Variation on Artin's vanishing theorem

A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,ω)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $φ:U\to X$ with $U$ an affine derived scheme, such that $(U,φ^*(ω))$ may be written explicitly in coordinat…

Lowest non-zero vanishing cohomology of holomorphic functions

We study the vanishing cycle complex $φ_fA_X$ for a holomorphic function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the monodromy eigenvalue decomposition may hold after a localization of $A$). Assuming the perversity of the shifted constant sheaf $A_X[d_X]$, we show that the lowest possibly-non-zero vanishing cohomology at $0\in X$ can be calculated by the restriction of $φ_fA_X$ to an appropriate nearby curve in the singular locus $Y$ of $f$, which is given by intersecting $Y$ with …

Further evaluation of Wahl vanishing theorems for surface singularities in characteristic $p$

Let $(\mathrm{Spec }R, \frak m)$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\geq 0$. We evaluate further the dimensions of the local cohomology groups which were treated by Wahl in 1975 as vanishing theorem C (resp. D) under the assumption that $p$ is a very good prime (resp. good prime) with respect to $(\mathrm{Spec }R, \frak m)$. We use Artin's classification of rational double points and completely determine the dimensions $\dim_k H_E^1(S_X)$, $\dim_k H_E^1(S_X\otimes \mathcal O_X(E))$, supplementing Wahl's theorems. In the proof we constr…

通过循环同调研究Chow群的变形

Deformations of Chow groups via cyclic homology

本文通过循环同调研究Chow群的变形，推广了Bloch等人的结果，解决了Chow群pro-可代表性与Hodge理论消失条件之间的联系问题。

Vanishing theorems for the negative K-theory of stacks

We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. is invertible on X). Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.

Effective behavior of multiple linear systems

We give optimal effective bounds for some well-known theorems on complex algebraic surfaces, which are respectively due to Serre, Zariski (1962), Castelnuovo (1897), Artin (1962, 1966), Benveniste (1984), Cutkosky and Srinivas (1993). These theorems are about Riemann-Roch problem (on the behavior of the function dim |nD| of n), vanishing theorems, base-point freeness and k-very ampleness of the linear systems |nD| and |nA+L|, where D is effective, A is nef and big and L is arbitrary. As a consequence, we obtain an effective version of Matsusaka's big theorem, and we give also examples to show …

Vanishing theorems for parabolic Higgs bundles

This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic Chern classes. This implies a general semipositivity theorem. This also implies a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.

Vanishing and injectivity theorems for Hodge modules

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.

Kodaira-Saito vanishing via Higgs bundles in positive characteristic

The goal of this paper is to give a new proof of a special case of the Kodaira-Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.

拟光滑栈爆破的广义消没定理

A Generalized vanishing theorem for Blow-ups of Quasi-smooth Stacks

该研究证明了拟光滑导出Artin栈导出爆破上拟凝聚层的广义消没定理，给出其在K理论、消奇、半正交分解及代数作用范畴化四方面的应用，并提出相关猜想。

Vanishing and semipositivity theorems for semi-log canonical pairs

We prove an effective vanishing theorem for direct images of log pluricanonical bundles of projective semi-log canonical pairs. As an application, we obtain a semipositivity theorem for direct images of relative log pluricanonical bundles of projective semi-log canonical pairs over curves, which implies the projectivity of the moduli spaces of stable varieties. It is worth mentioning that we do not use the theory of variation of (mixed) Hodge structure.

局部上同调的消失及其在霍奇理论中的应用

Vanishing of Local Cohomology with Applications to Hodge Theory

本文利用混合霍奇模理论与局部上同调研究极化霍奇结构变分在紧化下的延拓，分析其 de Rham 复形的分次部分及低次数上同调的霍奇结构。

Variation of numerical dimension of singular hermitian line bundles

The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension of singular hermitian line bundles. The other is an analytic injectivity theorem for log canonical pairs on surfaces, which can be seen as a partial answer for Fujino's conjecture.

韦伊猜想与仿射超曲面

Weil conjectures and affine hypersurfaces

本文通过将问题简化为超曲面情况，重新证明了有限域上光滑正则簇的黎曼假说，方法结合了变形到仿射超曲面、Artin消失定理及 perverse sheaves 基本性质。

混合霍奇结构的变化及其应用

Variation of mixed Hodge structure and its applications

本文研究复解析空间射影态射诱导的混合霍奇结构变化，并将其用于推广 Kollár 无挠定理与消失定理，为复解析空间射影态射的极小模型理论奠定基础。