Walter D. Neumann, Jonathan Wahl
The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a $\Q$-Gorenstein singularity.
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Irregular cusps of ball quotients
本文研究球商环面紧化边界的分支除子，提出低斜率尖点形式技巧以判别一般类型，分类判别核下的非正则尖点并构造例子，还分析复球嵌入 IV 型域时尖点正则性的映射关系。
The classification of smooth quotients of abelian surfaces
本文分类了由有限群作用在阿贝尔曲面上得到的光滑射影曲面，完整刻画了这类商面的几何结构。
We study finite quotients of abelian varieties (fqav for short) i.e. quotients of abelian varieties by finite groups. We show that Q-abelian varieties (i.e. fqav's with $\mathbb Q$-linearly trivial canonical divisors) are characterized by the existence of quasiétale polarized (or int-amplified) endomorphisms. We show that every fqav has a finite quasiétale cover by the product of an abelian variety and a $\mathbb Q$-Fano fqav. Using such coverings, we give a characterization of $\mathbb Q$-Fano fqav's, and show that $\mathbb Q$-Fano fqav's and Q-abelian varieties are ``building blocks'' of gen…
We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence needs only simple computations comparing with the results ever known.
We give an explicit characterization of all principally polarized abelian varieties $(A,Θ)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Θ$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobians of curves.
In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded $\mathbf{Q}$-resolutions of plane curve singularities and abstract $\mathbf{Q}$-resolutions of surfaces.
Let X' be the toroidal compactification of the quotient of the complex 2-ball by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is an abelian surface, birational to X'. The present work can be viewed as an illustration for the presence of a plenty of non-compact co-abelian torsion free toroidal compactifications. More precisely, it shows that all the admissible values for the volume of a torsion free quotient of the complex 2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers. The article provides three types of infinite series of finite unra…
In [3], it is proved that the quotient of an abelian variety $A$ by a finite order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$ is enough to guarantee that $A/\langle g\rangle$ has at least one rational curve.
We make precise conjectures relating the genus zero Gromov-Witten theory of a nonabelian GIT quotient X//G to that of the associated abelian quotient X//T by a maximal torus T in G.These conjectures imply in particular closed formulas for the J-functions (that is, the generating functions for 1-point Gromov-Witten invariants) of all generalized flag manifolds of classical A, B, C and D types. These formulas are proved in the second part of the paper.
We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.
The multiplicity of abelian covers of splice quotient singularities
本文提出通过解析图和伽罗瓦群计算splice quotient奇点的abelian覆盖的multiplicity的方法。
The quotient space of a $K3$ surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, we will study finite groups which act faithfully on $K3$ surfaces such that the quotient space are smooth. In particular, we will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a $K3$ surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.…
Fibrations associated to smooth quotients of abelian varieties
本文研究阿贝尔簇被保持原点的有限自同构群作用所得光滑商的结构，将其刻画为阿贝尔簇上以射影空间乘积为纤维的纤维化，并分类纤维粘合方式、证明一般情形为纤维积。
We consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $\mathbb{P}$-functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic cur…
Equivariant deformation of minimally elliptic singularities
本文研究了有限群作用下极小椭圆曲面奇点的等变变形分量，探讨了简单椭圆与尖点奇点的商的例子，并在局部完全交情形下识别出具有完美障碍理论的一般型曲面的等变变形分量。
The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a K3 surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its Néron--Severi group. This allows one to describe the coarse moduli space of the K3 surfaces which are (rationally) $G$-covered by Abelian or K3 surfaces (in the latter case $G$ is an Abelian group). If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result was already known, so we extend it to the other cases. Moreover, we p…
Quotients of abelian varieties by reflection groups
本文通过逐例分析证明复反射群作用下阿贝尔簇的商为加权射影空间，并将结果推广至有限特征及超奇异阿贝尔簇上的四元数反射群作用。
In this paper we prove a characterization of quotients of Abelian varieties by the actions of finite groups that are free in codimension-one via some vanishing conditions on the orbifold Chern classes. The characterization is given among a class of varieties with mild singularities that are more general than quotient singularities, namely among the class of klt varieties. Furthermore we show that over a projective klt variety, any semistable reflexive sheaf with vanishing orbifold Chern classes can be obtained as the invariant part of a locally-free sheaf on a finite Galois cover whose associa…
Let $A$ be an abelian surface and let $G$ be a finite group of automorphisms of $A$ fixing the origin. Assume that the analytic representation of $G$ is irreducible. We give a classification of the pairs $(A,G)$ such that the quotient $A/G$ is smooth. In particular, we prove that $A=E^2$ with $E$ an elliptic curve and that $A/G\simeq\mathbb{P}^2$ in all cases. Moreover, for fixed $E$, there are only finitely many pairs $(E^2,G)$ up to isomorphism. This fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origi…
We give a complete classification of smooth quotients of abelian varieties by finite groups that fix the origin. In the particular case where the action of the group $G$ on the tangent space at the origin of the abelian variety $A$ is irreducible, we prove that $A$ is isomorphic to the self-product of an elliptic curve and $A/G\simeq \mathbb P^n$. In the general case, assuming $\dim(A^G)=0$, we prove that $A/G$ is isomorphic to a direct product of projective spaces.
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
We propose an approach via Frobenius manifolds to the study (began in math.AG/0407254) of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding ``abelianized'' quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.
Let $(({\mathbb B} / Γ_1)', T(1))$ be á torsion free toroidal compactification with abelian minimal model $(A_1, D(1))$. An arbitrary principal isogeny $μ_a : A_2 \rightarrow A_1$, $a \in {\mathbb C}$ pulls-back $(A_1, D(1))$ to the abelian minimal model $(A_2, D(2))$ of a torsion free toroidal compactification $(({\mathbb B} / Γ_2)', D(2))$. The present work makes use of the isogeny pull-backs of abelian ball quotient models, towards the construction of infinite series of mutually non-birational co-abelian torsion free Galois covers $(({\mathbb B} / Γ_n)', T(n))$ of a ball quotient compactifi…
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
Stability conditions on free abelian quotients
研究光滑射影簇在有限阿贝尔群自由作用下的商簇上的斜率稳定向量丛和Bridgeland稳定性条件，证明了覆盖上的G不变几何稳定性条件与商簇上关于群表示的对偶群不变的几何稳定性条件之间的解析同构，并应用于描述自由阿贝尔商簇稳定性流形中的连通分支。
Let $M = \mathbb{B}^2 / Γ$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a l…
If $X'= ({\mathbb B} / Γ)'$ is a torsion free toroidal compactification of a discrete ball quotient $X_o={\mathbb B} / Γ$ and $ξ: (X', T = X'\setminus X_o) \rightarrow (X, D = ξ(T))$ is the blow-down of the $(-1)$-curves to the corresponding minimal model, then $G'= Aut (X',T)$ coincides with the finite group $G=Aut(X,D)$. In particular, for an elliptic curve $E$ with endomorphism ring $R = End(E)$ and a split abelian surface $X = A = E \times E$, $G$ is a finite subgroup of $Aut(A) = \mathcal{T}_A \leftthreetimes GL(2,R)$, where $(\mathcal{T}_A,+) \simeq (A,+)$ is the translation group of $A$…
By a K3-surface with nine cusps I mean a surface with nine isolated double points A_2, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown, that such a surface admits a cyclic triple cover branched precisely over the cusps. This parallels the theorem of Nikulin, that a K3-surface with 16 nodes is a Kummer quotient of a complex torus.
For Gorenstein quotient spaces $C^d/G$, a direct generalization of the classical McKay correspondence in dimensions $d\geq 4$ would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We us…
Abelian versus triangulated quotients
本文证明了三角范畴的局部化可诱导阿贝尔范畴的局部化，并由此构造了仅在更大格罗滕迪克宇宙中存在的阿贝尔商范畴的例子，同时将结果推广到任意范畴的阿贝尔包。
We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of quasi-coherent complexes over a derived quotient stack constructed from a motivic algebra of the abelian variety. We then study the structure of the motivic Galois groups of their mixed motives. We prove that the motivic Galois group is decomposed into a unipotent part constructed from the motivic algebra, and the reductive quotient which is the Tannaka dual of Grothend…
Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $…
Finite quotients of abelian varieties with a Calabi-Yau resolution
该研究针对阿贝尔簇的有限商是否存在Calabi-Yau解消的问题，证明4维情形下无此类解消，并在任意维数下按同源等价关系对相关阿贝尔簇进行了分类。
For which finite subgroups G of SL(r,C), r \geq 4, are there crepant desingularizations of the quotient space C^r/G? A complete answer to this question (also known as "Existence Problem" for such desingularizations) would classify all those groups for which the high-dimensional versions of McKay correspondence are valid. In the paper we consider this question in the case of abelian finite subgroups of SL(r,C) by using techniques from toric and discrete geometry. We give two necessary existence conditions, involving the Hilbert basis elements of the cone supporting the junior simplex, and an Up…
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we conside…
We study interactions between the categories of $\D$-modules on smooth and singular varieties. For a large class of singular varieties $Y$, we use an extension of the Grothendieck--Sato formula to show that $\D_Y$-modules are equivalent to stratifications on $Y$, and as a consequence are unaffected by a class of homeomorphisms, the {\em cuspidal quotients}. In particular, when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\D_Y$ and $\D_X$ and a Kashiwara theorem for $\D_Y$, thereby solving conjectures of Hart-Smith and Berest-Etingof-Ginzburg (generalizing re…
Let $M$ be a symplectic manifold, equipped with a Hamiltonian action of a torus $T$. We give an explicit formula for the rational cohomology ring of the symplectic quotient $M//T$ in terms of the cohomology ring of $M$ and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain cases these methods enable us to show that the cohomology of the reduced space is torsion-free.
Critical groups in harmonic abelian quotients
本文研究图调和 Galois 覆盖下临界群的前推映射，借助 Zaslavsky 偏置拟阵计算其核的阶，并建立两图生成树数目的关系。
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quo…
We investigate field theories on the worldvolume of a D3-brane transverse to partial resolutions of a $\Z_3\times\Z_3$ Calabi-Yau threefold quotient singularity. We deduce the field content and lagrangian of such theories and present a systematic method for mapping the moment map levels characterizing the partial resolutions of the singularity to the Fayet-Iliopoulos parameters of the D-brane worldvolume theory. As opposed to the simpler cases studied before, we find a complex web of partial resolutions and associated field-theoretic Fayet-Iliopoulos deformations. The analysis is performed by …
On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems
本文研究了兼容系统中系数、阿贝尔商与残余不可约性之间的关系，通过构造单束群并证明其阿贝尔商的独立性，推广了残余不可约性结果。
Let A_2 be the moduli stack of principally polarized abelian surfaces and V a smooth l-adic sheaf on A_2 associated to an irreducible rational finite dimensional representation of Sp(4). We give an explicit expression for the cohomology of V in any degree in terms of Tate type classes and Galois representations attached to elliptic and Siegel cusp forms. This confirms a conjecture of Faber and van der Geer. As an application we prove a dimension formula for vector-valued Siegel cusp forms for Sp(4,Z) of weight three, which had been conjectured by Ibukiyama.
Horospherical varieties with quotient singularities
研究了horospherical变种的商奇点特征，证明了所有具有商奇点的quasiprojective horospherical变种都是平滑变种对有限阿贝尔群的商。
A three-dimensional orbifold $(Σ, γ_i, n_i)$, where $Σ$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with th…
On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification
针对有限 Abel 子群作用于仿射空间所得商奇点，本文利用非交换 crepant 消解构造其非交换射影空间导出范畴与分次奇点范畴的 Efimov 光滑紧化，并给出核中典范经典生成元。
Exactness of the 2-categories of abelian and triangulated categories
本文引入$2$-同调范畴概念，证明双正合、三角、带点、加性及阿贝尔范畴的$2$-范畴均为$2$-同调，并给出统一商构造准则。
A description of Cohen-Macaulay modules over cusp surface singularities and over unimodule hypersurface singularities of type T is given. It is proved that among minimally elliptic singularities and their quotients only simple elliptic and cusps are tame, all others are wild (with respect to the classification of Cohen-Macaulay modules).
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