Virtual intersections on the Quot-scheme and Vafa-Intriligator formulas

线性空间半正商簇的Vafa-Intriligator公式

A Vafa-Intriligator formula for semi-positive quotients of linear spaces

本文针对形如$V/\!/G$的光滑射影目标，建立了拟映射不变量与极大子环面商的不变量之间的联系，证明了Vafa-Intriligator公式，将生成级数表示为有限显式贡献之和。

On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator

We observe a general structure theorem for quantum cohomology rings, a non-homogeneous version of the usual cohomology ring encoding information about (almost holomorphic) rational curves. An application is the rigorous computation of the quantum cohomology of Grassmannians. As purely algebraic consequence we prove a beautiful formula of Vafa and Intriligator for intersection numbers of certain compactifications of moduli spaces of maps from a Riemann surface (any genus) to G(k,n) which recently has excited many mathematicians. The formula generalizes to any Fano manifold whose cohomology ring…

曲线上的超商方案交理论

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。

The virtual intersection theory of isotropic Quot Schemes

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.

Counts of maps to Grassmannians and intersections on the moduli space of bundles

We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highest-degree asymptotics in formulas of Vafa-Intriligator type. In particular, we explicitly evaluate all intersection numbers appearing in the Verlinde formula. Our results are in agreement with previous computations of Witten, Jeffrey-Kirwan and Liu. Moreover, we prove the vanishing of certain intersections on a suitable Quot scheme which can be interpreted as giving equations between counts of maps to the Grassmannian.

Intersections numbers on the compact variety of rational ruled surfaces

We consider the Quot scheme, R_{d}, compactifying the space of degree d maps from the projective line to the Grassmannian of lines. We give an algorithm for computing the degree of R_{d} under a "generalized Plücker embedding", this is a certain Gromov-Witten invariant. The approach is to apply the Atiyah-Bott localization formula for the natural C^{*}-action on R_{d}. These numbers can be obtained directly from Vafa-Intriligator formula.