相关论文: The mirror formula for quintic threefolds

We prove the KKV conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K3-fibered…

The orientability problem in real Gromov-Witten theory is one of the fundamental hurdles to enumerating real curves. In this paper, we describe topological conditions on the target manifold which ensure that the uncompactified moduli spaces…

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK…

We prove the integrality and finiteness of open BPS invariants of toric Calabi-Yau 3-folds relative to Aganagic-Vafa outer branes, defined from open Gromov-Witten invariants by the Labastida-Mari\~no-Ooguri-Vafa formula. Specializing to…

We calculate the matrix of the Frobenius map on the middle dimensional cohomology of the one parameter family that is related by mirror symmetry to the family of all quintic threefolds.

We study the Gromov-Witten theory of $K_{\mathsf{P}^1\times\mathsf{P}^1}$ and some Calabi-Yau hypersurface in toric variety. We give a direct geometric proof of the holomorphic anomaly euqation for $K_{\mathsf{P}^1\times\mathsf{P}^1}$ in…

In this expository paper, we discuss how Fourier-Mukai-type transformations, which we call SYZ mirror transformations, can be applied to provide a geometric understanding of the mirror symmetry phenomena for semi-flat Calabi-Yau manifolds…

Let $\mathcal X=[(\mathbb C^r\setminus Z)/G]$ be a toric Fano orbifold. We compute the Fourier transform of the $G$-equivariant quantum cohomology central charge of any $G$-equivariant line bundle on $\mathbb C^r$ with respect to certain…

For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class…

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an…

In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle…

We show that prime Fano threefolds $Y$ of genus 10 have a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.

The zeroth line bundle cohomology on Calabi-Yau three-folds encodes information about the existence of flop transitions and the genus zero Gromov-Witten invariants. We illustrate this claim by studying several Picard number 2 Calabi-Yau…

We survey mirror symmetry of Calabi-Yau manifolds from the perspective of families of Calabi-Yau manifolds and their period integrals. Special emphasis is laid on distinguished properties of the hypergeometric series of Gel'fand, Kapranov,…

We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with…

The classification of Fano varieties is an important open question, motivated in part by the MMP. Smooth Fano varieties have been classified up to dimension three: one interesting feature of this classification is that they can all be…

In this paper, we propose a definition of genus one real Gromov-Witten invariants for certain symplectic manifolds with real a structure, including Calabi-Yau threefolds, and use equivariant localization to calculate certain genus one real…

We construct all quintic invariants in five variables with simple Non-Abelian finite symmetry groups. These define Calabi-Yau three-folds which are left invariant by the action of A_5, A_6 or PSL_2(11).

This is the second part of our ongoing project on the relations between Gopakumar-Vafa BPS invariants (GV) and quantum K-theory (QK) on the Calabi--Yau threefolds (CY3). We show that on CY3 a genus zero quantum K-invariant can be written as…

We study a certain family of determinantal quintic hypersurfaces in $\mathbb{P}^{4}$ whose singularities are similar to the well-studied Barth-Nieto quintic. Smooth Calabi-Yau threefolds with Hodge numbers $(h^{1,1},h^{2,1})=(52,2)$ are…