We compute the Small Quantum Cohomology of Hilb^2(P^1xP^1) and determine recursively most of the Big Quantum Cohomology. We prove a relationship between the invariants so obtained and the enumerative geometry of hyperelliptic curves in P^1xP^1. This extends the results obtained by Graber for Hilb^2(P^2) and hyperelliptic curves in P^2.
We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains.
For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal $Z_{A,G}^{-1}$ is a modular form of weight $\frac{1}{2}e(A/G)$ for the congruence subgroup $Γ_{0}(|G|)$, and give explicit expressions in terms of eta products. Refined formulas for the $χ_{y}$-genera of $\text{Hilb}(A)^{G}$ are also given. For the group generated by the standard involution $τ: A \to A$, our formulas arise from the enumerative geometry of t…
We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, \ldots, x_n) \in (\mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, \ldots, x_n \in \mathbb{P}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $d\times n$ matrix $M$ an $n$-ary operation $[M]_\hbar$ on the small equivariant quantum cohomology ring of $\mathbb{P}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M \mapsto [M]_\hbar$ is a …
Quantum cohomology of the Hilbert scheme of points on A_n-resolutions
本文确定了与A_n型奇点有关的表面分解点的Hilbert方案的等变量子上同调的两点不变量。编码这些不变量的算子表示为仿射李代数\hat{gl}（n+1）在其基本表示上的作用。假设一定的非简并性猜想，这些算子决定了量子上同调环的完整结构。证明了量子上同调与A_n x P^1的Gromov-Witten/Donaldson-Thomas理论之间的关系。最后，我们讨论了相关量子微分方程的单值性质，并推广到D型和E型奇点。
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…
Equivariant cohomology, symmetric functions and the Hilbert schemes of points on the total space of the invertible sheaf O(-2) over the projective line
Orbifold quantum cohomology of the symmetric product of A_r
设A_r是循环商奇点C^2/ Z_{r+1}的最小分解.研究了A_r的n重对称积栈[Sym^n（A_r）]的等变量子上同调环.我们用除数类来计算量子乘法的算符。在非减损猜想的假设下，这些算子完全决定了环的结构，从而对[Sym^n（A_r）]和Hilb^n（A_r）上的Crepant归结猜想给出了肯定的回答.更引人注目的是，这使我们能够完成一个四面体的等价关系，它将[Sym^n（A_r）]/Hilb^n（A_r）的格罗莫夫-威滕理论与A_r x P^1的相关格罗莫夫-威滕/唐纳森-托马斯理论联系起来。
In algebraic geometry, Gromov--Witten invariants are enumerative invariants that count the number of complex curves in a smooth projective variety satisfying some incidence conditions. In 2001, A. Givental and Y.P. Lee defined new invariants, called $K$-theoretical Gromov--Witten invariants. These invariants are obtained by replacing cohomological constructions used in the definition of the usual Gromov--Witten invariants by their $K$-theoretical analogues. Then, an essential question is to understand how these two invariants are related. In 2013, Iritani-Givental-Milanov-Tonita show that $K$-…
