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Let $X$ be a smooth projective surface and $D$ a smooth rational ample divisor in $X$. We prove an all-genus generalization of the genus $0$ WDVV equation for primary Gromov--Witten invariants of the local 3-fold $\mathcal{O}_X(-D)$. The…

We investigate a supersymmetric generalisation of topological recursion from two perspectives: algebraic and geometric. The algebraic side concerns a recursive structure encoded in modules of a super Virasoro algebra, and the geometric…

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to G\"ottsche and Pandharipande in the zero genus case,…

We prove a structure theorem for the Gromov-Witten invariants of compact Kahler surfaces with geometric genus $p_g>0$. Under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components, the…

An algorithm to determine all the Gromov-Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro…

BPS invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch…

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in…

There is a set of remarkable physical predictions for the structure of BCOV's higher genus B-model of mirror quintic 3-folds which can be viewed as conjectures for the Gromov-Witten theory of quintic 3-folds. They are (i) Yamaguchi--Yau's…

We describe generating functions for arbitrary-genus Gromov-Witten invariants of the projective space with any number of marked points explicitly. The structural portion of this description gives rise to uniform energy bounds and vanishing…

Belmans, Oberdieck, and Rennemo asked whether natural automorphisms of Hilbert schemes of points on surfaces can be characterized by the fact that they preserve the exceptional divisor of non-reduced subschemes. Sasaki recently published…

In this paper, we directly derive generalized mirror transformation of projective hypersurfaces up to degree 3 genus 0 Gromov-Witten invariants by comparing Kontsevich localization formula with residue integral representation of the virtual…

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over the complex numbers. Then the primitive cohomology $H^{1,1}_{prim}(X)$ has, up to a sign, a natural orthonormal basis $(\eta_i)_{i\in [1, N]}$ given by certain meromorphic…

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…

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten…

We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on…

We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…

In this paper, we study holomorphic discs in K3 surfaces and defined the open Gromov-Witten invariants. Using this new invariant, we can establish a version of correspondence between tropical discs and holomorphic discs with non-trivial…

We classify complex smooth projective surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal. This completely answers a question raised by Belmans, Oberdieck and Rennemo, and extends previous works…

For each elliptic curve A over the rational numbers we construct a 2-periodic S^1-equivariant cohomology theory E whose cohomology ring is the sheaf cohomology of A; the homology of the sphere of the representation z^n is the cohomology of…

We compute the genus zero family Gromov-Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau-Zaslow formula for…