G. van der Geer, T. Katsura
We introduce an invariant of varieties in positive characteristic which generalizes the a-number of an abelian variety. We calculate it in some examples and discuss its meaning for moduli.
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We prove the equality of the Makar-Limanov invariant and the modified Makar-Limanov invariant in the case of not necessary normal affine toric varieties. Also we give a combinatorial description of these invariants.
In this paper we show that quotients of smooth projective toric varieties by $μ_p$ in positive characteristics $p$ are toric varieties.
We construct a theory of motivic integration for smooth rigid varieties. As an application new invariants of degenerations are obtained.
Generically Flexible Varieties with an Invariant Divisor
本文构造了维数至少为3的光滑仿射簇，它们通用柔性但非柔性，且具有特殊自同构子群下的非平凡不变除数。
We study invariants of Calabi-Yau varieties in positive characteristic, especially the height of the Artin-Mazur formal group. We illustrate these results by Calabi-Yau varieties of Fermat and Kummer type.
We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfaces
We describe the quantum cohomology rings of a class of toric varieties. The description includes, in addition to the (already known) ring presentations, the (new) analogues for toric varieties of the sorts of quantum Giambelli formulas which exist already for Grassmannian varieties, flag varieties, etc.
We show that the Picard rank is constant in families of Fano varieties (in arbitrary characteristic) and we moreover investigate the constancy of the index.
We introduce two new general methods to compute the Chow motives of twisted flag varieties and settle a 20-year-old conjecture of Markus Rost about the Rost invariant for groups of type E7.
We present a recursive algorithm computing all the genus-zero Gromov-Witten invariants from a finite number of initial ones, for Fano varieties with generically tame semi-simple quantum (and small quantum) (p, p)- type cohomology, whose first Chern class is a strictly positive combination of effective integral basic divisors.
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.
By showing a structural result on how the Rozansky-Witten invariants of Hilbert schemes of points on a K3 surface and the Generalised Kummer varieties depend on the dimension, we give a method how to compute a large class of Rozansky-Witten invariants in terms of Chern numbers.
Generically flexible affine varieties with invariant divisors
本文构造了维数至少为4的正规仿射簇，其特殊自同构群具有开轨道，且轨道补集为余维一不变除子。
We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.
Integrality of genus zero Gopakumar-Vafa type invariants of semi-positive varieties
本文通过代数几何方法，将半正簇上亏格零 Gopakumar-Vafa 型不变量与量子 K-不变量建立联系，并借助后者的整性给出该整性猜想的另一证明。
For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version of the $δ$-threshold. We prove that for a Fano variety $(X, -K_X)$ and a connected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.
We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.
Nondefectivity of invariant secant varieties
本文证明一大类不变割线簇的非缺陷性，解决大维数 Grassmannian 割线猜想的多数情形，并改进 Chow、Segre-Veronese、Gaussian moment 等簇的非缺陷性、可辨识性与一般秩界。
Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant higher-genus situations. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.
Unlike in characteristic 0, there are no non-trivial smooth varieties over an algebraically closed field k of characteristic p>0 that are contractible in the sense of etale homotopy theory.
Tangent Weights and Invariant Curves in Type A Bow Varieties
本文对 A 型 Cherkis 弓形簇中的环面不变曲线进行了完整分类，提出基于蝶形图、杨图和二元列联表的组合编码，并建立新的切权公式，进而用于计算弓形簇的 1-骨架。
We show that every 3-dimensional affine normal quasihomogeneous $SL(2)$-variety has an equivariant resolution of singularities given by an invariant Hilbert scheme. This article treats the case where such $SL(2)$-variety is toric. The non-toric case is considered in the forthcoming article [Kub18].
Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain 'motivic integral', living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization t…
We prove Bogomolov's inequality for semistable sheaves on product type varieties in arbitrary characteristic. This gives the first examples of varieties with positive Kodaira dimension in positive characteristic on which Bogomolov's inequality holds for semistable sheaves of any rank. The key ingredient in the proof is a high rank generalization of the slope inequality established by Xiao and Cornalba-Harris. This Bogomolov's inequality is applied to study the positivity of linear systems and semistable sheaves and construct Bridgeland stability conditions on product type surfaces in positive …
We study invariant divisors on the total spaces of the homogeneous deformations of rational complexity-one T-varieties constructed by Ilten and Vollmert. In particular, we identify a natural subgroup of the Picard group for any general fiber of such a deformation, which is canonically isomorphic to the Picard group of the special fiber. This isomorphism preserves Euler characteristic, intersection numbers, and the canonical class.
We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4K_X| induces a birational map.
Isolated torus invariants and automorphism groups of rigid varieties
该研究针对刚性仿射簇自同构群单位分支为环面的猜想，在环面簇与复杂度1环面作用簇上完成证明，给出刚性簇上m-悬垂的刚性判别准则并研究对应自同构群。
The Konno invariant of a projective variety X is the minimum geometric genus of the fiber of a rational pencil on X. It was computed by Konno for surfaces in P^3, and in general can be viewed as a measure of the complexity of X. We estimate Konno(X) for some natural classes of varieties, including sharp asymptotics for polarized K3 surfaces. In an appendix, we give a quick proof of a classical formula due to Deligne and Hoskin for the colength of an integrally closed ideal on a surface.
For any smooth projective variety with a C* action, we reduce the problem of computing its Gromov-Witten invariants to the similar problem for its fixed locus. Starting from the stacky version of variation of GIT for our variety, we construct the building blocks for the fixed loci of the moduli space of stable maps. We use this construction to compute their contribution to the virtual fundamental class.
In this paper, we study the set $R_g^{(p)}$ of possible Picard numbers of abelian varieties of dimension $g$ over algebraically closed fields of characteristic $p>0$. We show that many of the results for complex abelian varieties have analogues in positive characteristic: non-completeness in dimension $g \geq 2$, asymptotic completeness as $g \rightarrow +\infty$, structure results for abelian varieties of large Picard number. On the way, we highlight and discuss new characteristic $p>0$ features and pathologies: non-additivity of the range of Picard numbers, supersingularity index of an abeli…
The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset with a G_m-action or G_a-action. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincare characteristic for the Chow var…
We prove a stronger version of a criterion of rationality given by Ionescu and Russo. We use this stronger version to define an invariant for rational varieties (we call it rationality degree), and we classify rational varieties for small values of the invariant.
In characteristic zero, if a quotient variety has a crepant resolution, the Euler characteristic of the crepant resolution is equal to the number of conjugacy classes of the acting group, by Batyrev's theorem. This is one of the McKay correspondence. It is natural to consider the analogue statement in the positive characteristic. In this paper, we present sequences of crepant resolutions of quotient varieties in the positive characteristic and show that one of the sequences gives a counterexample to the analogue statement of Batyrev's theorem.
We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a generalized version of Hochschild homology. Furthermore, using techniques coming from birational geometry, we establish the derived invariance of the Albanese dimension for varieties having non-negative Kodaira dimension. We apply our machinery to study the derived invariance of the holomorphic Euler characteristic and of certain Hodge numbers for special classes of varieties. Further applications con…
Nash blowups of 2-generic determinantal varieties in positive characteristic
本文通过显式描述toric结构并证明Nash吹胀组合学无特征依赖，得出正特征域上2-generic行列式簇的Nash吹胀非奇异。
Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its singular locus and, in the hypersurface case, Milnor numbers. We obtain simple proofs of these formulas, recovering along the way several classically known results.
We show that a holomorphic two-form $θ$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\mgn(X,β)$ to the locus where $θ$ degenerates; it then enables us to define the localized GW-invariant, an algebro-geometric analogue of the local invariant of Lee and Parker in symplectic geometry, which coincides with the ordinary GW-invariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GW-invariants of minimal general type surfaces with p_g>0 conjectured by Maulik and Pandharipande.
Computing positive tropical varieties and lower bounds on the number of positive roots
本研究提出两种计算代数簇正热带化的有效工具，给出热带几何基本定理与横截相交定理的实数版本，据此提出计算参数化多项式方程组正实根组合界的算法，并探讨其在化学反应网络正稳态数量研究中的应用。
Segre classes and invariants of singular varieties
本文综述了Segre类在定义和研究奇异空间不变量方面的应用，重点讨论了数值不变量、奇异簇特征类及Le循环类，并回顾了代数几何与相交理论的相关背景。
In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an étale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization of toric varieties. We further prove that, without assumption of global $F$-splitting, a smooth projective separably rationally connected variety of arbitrary characteristic with numerically flat log cotangent bundle is a toric variety.
Bogomolov's inequality and Higgs sheaves on normal varieties in positive characteristic
本文证明了正特征下正规射影簇上的Bogomolov不等式，并由此得出新限制定理与有界性结果，同时重新定义正规簇上的Higgs层，证明了任意特征下半稳定对数Higgs层的限制定理及Bogomolov型不等式。
We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and classify Fano varieties whose anti-canonical divisors have Seshadri constants $n$. In characteristic $p>5$ and dimension $3$, we also show that Fano varieties $X$ with Seshadri constants $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ (for some fixed $ε>0$) have bounded anti-canonical degrees.
We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $Φ(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $α(X,B)^{d-1}\mathrm{vol}(-(K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.
We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general type so that $X$ is canonically fibred by $d$-dimensional varieties $F$ with $p_g(F)\geq N$. This disproves the desired boundedness parallel to Beauville's boundedness theorem in the surface case.
We find all homogeneous symplectic varieties of connected reductive algebraic groups that admit an invariant linear connection.
Invariants of the singularities of secant varieties of curves
本文研究光滑射影曲线的 $k$-平面割线簇的奇点不变量，计算了任意曲线的交点上同调，并证明有理正规曲线的所有割线簇均为满足 Poincaré 对偶的有理同调流形，同时计算了最大非平凡割线簇的邻近圈与消没圈。
Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by nai…
Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.
We prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of Kawamata's theorems: a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the Albanese map when the Frobenius stable Kodaira dimension is zero.
This article is a continuation of [Kub18], which proves that if a $3$-dimensional affine normal quasihomogeneous $SL(2)$-variety $E$ is toric, then it has an equivariant resolution of singularities given by an invariant Hilbert scheme $\mathcal H$. In this article, we consider the case where $E$ is non-toric and show that the Hilbert-Chow morphism $γ: \mathcal H \to E$ is a resolution of singularities and that $\mathcal H$ is isomorphic to the minimal resolution of a weighted blow-up of $E$.
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
Gromov-Witten and Welschinger invariants of del Pezzo varieties
通过比较二维与三维 del Pezzo 簇，建立了计算某些三维 del Pezzo 簇的亏格-0 Gromov-Witten 和 Welschinger 不变量的公式，推广了 E. Brugallé 和 P. Georgieva 在 2016 年对三维射影空间给出的结果。
Effective local differential topology of algebraic varieties over local fields of positive characteristics
本文研究了代数簇上局部域的微分拓扑性质，证明了隐函数定理的类比结果，并分析了光滑测度在子浸入下的行为。
A note on Fourier-Mukai partners of abelian varieties over positive characteristic fields
本文研究正特征域上阿贝尔簇的Fourier-Mukai伙伴，将Krug-Sosna关于Kummer栈的导出等价结果推广到奇特征情形，证明阿贝尔曲面导出等价与Kummer曲面同构等价，并建立超奇异阿贝尔簇的导出Torelli定理。
Let $q$ be a nondegenerate quadratic form on $V$. Let $X\subset V$ be invariant for the action of a Lie group $G$ contained in $SO(V,q)$. For any $f\in V$ consider the function $d_f$ from $X$ to $C$ defined by $d_f(x)=q(f-x)$. We show that the critical points of $d_f$ lie in the subspace orthogonal to ${\mathfrak g}\cdot f$, that we call critical space. In particular any closest point to $f$ in $X$ lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we c…
On positive cones of finite quotients of a normal variety
研究正规射影簇的有限平坦商簇的正性性质，建立了商簇数值群与正性锥与原簇的关系。
Stabilized measures of association: comparing the birational geometry of varieties having different dimensions
本文推广了Lazarsfeld-Martin的关联度量，得到比较不同维数簇双有理几何的不变量，并给出实例与计算。
Given the Luna-Vust invariants of a spherical variety, we determine the Luna-Vust invariants of the spectrum of its Cox ring. In particular, we deduce an explicit description of the divisor class group of the Cox ring. It follows that every spherical variety admits iteration of Cox rings. Moreover, we obtain a combinatorial proof of the fact that the Cox ring is determined by the spherical skeleton, which is a known result following from the description of the Cox ring due to Brion. Finally, we show that a conjectural combinatorial smoothness criterion can be reduced to the case of a factorial…
This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kähler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kähler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number o…
Non-cancellative varieties, maximal tori, and the Makar-Limanov invariant
研究广义扎里斯基消去问题，构建新反例，通过反例上柱体的极大环面及非刚性三项式簇上柱体的性质研究，证明相关结论并得出其马卡洛夫 - 利马诺夫不变量平凡的结果。
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