Mihnea Popa
Given a derived equivalence of orbifolds associated to projective varieties with (not necessarily Gorenstein) quotient singularities, we deduce consequences related to the behavior of orbifold Hodge numbers and the Picard variety, extending what is known in the smooth case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an $n$-dimensional simplicial Fano toric variety and then explicitly compute $h^{1,1}_{orb}$ and $h^{n-2,1}_{orb}$ for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.
We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''. Furthermore, the total cohomology $H_{orb}^{*,*}(X)$ forms a mixed Hodge structure that is polarized by every element of the Kähler cone of $X$. Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of $X$. This construction should be considered as the analogue of the abstract polarized variation of Hodge s…
We shall reproof formulas for the Hodge numbers of Calabi-Yau threefolds of Borcea-Voisin type constructed by A. Cattaneo and A. Garbagnati, using the orbifold cohomology formula and the orbifold Euler characteristic.
We prove that the wreath product orbifolds studied earlier by the first author provide a large class of higher dimensional examples of orbifolds whose orbifold Hodge numbers coincide with the ordinary ones of suitable resolutions of singularities. We also make explicit conjectures on elliptic genera for the wreath product orbifolds.
We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When $X$ is the Fermat quintic in $¶^4$, and $\tilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $Γ$ of the alternating group $A_5$, the $Γ$-Hilbert schemes $Γ$-$\Hilb(X)$ and $Γ$-$\Hilb(\tilde{X}^*)$ are smooth Calabi…
We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .
Mixed Hodge structures for vanishing cycles and orbifold cohomology
研究消失环的混合Hodge结构与orbifold上同调的极化关系，探讨Hodge-Tate情况下的上同调结构。
Stringy Irregular Hodge Numbers for Non-Simplicial Toric Landau--Ginzburg Data
将轨道折叠不规则Hodge理论推广至非单纯环带数据，构造滤过Koszul--de Rham复形，证明其射影crepant单纯细分模型共享同一Hodge多项式，从而定义弦论不规则Hodge数。
Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.
We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
Starting from the Kähler moduli space of the rigid orbifold Z=E^3/\mathbb{Z}_3 one would expect for the cohomology of the generalized mirror to be a Hodge structure of Calabi-Yau type (1,9,9,1). We show that such a structure arises in a natural way from rational Hodge structures on Λ^3 \mathbb{K}^6, \mathbb{K}=\mathbb{Q}[ω], where ωis a primitive third root of unity. We do not try to identify an underlying geometry, but we show how special geometry arises in our abstract construction. We also show how such Hodge structure can be recovered as a polarized substructure of a bigger Hodge structure…
We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
Orbifold Aspects of Certain Occult Period Maps
本文刻画了三次三维簇和四次簇的霍奇结构自同构群，并确定了小维数射影流形自同构群在循环覆盖霍奇结构上的作用，证实了Kudla和Rapoport的猜想。
We define and construct the real analytic moduli stack of pluriharmonic bundles on a compact Kaehler manifold X, and show how this is equipped with Hodge and quaternionic structures. This stack maps to the de Rham moduli stack, giving rise to preferred sections of the Deligne-Hitchin twistor stack. We then show how the non-abelian mixed Hodge structures on Malcev homotopy types X can be extended to objects over the pluriharmonic moduli stack.
Mixed Hodge modules on stacks
本文借助混合 Hodge 模的 ∞-范畴增强，将其连同六运算与权推广到代数栈，并证明 Drew 的 motivic Hodge 模范畴全忠实嵌入且像为几何来源的混合 Hodge 模。
In this paper we study the integral Chow ring of toric Deligne-Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne-Mumford stack is isomorphic to the Stanley-Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.
Logarithmic de Rham Stacks and Non-Abelian Hodge Theory
本文提出了一种新的对数de Rham堆栈，并基于此推导出对数非阿贝尔Hodge定理，扩展了此前的理论成果。
Irregular Hodge numbers of stacky Clarke mirror pairs
本文证明堆叠 Clarke 镜面对扭曲上同调的不规则 Hodge 滤过分次片段对偶，并推广相关猜想与 Hodge 数对偶结果。
Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some examples related to crepant resolutions are discussed.
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.
We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the genus $0$ Gromov-Witten theory of a toric stack bundle.
We construct relative moduli spaces of semistable pairs on a family of projective Deligne-Mumford stacks. We define moduli stacks of stable orbifold Pandharipande-Thomas pairs on stacks of expanded degenerations and pairs, and then show they are separated and proper Deligne-Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande-Thomas invariants.
Positivity in the context of Hodge modules and Higgs bundles on Deligne-Mumford stacks
将Popa-Wu和Popa-Schnell关于光滑射影簇上Hodge模和Higgs丛的正性结果推广到具有射影粗模空间的光滑真DM栈上。
A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. (i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the 1-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-actio…
Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces
本文为刚性空间上的 v-完备复形构造了从 Hodge-Tate 叠到 v-站点的全忠实函子，并用 Higgs 模导出范畴刻画完美复形，从而建立导出的 $p$-进 Simpson 函子。
Higher dimensional analogon of Borcea-Voisin Calabi-Yau manifolds, their Hodge numbers and $L$-functions
本文构造了任意维度的 Calabi-Yau 流形示例，推广了 Borcea-Voisin 三流形，并提出利用 Frobenius 态射与 orbifold Chen-Ruan 上同调计算局部 zeta 函数与 Hodge 数的方法。
Stringy Hodge numbers via crepant resolutions by Artin stacks
本研究证明簇可通过光滑Artin栈实现crepant消解等价于其具有对数典范奇点，给出弦霍奇数的扭弧积分公式，还定义了光滑Artin栈的弦霍奇数并复现轨形霍奇数，为弦霍奇数的上同调解释提供了可行方向。
A compact moduli of orbifold projective curves
该研究提出稳定轨形射影曲线概念，证明其模栈与由栈点自同构群确定权重的Hassett加权带点稳定曲线模栈同构，构建了轨形射影曲线的紧致模空间。
$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks
针对粗空间拟射影的光滑分离德利涅-芒福德栈，利用Kontsevich复形的有限平展下降，将射影栈情形归约为Esnault–Sabbah–Yu的定理，证明其非正则霍奇滤子在所有有理指标处$E_1$退化，结果适用于各类栈紧化。
Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational (k,k)-cohomology class is algebraic, i.e., the Hodge conjecture holds on them.
In \cite{TY18}, higher genus Gromov--Witten invariants of the stack of $r$-th roots of a smooth projective variety $X$ along a smooth divisor $D$ are shown to be polynomials in $r$. In this paper we study the degrees and coefficients of these polynomials.
On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we put in correspondence a pro-algebraic group on whose functional algebra we introduce a mixed Hodge structure. This construction, in fact, works for any smooth algebraic variety X which, considered as an analytic space, has a nonabelian first homotopy group, and the rest are trivial. The Hodge structure defined in this way can be expressed in terms of iterated integrals. Furthermore, considered in the…
We investigate moduli stacks of pointed chains of projective lines related to the Losev-Manin moduli spaces and show that these moduli stacks coincide with certain toric stacks which can be described in terms of the Cartan matrices of root systems of type A. We also consider variants of these stacks related to root systems of type B and C.
In this paper, we study the bigraded vector space structure of Landau-Ginzburg orbifolds. We prove the formula for the generating function of the Hodge numbers of possibly nonabelian Landau-Ginzburg orbifolds. As an application, we calculate the Hodge numbers for all nondegenerate quintic homogeneous polynomials with five variables. These results yield an evidence for the Calabi-Yau/Landau-Ginzburg correspondence between the Calabi-Yau geometries and the Landau-Ginzburg B-models.
A Brill-Noether stack is an algebraic very presentable stack whose homotopy type has two nontrivial homotopy groups. We consider one with a fundamental group --- a reductive algebraic group-scheme S and one higher homotopy group, represented by a vector space V. The homotopy type also defines an action of S on V. This stack is used as coefficient space for nonabelian cohomological space on a smooth algebraic variety X. We define nonabelian MHS on cohomological spaces of this type in the context of the work of C. Simpson related to MHS on the space of local systems. It is defined via an action …
An orbifold formula for algebraic stacks
本文推广orbifold公式至与光滑代数栈双有理的簇，提供计算弦论E-函数的新方法，并应用于klt奇点研究，涉及动机积分及Bruhat-Tits理论。
A Conjecture of Bhatt--Lurie and weakly $p$-nilpotent Hodge--Tate stacks
本文针对特征p光滑簇上Hodge-Tate晶体子范畴的刻画问题，构造弱p-幂零Hodge-Tate栈建立对应关系，并证明其为带特定结构的gerbe且障碍类匹配Frobenius提升障碍，证明了Bhatt与Lurie的相关猜想。
Arithmetic Hodge-Iwasawa Moduli Stacks
本文在已有Hodge-Iwasawa理论基础上构造G-丛的算术模叠，用以连贯参数化Hodge-Iwasawa结构。
Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks
研究德利涅-芒福德栈上由Yu方法定义的非正则霍奇滤子，证明其不依赖紧化，该结果适用于Harder和Lee的orbifold非正则霍奇数研究，仅依赖栈Landau–Ginzburg模型。
We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $λ$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $M_{DR}(X,Perf)$, parametrizing complexes of $D_X$-modules which are $O_X$-perfect. We apply the result of Toen-Vaquie that $Perf(X)$ is locally geometric. The proof of geometricity of the map $M_{Hod}(X,Perf) \to Perf(X)$ uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a s…
Birational classification of toric orbifolds
本文借助相伴堆叠扇，对平凡一般稳定子的光滑真环面 Deligne–Mumford 堆叠的环面等变双有理等价类给出完整分类。
We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and pairs and prove the properness of their moduli's. On 3-dimensional smooth projective DM stacks this leads to a definition of relative Donaldson-Thomas invariants and the associated degeneration formula.
Twisted jets, motivic measure and orbifold cohomology
本文通过引入Deligne-Mumford栈上的扭曲射流并研究其母题测度，证明了具有Gorenstein商奇点的双有理极小模型具有相同的带霍奇结构的轨形上同调。
Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces
本文通过轨形Mukai配对导出新的轨形HRR公式，用于计算K3曲面上有限群作用下稳定层等变模空间的维数，并证明这些模空间是形变等价于点Hilbert概型的不可约辛流形。
Starting from a collection of line bundles on a projective toric orbifold X, we introduce a stacky analogue of the classical linear series. Our first main result extends work of King by building moduli stacks of refined representations of labelled quivers. We associate one such stack to any collection of line bundles on X to obtain our notion of a stacky linear series; as in the classical case, X maps to the ambient stack by evaluating sections of line bundles in the collection. As a further application, we describe a finite sequence of GIT wall crossings between [A^n/G] and G-Hilb(A^n) for a …
Orbifold pseudo-effective cones of toric stacks
本文研究了分裂折线堆叠的折线伪有效锥的结构与性质。
On the Kähler cone of irreducible symplectic orbifolds
本文在不可约辛轨形上推广了全局Torelli定理及Kähler锥相关结果，并新定义了镜像对称作为模空间上的对合。
$p$-adic non-abelian Hodge theory for curves via moduli stacks
本文通过模堆的典范扭转，将曲线上的 $p$-adic 非阿贝尔 Hodge 理论几何化，并恢复 Faltings 对应为同胚，同时将表示堆映射到半稳定 Higgs 丛的开子堆。
Let X be a smooth and tame stack with finite inertia. We prove that there is a functorial sequence of blow-ups with smooth centers after which the stabilizers of X become abelian. Using this result, we can extend the destackification results of the first author to any smooth tame stack. We give applications to resolution of tame quotient singularities, prime-to-l alterations of singularities and weak factorization of Deligne-Mumford stacks. We also extend the abelianization result to infinite stabilizers in characteristic zero, generalizing earlier work of Reichstein-Youssin.
The existence of interesting multiplicative cohomology theories for orbifolds was first suggested by string theorists. Orbifold products have been intensely studied by mathematicians for the last fifteen years. In this paper we focus on the virtual orbifold product that was first introduced in Lupercio et al. (2007). We construct a categorification of the virtual orbifold product that leverages the geometry of derived loop stacks. By work of Ben-Zvi, Francis and Nadler, this reveals connections between virtual orbifold products and Drinfeld centers of monoidal categories, thus answering a ques…
We conjecture an evaluation of three-partition cyclic Hodge integrals in terms of loop Schur functions. Our formula implies the orbifold Gromov-Witten/Donaldson-Thomas correspondence for toric Calabi-Yau threefolds with transverse type A singularities. We prove the formula in the case where one of the partitions is empty, and thus establish the orbifold Gromov- Witten/Donaldson-Thomas correspondence for local toric surfaces with transverse type A singularities.
In this paper, we present some Hurwitz-Hodge integral identities which are derived from the Laplace transform of the cut-and-join equation for the orbifold Hurwitz numbers. As an application, we prove a conjecture on Hurwitz-Hodge integral proposed by J. Zhou in 2008.
We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.
Nonabelian Hodge isomorphisms for stacks and cohomological Hall algebras
本文通过构造联络范畴的2-Calabi-Yau dg-范畴实现并定义联络栈Borel-Moore同调上的上同调霍尔代数，完成了曲线非阿贝尔霍奇理论三侧同调及其CoHA结构的一致性比较。
We compute the Hodge and de Rham cohomology of the classifying space BG (defined as etale cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.
Starting with a vertex algebra $V$, a finite group $G$ of automorphisms of $V$, and a suitable collection of twisted $V$--modules, we construct (twisted) $D$--modules on the stack of pointed $G$--covers, introduced by Jarvis, Kaufmann, and Kimura. The fibers of these sheaves are spaces of orbifold conformal blocks defined in joint work with Edward Frenkel. The key ingredient is a $G$--equivariant version of the Virasoro uniformization theorem.
We consider the question of how geometric structures of a Deligne-Mumford stack affect its Gromov-Witten invariants. The two geometric structures studied here are {\em gerbes} and {\em root constructions}. In both cases, we explain conjectures on Gromov-Witten theory for these stacks and survey some recent progress on these conjectures.
A stacky comparison of the Hodge and Nygaard filtrations
本文通过Drinfeld与Bhatt–Lurie的栈化方法，将棱镜上同调的Nygaard滤链与de Rham上同调的Hodge滤链的比较结果推广到任意规范系数情形，并发展出能同时捕捉共轭滤链与Sen算子的衍射Hodge上同调栈化理论。
The moduli stacks of Calabi-Yau varieties are known to enjoy several hyperbolicity properties. The best results have so far been proven using sophisticated analytic tools such as complex Hodge theory. Although the situation is very different in positive characteristic (e.g. the moduli stack of principally polarized abelian varieties of dimension at least 2 contains rational curves), we explain in this note how one can prove many hyperbolicity results by reduction to positive characteristic, relying ultimately on the nonabelian Hodge theory in positive characteristic developed by Ogus and Volog…
We compute some Hodge and Betti numbers of the moduli space of stable rank $r$ degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles on a curve. Our methods also yield a formula for the Poincare polynomial of the moduli stack that is valid over any ground field.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。