Young-Hoon Kiem, Hyeonjun Park
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.
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We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections are computed by the same method. Finally, we present an application to the non-vanishing of the Pontrjagin ring of the moduli space of bundles.
We compute the quantum Euler class of Fano complete intersections X in a projective space. In particular, we prove a recent conjecture of A. Buch and R. Pandharipande. Finally we apply our result to obtain formulas for the virtual Tevelev degrees of X. An algorithm computing all genus 0 two-point Hyperplane Gromov Witten invariants of X is illustrated along the way.
We study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of $C^n$ by using the theory of motivic Milnor fibers. The results will be explicitly described by the mixed volumes of the faces of Newton polyhedrons.
Isotropic Quot schemes parameterize rank $r$ isotropic subsheaves of a vector bundle equipped with symplectic or symmetric quadratic form. We define a virtual fundamental class for isotropic Quot schemes over smooth projective curves. Using torus localization, we prescribe a way to calculate top intersection numbers of tautological classes, and obtain explicit formulas when $r=2$. These include and generalize the Vafa-Intriligator formula. In this setting, we compare the Quot scheme invariants with the invariants obtained via the stable map compactification.
In this paper, we construct proper pushforwards and flat pullbacks in Chow groups of coherent sheaf stacks over a Deligne-Mumford(DM) stack. When there is a relative semi-perfect obstruction theory for a DM-type morphism $X \to Y$, $X$ is a DM stack and $Y$ is a DM stack or a smooth Artin stack, we define a virtual pull-back as a bivariant class. This is an analogue of virtual pull-backs defined by Manolache.
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.
We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement and new proof of R. Thomason's formula.
Intersection Theory of Hyperquot Schemes on curves
本文研究曲线上的超商方案虚拟交理论，推广了Vafa-Intriligator公式，并为曲线到部分旗流形的映射提供闭合计数公式。
We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of $p$-fields to the setting of (quasi-)maps to complete intersections in arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In particular, we show that the virtual cycle of stable (quasi-)maps to a complete intersection can be recovered by the cosection localized virtual cycle of the moduli of $p$-fields of the ambient space.
We construct algebraic virtual cycles that give us the cohomological field theories of Fan-Jarvis-Ruan invariants by integral transformations.
The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.
Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space
本文研究了加权射影空间中完备交集的椭圆虚拟结构常数及其与格罗莫夫-王特征值的关系。
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.
For a nonsingular projective variety $X$, the virtual Tevelev degree in Gromov-Witten theory is defined as the virtual degree of the morphism from $M_{g,n}(X,d)$ to the product $M_{g,n} \times X^n$. After proving a simple formula for the virtual Tevelev degree in the (small) quantum cohomology ring of $X$ using the quantum Euler class, we provide several exact calculations for flag varieties and complete intersections. In the cominuscule case (including Grassmannians, Lagrangian Grassmannians, and maximal orthogonal Grassmannians), the virtual Tevelev degrees are calculated in terms of the eig…
In this paper, we derive the virtual structure constants used in mirror computation of degree k hypersurface in CP^{N-1}, by using localization computation applied to moduli space of polynomial maps from CP^{1} to CP^{N-1} with two marked points. We also apply this technique to non-nef local geometry O(1)+O(-3)->CP^{1} and realize mirror computation without using Birkhoff factorization.
We approach Gathmann's virtual pushforward property from the perspective of bivariant intersection theory, extend a virtual pushforward result of Manolache, and use our extension to deduce a result of Gathmann relating relative and rubber GW invariants of a $P^1$ bundle with invariants of its base.
Recently H.-L. Chang and J. Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization, the cosection localization and their combination, to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometr…
Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and $K$-theoretic invariants for many moduli stacks of interest, including $K$-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the $K$-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory…
Virtual Riemann-Roch Theorems for Almost Perfect Obstruction Theories
本文在几乎完美障碍理论框架下证明虚拟黎曼-罗赫定理，涵盖非等变与等变情形及余截面局部化版本，推广并简化了既有结果，为K-理论不变量提供坚实基础。
We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.
We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These…
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.
This is the first article in an upcoming series of papers. They have arisen through an attempt to answer open questions of Deligne proposed in "Le determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf. "Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm. Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary results on virtual categories which are the categories where the Deligne…
We obtain a simple, recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus zero and one in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, even, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann…
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…
The moduli space of stable maps with divisible ramification uses $r$-th roots of a canonical ramification section to parametrise stable maps whose ramification orders are divisible by a fixed integer $r$. In this article, a virtual fundamental class is constructed while letting domain curves have a positive genus; hence removing the restriction of the domain curves being genus zero. We apply the techniques of virtual localisation and obtain the $r$-ELSV formula as an intersection of the virtual class with a pullback via the branch morphism.
The point insertion technique and open $r$-spin theories II: intersection theories in genus-zero
本文利用点插入技术定义了新的开$r$-spin与开FJRW相交理论，构造了具有多种边界态的开$r$-spin理论并计算相交数，同时构造了五次费马FJRW理论，为先前猜想提供了证据。
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, …
Virtual Localisation Formula for $ SL_η $-Oriented Theories
本文将 Levine 虚拟局部化公式推广至 motivic 环谱类，获得 Witt 理论中虚拟基本类的局部化公式，并建立等变 Atiyah-Bott 局部化定理及 $N$-等变虚拟局部化公式，同时描述 $\mathrm{KW}(BN)$ 的环结构。
Hori's Equation for Gravitational Virtual Structure Constants of Calabi-Yau Hypersurface in CP^{N-1}
本文通过局部化技术证明了Hori方程，该方程涉及CP^{N-1}中Calabi-Yau超曲面的引力虚拟结构常数，具体为带(2+1)个标记点的CP^{1}到CP^{N-1}的拟映射模空间交点数。研究问题为验证该方程，核心方法是局部化技术，主要贡献是给出了严格证明。
We use Picard-Lefschetz theory to prove a new formula for intersection numbers of twisted cocycles associated to a given arrangement of hyperplanes. In a special case when this arrangement produces the moduli space of punctured Riemann spheres, intersection numbers become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism. We introduce several kinds of bivariant theory associated with a suitable ring spectrum and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. This fundamental classes satisfies all the expected properties from classical intersection theory and lead to Gysin morphisms, Rieman…
We define a notion of formal quantum field theory and associate a formal quantum field theory to K-theoretical intersection theories on Hilbert schemes of points on algebraic surfaces. This enables us to find an effective way to compute K-theoretical intersection theories on Hilbert schemes via a connection to Macdonald polynomials and vertex operators.
Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa-Witten and reduced local DT theories. The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkov operators. This proves and extends a conjecture of Gholamp…
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.
Sheaves on surfaces and virtual invariants
本文综述了作者关于光滑射影曲面上稳定层模空间虚拟不变量的一组新猜想，涵盖虚拟欧拉示性数、Verlinde数与Segre数，并讨论其在Vafa-Witten理论、K-理论S-对偶等方向的应用，核心工具为Mochizuki的下降Donaldson不变量公式。
We define "BPS graphs" on punctured Riemann surfaces associated with $A_{N-1}$ theories of class $\mathcal{S}$. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with cer…
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of $\ell$ results for traces and zeta-functions, and constructions of characteristic classes (as $0$-cycles). This leads to the Grothendieck-Lefschetz formula, of which we give a new motivic proof. There are also a few additions to what have been told in …
Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections
该研究在射影空间光滑法诺完全交的ambient状态空间上证明了亏格一Virasoro猜想，推广了拟映射不变量的跨壁公式，为完全交与射影空间的Gromov-Witten理论建立了关键桥梁。
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