Equivariant Resolution of Points of Indeterminacy

Factorial Schur and Grothendieck polynomials from Bott Samelson varieties

We use Bott-Samelson resolutions of Schubert varieties in Grassmannians along with equiariant localization techniques to show that the factorial Schur functions and the factorial Grothendieck polynomials represent Schubert classes in equivariant cohomology and equivariant K-theory respectively.

等变非交换平展分解与G-极大修改模

Equivariant noncommutative crepant resolutions and $G$-maximal modifying modules

本文针对含特征零域上有限群作用的Cohen-Macaulay正规环，研究等变非交换平展分解与极大修改代数的导出等价性，证明三维指数2终端奇点的MM模存在性及对应MMA的导出等价。

On an equivariant version of the zeta function of a transformation

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W.Lück and J.Rosenberg. Here we offer another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta …

Springer解消的量子上同调自同构及其应用

Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications

该研究引入作用于Springer解消等变量子上同调的量子Demazure–Lusztig算子，给出其环面等变量子上同调的生成元关系表示与典型类型显式描述，并通过Toda极限复现了Kim关于完全旗簇的相关结果。

Twisted equivariant HKR theorem for torus action and the small quantum group

We show that when a torus $T$ acts on a smooth variety $X$, the twisted HKR isomorphism is equivariant. The main consequence is that the Bezrukavnikov- Lachowska isomorphism, relating the Hochschild cohomology of the principal block of the small quantum group to certain sheaf cohomology groups on the Springer resolution $\widetilde{\mathcal{N}}$ , can be upgraded to a ring isomorphism by a twist.

Hilbert schemes of points on the minimal resolution and soliton equations

The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C^2//G for cyclic G are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C^2//G are tau functions of 2-Toda hierarchies.

Exotic t-structure for partial resolutions of the nilpotent cone

We define and study an exotic t-structure on the bounded derived category of equivariant coherent sheaves on partial resolutions of the nilpotent cone.

On Atiyah-Segal completion for T-equivariant Hermitian K-theory

We show how derived completion can be used to prove an analogue of Atiyah-Segal completion for the $T$-equivariant Hermitian K-theory of a scheme $X$ with a trivial $T$-action, containing $\tfrac{1}{2}$ and satisfying the resolution property, where $T$ is a split torus of rank $t$. This result is an important first step towards a more general Atiyah-Segal completion theorem for Hermitian K-theory.

K3曲面上的等变线性化与旋转域

Equivariant Linearization and Rotation Domains on K3 Surfaces

本文结合商奇点解消下的等变线性化理论与超几何群方法，构造了大量具有正熵及秩1、秩2旋转域的K3曲面自同构。

关于克莱因奇点极小分解的某些有限循环商的折叠的orbifold量子上同调

On Orbifold Quantum Cohomology of Foldings of $ADE$ Resolutions

该研究计算克莱因奇点极小分解有限循环商（折叠）的\(\mathbb{C}^\times\)等变orbifold量子上同调非扭曲部分，受crepant分解猜想启发对完整的提出猜想并提供证据，还识别了所得弗罗贝尼乌斯结构与已知结构。

广义空间的单子分解

Monadic resolutions for generalized spaces

本文将 Bousfield–Kan 的单子分解方法推广到 ∞-拓扑斯，证明主纤维化引理，并据此建立不稳定等变与母题同伦理论中多类单子分解的收敛性，特别证明了代数闭域上不稳定 Adams–Novikov 谱序列对幂零连通 2-有效母题空间的收敛性。

辛分解的量子Steenrod运算

Quantum Steenrod operations of symplectic resolutions

本文研究了辛分解的量子Steenrod运算，通过验证Springer分解案例，揭示了其与mod p等效量子连接p曲率的一致性。

Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves

We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle…

Derived categories of resolutions of cyclic quotient singularities

For a cyclic group $G$ acting on a smooth variety $X$ with only one character occurring in the $G$-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold $[X/G]$ and the blow-up resolution $\widetilde Y \to X/G$. Some results generalise known facts about $X = A^n$ with diagonal $G$-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals $|G|$, we study the induced tensor products under the equivalence $D^b(\widetilde Y) \cong D^b([X/G])$ and give a 'flop-flop…

Equivariant slices for symplectic cones

The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqu…

Crepant Resolutions of Gorenstein Toric Singularities and Upper Bound Theorem

A necessary condition for the existence of torus-equivariant crepant resolutions of Gorenstein toric singularities can be derived by making use of a variant of the classical Upper Bound Theorem which is valid for simplicial balls.

Resolving mixed Hodge modules on configuration spaces

Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a f…

The open string McKay correspondence for type A singularities

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Calabi-Yau orbifolds by viewing the open theories as sections of Givental's symplectic vector space and the correspondence as a linear map of Givental spaces which identifies them. We deduce a Bryan-Graber-type statement for disk invariants and extend it to arbitrary genus zero topologies in the Hard Lefschetz case. Upon leveraging Iritani's theory of integral structures to equivariant quantum cohomology, we conjecture a general form of the symplectomorphism entering the OCRC which arises from a …

$\mathfrak S_5$-equivariant syzygies for the Del Pezzo Surface of Degree 5

The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\ma…

Root Systems and the Quantum Cohomology of ADE resolutions

We compute the C*-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C^2/G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov-Witten potential of [C^2/G].

Quantum cohomology of the Hilbert scheme of points on A_n-resolutions

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra \hat{gl}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A_n x P^1. We close with a discussion of the monodromy prope…

Enumerative geometry and geometric representation theory

This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.