on surfaces

We determined the first two Betti numbers of the moduli of rank two stable sheaves on an arbitrary algebraic surface

On the geometry of the moduli space of sheaves supported on curves of genus four contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of bidegree (3, 3) contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs.

Betti 数的估计及反常层的相对 Hermite-Minkowski 定理

Estimates for Betti numbers and relative Hermite-Minkowski theorem for perverse sheaves

本文建立特征 p 下可构造层 Betti 数仅依赖秩、分层与野性分歧的估计，并推广 Deligne 有限性至奇异概形上的反常层。

Moduli of sheaves supported on curves of genus two in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We give a classification of the stable sheaves involving locally free resolutions or extensions. We compute the Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs.

Desingularized moduli spaces of sheaves on a K3, II

In preprint alg-geom/9708009 we have constructed a (ten-dimensional) symplectic desingularization of the moduli space of rank-two torsion-free semistable sheaves on a $K3$, with trivial determinant and second Chern class equal to 4. In the present paper we prove that (every deformation of) this desingularization is a new irreducible symplectic variety. More precisely we show that the desingularization is irreducible symplectic, and that its second Betti number is at least 24.

Invariants of moduli spaces of stable sheaves on ruled surfaces

We compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the conjecture of Manschot for arbitrary rank sheaves on the Hirzebruch surfaces.

On the moduli scheme of stable sheaves supported on cubic space curves

We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the relation between M and the Hilbert scheme Hilb_H(P_3) of cubic space curves. We also compute the Betti numbers of the Chow groups of the moduli space M.

Intersection Poincare polynomial for Nagaraj-Seshadri moduli space

We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.

Betti数的均匀逼近

Uniform approximation of Betti numbers

本文将Lefschetz关于仿射概形上同调逼近的原理实现为均匀形式，利用Jouanolou技巧、Deligne一般基变换公式的动形版本及反常层理论，提出了一种独立于上同调界定域上拟射影概形Betti数的新方法。

Motivic measures of the moduli spaces of pure sheaves on $\mathbb{P}^2$ with all degrees

Let $\mathcal{M}(d,χ)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $χ$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We compute the $A$-valued motivic measure $μ_A(\mathcal{M}(d,χ))$ of $\mathcal{M}(d,χ)$ and get explicit formula in codimension $D:=ρ_d-1$, where $ρ_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,χ)$ when $d$ is coprime to $χ$.

关于局部$\mathbb{P}^2$的领先Gopakumar--Vafa不变式及$\mathbb{P}^2$上一维层模空间的领先Betti数

On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$

本文用拓扑顶点形式推导局部$\mathbb{P}^2$的Gromov--Witten不变式领先项公式，并应用它推广Guo--Zhou公式及证明关于$\mathbb{P}^2$上一维层模空间领先Betti数的猜想。

Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

The geometry of the moduli space of one-dimensional sheaves

Let $\mathbf{M}_d$ be the moduli space of stable sheaves on $\mathbb{P}^2$ with Hilbert polynomial $dm+1$. In this paper, we determine the effective and the nef cone of the space $\mathbf{M}_d$ by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem. We also present the stable base locus decomposition of the space $\mathbf{M}_6$. As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction i…

驯化局部Betti几何Langlands对应

Tame local Betti geometric Langlands

通过证明普遍仿射Hecke范畴的谱与自守实现的幺半等价，证明了驯化分歧局部Betti几何Langlands对应，并为Bezrukavnikov定理提供了新论证。

Modular varieties of D-elliptic sheaves and the Weil-Deligne bound

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.

通过插值与Bridgeland稳定性研究$\mathbb{P}^2$上层化（syzygies）的几何

Geometry of syzygies of sheaves on $\mathbb{P}^2$ via interpolation and Bridgeland stability

本文证明$\mathbb{P}^2$上一般半稳定层的极小自由消解中的子复形决定模空间有效除子锥的极值射线，并利用层化给出若干层模空间的可移动锥与Mori分解的新计算。

射影平面上单维层模空间的庞加莱多项式

Poincaré polynomials of moduli spaces of one-dimensional sheaves on the projective plane

本文通过 Bousseau 的精细层/Gromov-Witten 对应证明了射影平面上单维层模空间庞加莱多项式的整除性猜想，并提出利用局部几何的 Gromov-Witten 不变量计算庞加莱多项式的新方法，给出了低次数显式结果及主导 Betti 数的闭公式和猜想。

$\bP^2$ 上一维层模空间环结构的上同调 $\u03c7$ 依赖性

Cohomological $χ$-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb{P}^2$

本文证明 $\bP^2$ 上一维层模空间 $M_{d,\u03c7}$ 的上同调环通常随 Euler 特征 $\u03c7$ 改变，并据此在排除显然对称性后推出其拓扑差异，强化了 Woolf 的代数簇区分结果。

$O(-2)$ Blow-up Formula via Instanton Calculus on $\hat{C^2/Z_2}$ and Weil Conjecture

We calculate Betti numbers of the framed moduli space of instantons on $\hat{{\bf C}^2/{\bf Z}_2}$, under the assumption that the corresponding torsion free sheaves $E$ have vanishing properties ($Hom(E,E(-l_\infty))=Ext^2(E,E(-l_\infty))=0$). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers of the moduli spaces of stable sheaves on $X$ with an $A_1$-singularity and that on $\hat{X}$ blow-uped at the singularity, by using Weil conjecture. We call this the $O…

Relative compactified Jacobians of linear systems on Enriques surfaces

We study certain moduli spaces of sheaves on Enriques surfaces thereby obtaining, in every odd dimension, new examples of Calabi-Yau manifolds. We describe the geometry (canonical bundle, fundamental group, second Betti number and certain Hodge numbers) of these moduli spaces showing, in partial analogy to the well-known case of sheaves on K3 or Abelian surfaces, how the geometry of the surface reflects that of the moduli space itself.

Betti数界与指数和

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。

Compactified moduli spaces of rational curves in projective homogeneous varieties

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.

具有自由切层的代数簇

Varieties with free tangent sheaves

提出T-平凡代数簇概念，研究特征p下其与阿贝尔簇的关联，证明可分支配的T-平凡簇仅在p≤3存在，还得到有限平展覆盖后的纤维化结果及提升特征零情形的覆盖结论。

Multiplicities and Betti numbers in local algebra via lim Ulrich points

This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. T…

Symmetries and stabilization for sheaves of vanishing cycles

Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $Φ:U\to U$ is an isomorphism with $f\circΦ=f$ and $Φ\vert_X=$id$_X$. Then $Φ$ induces an isomorphism $Φ_*:PV_{U,f}\to PV_{U,f}$. We show that $Φ_*$ is multiplication by det$(dΦ\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-o…

Moduli spaces of reflexive sheaves of rank 2

Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,σ), σ\in H^0(\sF), and the functor which deforms the corresponding pair (X,ξ) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without a…

On a compactification of the moduli space of the rational normal curves

For any odd $n$, we describe a smooth minimal (i.e. obtained by adding an irreducible hypersurface) compactification $\tilde S_n$ of the quasi-projective homogeneous variety $S_{n}=PGL(n+1)/SL(2)$ that parameterizes the rational normal curves in $P^n$. We show that $\tilde S_{n}$ is isomorphic to a component of the Maruyama scheme of the semi-stable sheaves on $P^n$ of rank $n$ and Chern polynomial $(1+t)^{n+2}$ and we compute its Betti numbers. In particular $\tilde S_{3}$ is isomorphic to the variety of nets of quadrics defining twisted cubics, studied by G. Ellinsgrud, R. Piene and S. Strøm…

双椭圆曲面上半稳定层模空间的拓扑与几何

Topology and geometry of moduli spaces of semistable sheaves on bielliptic surfaces

研究双椭圆曲面上半稳定层模空间的拓扑与低次霍奇理论，通过特定纤维化等方法构造满同态计算相关贝蒂数，扩展计算到正秩向量，证明\(H^2(F,\mathbb{C})\)性质，在一定条件下得出\(F\)的相关结论及\(M^\circ_{H,S}(\mathbf{v})\)的性质。

关于$(\u0001mathbb{P}^2,E)$的N. Takahashi猜想的证明及一个精细的层/Gromov–Witten对应

A proof of N.Takahashi's conjecture for $(\mathbb{P}^2,E)$ and a refined sheaves/Gromov-Witten correspondence

本文证明相对$\u0001mathbb{P}^2$极大接触理论中的N. Takahashi猜想，并借助热带几何、导出范畴穿墙与半稳定层模空间，建立新的精细层/Gromov–Witten对应。

导出二项环 I：对数方案的整数贝蒼威上同调

Derived binomial rings I: integral Betti cohomology of log schemes

本文引入并研究了在无界导出范畴中的一种导出二项单子，用于计算自由导出二项环，并展示了奇异上同调在连接 nilpotent 空间中的全忠实嵌入。

来自模空间构造的极大实代数簇

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。

刚性同痕类的拓扑学图不变量

Graph invariants from the topology of rigid isotopy classes

本文定义了一类新的图不变量，通过研究图在欧氏空间中几何实现模空间的拓扑，利用类似Floer同调的极限过程，并给出刚性同痕类数量的渐近估计。