F. Qu
We show that a perfect obstruction theory for a $\mathbb{G}_\text{m}$-gerbe determines a semi-perfect obstruction theory for its base, which is perfect if the gerbe is quasi-compact and affine-pointed. These results streamline the construction of a semi-perfect obstruction theory for the base, and allow us to relate virtual classes of the gerbe and its base.
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The Brauer group of $BG$ and gerbe structures of moduli spaces
本文研究了具有阶$N$自同构的曲线的$μ_N$-gerbe结构，并确定了其Brauer类，关键在于计算$BG$的Brauer群。
This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $μ_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual fundamental classes. Our result verifies the genus 0 part of the so-called decomposition conjecture which compares Gromov-Witten theory of étale gerbes with that of the bases. We also verify this conjecture in all genera for toric gerbes over toric Deligne-Mumford stacks.
We study $S^1$-bundles and $S^1$-gerbes over differentiable stacks in terms of Lie groupoids, and construct Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvature.
Torsors and gerbes
本文研究有限群作用下阿贝尔层上的 torsor 与 gerbe 何时可由不变量诱导，通过定义阻碍类并利用谱序列描述低次上同调态射，给出判定条件。
Formal GAGA for gerbes
本文研究真A-概形与其模幂纤维的Brauer群关系，解答Grothendieck问题，刻画正/混合特征下非平凡Brauer类限制为零的现象，证明零特征下无此情况并推出形式GAGA结论。
Moduli of vector bundles on $μ_n$-gerbes over genus 2 curves and the period-index problem
研究通过构造 genus 2 曲线上的向量丛模空间，证明基域上存在周期等于指数的 Brauer 类，并展示 C_1-域上 2- torsion Brauer 类满足周期-指数问题，为该猜想提供新证据。
This note is intended to be a friendly introduction to virtual classes. We review virtual classes and we give a number of properties and applications. We also include a new virtual push-forward theorem and many computations of virtual classes in simple examples.
We construct virtual fundamental classes of Artin stacks over a Dedekind domain endowed with a perfect obstruction theory.
The paper has been withdrawn.
In this paper, we compare the virtual fundamental classes of the stacks of $(g,β,μ)$-stable ramified maps $\bar{\mathfrak U}_{g,μ}(X,β)$ and of $(g,β,μ)$-log stable ramified maps $\bar{\mathfrak U}^{\mathrm{log}}_{g,μ}(X,β)$. For that we will see how they are identified via virtual normalization and then apply Costello's push-forward formula.
Lagrangian correspondences and pullbacks of virtual fundamental classes
本文提出拉格朗日对应是研究源自$-2$-辛导出结构的虚拟基本类函子性的正确框架。
Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.
We give a new definition of an obstruction theory for infinitesimal deformation theory and relate it to earlier definitions of Artin, Fantechi-Manetti, Li-Tian, and Behrend-Fantechi.
This paper, first written in 1997, aims at a simplified and more elementary construction of algebraic-geometric virtual fundamental classes as defined by Li/Tian and Behrend/Fantechi. We replace the use of Artin stacks in the latter approach by a yoga of cone bundles of possible independent interest. We also observe a formula for virtual fundamental classes involving only the total Chern class of the index bundle and Fulton's canonical class of the moduli space.
Virtual classes: An introduction with exercises
本文介绍Behrend-Fantechi虚拟类，推广Fulton局部化顶部陈类，并推导具有全局分解的完美障碍理论的Deligne-Mumford栈上虚拟类的Siebert公式。
Virtual classes via vanishing cycles
本文提出了一种利用消没圈反常层为拟光滑导出概形构造虚拟基类的新方法，并证明其与现有类一致，同时探讨了$-2$-移位辛导出概形的DT4虚拟类构造方法。
In [DKO] we constructed virtual fundamental classes $[[ Hilb^m_V ]]$ for Hilbert schemes of divisors of topological type m on a surface V, and used these classes to define the Poincare invariant of V: (P^+_V,P^-_V): H^2(V,Z) --> Λ^* H^1(V,Z) x Λ^* H^1(V,Z) We conjecture that this invariant coincides with the full Seiberg-Witten invariant computed with respect to the canonical orientation data. In this note we prove that the existence of an integral curve $C \subset V$ induces relations between some of these virtual fundamental classes $[[Hilb^m_V ]]$. The corresponding relations for the Poinca…
For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other $3$-folds. Exploiting the critical structure on $\textrm{Quot}_{\mathbb A^3}(\mathscr O^r,n)$, we construct a virtual motive (in the sense of Behrend-Bryan-Szendrői) for $\textrm{Quot}_Y(F,n)$ for an arbitrary vector bundle $F$ on a smooth $3$-fold $Y$. We compute the associated motivic partition function. We obtain new examples of higher rank (motivic) Donaldson-Th…
We construct virtual fundamental classes for dg-manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend-Fantechi [BF] or Li-Tian [LT]. Our class is initially defined in K-theory, as the class of the structure sheaf of the dg-manifold. We compare our construction with that of [BF] as well as with the original proposal of Kontsevich. We prove a Riemann-Roch type result for dg-manifolds which involves integration over the virtual class. We prove a localization theorem for our v…
We construct the étale motivic Borel-Moore homology of derived Artin stacks. Using a derived version of the intrinsic normal cone, we construct fundamental classes of quasi-smooth derived Artin stacks and demonstrate functoriality, base change, excess intersection, and Grothendieck-Riemann-Roch formulas. These classes also satisfy a general cohomological Bézout theorem which holds without any transversity hypotheses. The construction is new even for classical stacks and as one application we extend Gabber's proof of the absolute purity conjecture to Artin stacks.
Desingulization of Quasi-smooth Derived Schemes and Virtual Fundamental Classes
证明拟光滑导出概形的去奇异化定理，并提出其虚拟基本类K理论积分与光滑概形上完美复形楔幂积分等价的猜想。
Let V be a convex vector bundle over a smooth projective manifold X, and let Y be the subset of X which is the zero locus of a regular section of V. This mostly expository paper discusses a conjecture which relates the virtual fundamental classes of X and Y. Using an argument due to Gathmann, we prove a special case of the conjecture. The paper concludes with a discussion of how our conjecture relates to the mirror theorems in the literature.
We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697). We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas fo…
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Co…
We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i-g+1 genus 0 components. (The proof of this result is quite short and naive.) As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the r…
Virtual invariants from the non-associative Hilbert scheme
引入非结合希尔伯特方案作为经典嵌套希尔伯特方案的光滑环境空间，通过结合性丛的显式截面定义完美障碍理论和虚拟基本类，并利用虚拟局部化得到封闭公式。
We study the cycle-valued reduced Gromov-Witten theory of a nonsingular projective K3 surface. For primitive curve classes, we prove that the correspondence induced by the reduced virtual fundamental class respects the tautological rings. Our proof uses monodromy over the moduli space of K3 surfaces, degeneration formulae and virtual localization. As a consequence of the monodromy argument, we verify an invariance property for Gromov-Witten invariants of K3 surfaces in primitive curve class conjectured by Oberdieck-Pandharipande.
We study classes $P_{g,T}(α;β)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. A comparison with classes $Q_{g,T}$ arising from sections of the universal Jacobian shows the classes $P_{g,T}(α;β)$ are polynomial in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.
We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.
We establish cosection localization and vanishing results for virtual fundamental classes of derived manifolds, combining the theory of derived differential geometry by Joyce with the theory of cosection localization by Kiem-Li. As an application, we show that the stable pair invariants of hyperkähler fourfolds, defined by Cao-Maulik-Toda, are zero.
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.
We consider virtual pullbacks in $K$-theory, and show that they are bivariant classes and satisfy certain functoriality. As applications to $K$-theoretic counting invariants, we include proofs of a virtual localization formula for schemes and a degeneration formula in Donaldson-Thomas theory.
Fulton defined classes in the Chow group of a quasi-projective scheme $M$ which reduce to its Chern classes when $M$ is smooth. When $M$ has a perfect obstruction theory, Siebert gave a formula for its virtual cycle in terms of its total Fulton class. We describe K-theory classes on $M$ which reduce to the exterior algebra of differential forms when $M$ is smooth. When $M$ has a perfect obstruction theory, we give a formula for its K-theoretic virtual structure sheaf in terms of these classes.
Let $({\bf X},ω_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that $X_{\rm an}$ can be given the structure of a derived smooth manifold ${\bf X}_{\rm dm}$, of real virtual dimension ${\rm vdim}_{\mathbb R}{\bf X}_{\rm dm}=n$. This ${\bf X}_{\rm dm}$ is not canonical, but is independent of choices up to bordisms fixing the underlying topological …
Virtual $χ_{-y}$-genera of Quot schemes on surfaces
本文通过将曲面上 Quot 概型的等变虚拟 χ_{-y}-亏格用 Seiberg-Witten 不变量表示，并在零曲线类、椭圆曲面及一般型极小曲面等情形给出完整计算，证明了 p_g>0 时生成级数为有理函数，解决了 Oprea-Pandharipande 猜想。
In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generating a recursive pattern.
The aim of this note is to give a simple definition of genus zero virtual orientation classes (or fundamental classes) for projective complete intersections or, more generally, for complete intersections in convex varieties, and to prove a push forward formula for them.
Virtual Classes of Character Stacks
本文将计算闭可定向曲面G-表示簇虚类的拓扑量子场论推广到特征栈情形，定义可表栈格罗滕迪克环，得到特征栈虚类的动机分解，并在两类群中验证了方法的有效性与优势。
In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the $G$-representation variety of surface groups $\mathfrak{X}_G(Σ_g)$ of arbitrary genus for $G$ being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Grothendieck ring of varieties of the $G$-representation variety and the moduli space of $G$-representations of surface groups for $G$ being the group of complex upper triangular matrices of rank $2$, $3$, and $4$ via constructing a topological quantum field theory. Furthermore, …
These expository notes are based on a series of lectures given at the May 2018 Snowbird workshop, Crossing the Walls in Enumerative Geometry. We give an introductory treatment of the notion of a virtual fundamental class in algebraic geometry, and describe a new construction of the virtual fundamental class for Gromov-Witten theory of a hypersurface. The results presented here are based on joint work with I. Ciocan-Fontanine, D. Favero, J. Guéré, and B. Kim.
In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fundamental classes defined in the two categories. We prove this statement in an interesting case, namely the moduli problem associated with the symplectic factorization problem which yields the complete toric varieties. Therefore, our main objects are the moduli spaces of "twisted"…
We prove that the pullbacks of the virtual fundamental classes of the Brill-Noether loci under any Abel-Jacobi section lie in the tautological ring of the moduli space of stable curves. This resolves a conjecture of Pagani, Ricolfi and van Zelm, and is part of a broader program to understand the logarithmic intersection theory of $\Mgn$.
In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such t…
Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics was conjectured to be a rational function when X is simply connected. We conjecture here the rationality of more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. We prove the rationality of descendent series in Hilbert scheme cases for all curve c…
We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira--Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.
We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating $λ_g$ integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibered rational surfaces. New formulas relating reduced virtual c…
Cosection Localization for D-Manifolds and $(-2)$-Shifted Symplectic Derived Schemes, Revisited
本文延续作者先前工作，为具有满射余截面的导出流形构造约化虚基本类，为$(-2)$-移位辛导出概形构造余截面局部化虚基本类，并利用Oh-Thomas结果证明代数与微分几何构造同调等价，应用于Calabi-Yau四维流形Donaldson-Thomas理论中约化不变量的构造与整性。
This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge theoretical Gromov-Witten type invariants of arbitrary genus associated with certain symplectic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects. The main tool for describing moduli spaces associated with…
In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variet…
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