Related papers: Exposition: Enumerative Geometry and Tree-Level Gr…

We study genus 1 Gromov-Witten invariants of Fano complete intersections in the projective spaces. Among other things, we show a reconstruction theorem for genus 1 invariants with only ambient insertions, and compute the genus 1 invariants…

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…

We construct a fully equivariant correspondence between Gromov-Witten and stable pairs descendent theories for toric 3-folds X. Our method uses geometric constraints on descendents, A_n surfaces, and the topological vertex. The rationality…

Via correspondence theorems, rational log Gromov--Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalized)…

We study the geometry of the Kontsevich compactification of stable maps to the Grassmannian of lines in the projective space. We consider a stratification of this space. As an application we compute the degree of the variety parametrizing…

Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it…

Making use of large-order techniques in asymptotics and resurgent analysis, this work addresses the growth of enumerative Gromov-Witten invariants---in their dependence upon genus and degree of the embedded curve---for several different…

In [LP] the authors defined symplectic "Local Gromov-Witten invariants" associated to spin curves and showed that the GW invariants of a Kahler surface X with p_g>0 are a sum of such local GW invariants. This paper describes how the local…

We introduce a geometric completion of the stack of maps from stable marked curves to the quotient stack [point/GL(1)], and use it to construct some gauge-theoretic analogues of the Gromov-Witten invariants. We also indicate the…

We establish new universal equations for higher genus Gromov-Witten invariants of target manifolds, by studying both the Chern character and Chern classes of the Hodge bundle on the moduli space of curves. As a consequence, we find new…

We consider the interplay of point counts, singular cohomology, \'etale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked,…

Let X=G/P be a homogeneous space and e_k be the class of a simple coroot in H_2(X). A theorem of Strickland shows that for almost all X, the variety of pointed lines of degree e_k, denoted Z_k(X), is again a homogeneous space. For these X…

We compute the Welschinger invariants of blowups of the projective plane at an arbitrary conjugation invariant configuration of points. Specifically, open analogues of the WDVV equation and Kontsevich-Manin axioms lead to a recursive…

An effective algorithm of determining Gromov--Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space is proposed. The Appendix is joint with E. Schulte-Geers.

In This paper, we survey recent progress on the theory of Gromov- Witten invariants on Hilbert schemes of points mainly on elliptic surfaces and simply connected minimal surface of general type. In particular, we focus on the aspects of…

A difference equation is proved for the Gromov-Witten potential of the resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten invariants of any Calabi-Yau threefold, it is further shown that similar difference…

We prove that stationary Gromov-Witten invariants of $\bp^1$ arise as the Eynard-Orantin invariants of the spectral curve $x=z+1/z$, $y=\ln{z}$. As an application we show that tautological intersection numbers on the moduli space of curves…

We introduce twisted K-theoretic Gromov-Witten invariants - in the frameworks of both "ordinary" and permutation-equivariant K-theoretic GW theory defined recently by Givental. We focus on the case when the twisting is given by the Euler…

Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich…

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…