相关论文: On D. Peterson's comparison formula for Gromov-Wit…
This note compares the usual (absolute) Gromov-Witten invariants of a symplectic manifold with the invariants that count the curves relative to a (symplectic) divisor D. We give explicit examples where these invariants differ even though it…
We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the equivariant Pontryagin product on the homology of the based loop group \Omega K.
Given a smooth projective variety $X$ and a smooth nef divisor $D$, we identify genus zero relative Gromov--Witten invariants of $(X,D)$ with $(n+1)$ relative markings with genus zero orbifold Gromov--Witten invariants of multi-root stacks…
We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and…
In this paper, we establish formulas for computing genus-$0$ Gromov-Witten and Welschinger invariants of some del Pezzo varieties of dimension three by comparing to that of dimension two. These formulas are generalizations of that given in…
We study the Abramovich--Vistoli moduli space of genus zero orbifold stable maps to [Sym^2 P^2], the stack symmetric square of P^2. This compactifies the moduli space of stable maps from hyperelliptic curves to P^2, and we show that all…
We prove that genus zero Gromov--Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus zero orbifold Gromov--Witten invariants of an appropriate root stack construction along the divisor.
The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing…
Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases…
We compute the relative orbifold Gromov-Witten invariants of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$, with respect to vertical fibers. Via a vanishing property of the Hurwitz-Hodge bundle, 2-point rubber invariants are…
We give a graph-sum algorithm that expresses any genus-$g$ Gromov-Witten invariant of the symmetric product orbifold $\mathrm{Sym}^d\mathbb{P}^r:=[(\mathbb{P}^r)^d/S_d]$ in terms of "Hurwitz-Hodge integrals" -- integrals over (compactified)…
We consider the decision problem of whether a particular Gromov--Witten invariant on a partial flag variety is zero. We prove that for the $3$-pointed, genus zero invariants, this problem is in the complexity class ${\sf AM}$ assuming the…
In our previous work, we provided an algebraic proof of the Zinger's comparison formula between genus one Gromov-Witten invariants and reduced invariants when the target space is a complete intersection of dimension two or three in a…
Let $(X,E)$ be a smooth log Calabi-Yau pair consisting of a smooth Fano surface $X$ and a smooth anticanonical divisor $E$. We obtain certain higher genus local Gromov-Witten invariants from the projectivization of the canonical bundle $Z…
A. Zinger defined reduced Gromov-Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). H. -L. Chang and J. Li provided a proof of the…
We study the equivariant Gromov-Witten and Donaldson-Thomas theories of $\mathbf{P}^2$-bundles over curves. We show the equivariant GW/DT correspondence holds to first order for certain curve classes
A homology class $d \in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $\overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals…
In this paper, we define genus-zero relative Gromov--Witten invariants with negative contact orders. Using this, we construct relative quantum cohomology rings and Givental formalism. A version of Virasoro constraints also follows from it.
We prove a conjecture of Artur Elezi in a generalized form suggested by Givental. Namely, our main result relates genus-0 Gromov--Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric…
We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $\Sigma$ and the Donaldson invariants of the algebraic surface $\Sigma \times P^1$. We…