Related papers: Counting curves with local tangency constraints
This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of…
We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…
We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov-Witten theory, we extend the degeneration formula to the logarithmically…
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we…
We define a signed count of real rational pseudo-holomorphic curves appearing in a one-parameter family of real Spin symplectic K3 surfaces. We show that this count is an invariant of the deformation class of the family. In the case of a…
We study the geometry of fanning curves in the Grassmann manifold of n-dimensional subspaces of $\mathbb{R}^{kn}$; we construct a complete system of invariants which solve the congruence problem. The geometry of the invariants themselves…
This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kahler surfaces with geometric genus p_g>0, for which the usual GW invariants…
In this paper, we propose a conjecture that clarifies the relationship between the number of degree d elliptic curves in complex four-dimensional projective Fano hypersurfaces and their degree d elliptic Gromov-Witten (GW) invariants. The…
Hirzebruch surfaces provide an excellent example to underline the fact that in general symplectic manifolds, Gromov-Witten invariants might well count curves in the boundary components of the moduli space. We use this example to explain in…
We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated…
Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a…
In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R.Pandharipande using Gromov-Witten theory. In this article we treat the tropical analogue of…
We use Gromov-Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general…
We recursively compute the Gromov-Witten invariants of the Hilbert scheme of two points in the plane. By studying the space of stable maps and computing virtual contributions, we use these invariants to enumerate hyperelliptic plane curves…
We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta, we compute these invariants in many cases that were…
The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of P^1. A TQFT formalism is defined via…
In this note, genus zero Gromov-Witten invariants are reviewed and then applied in some examples of dimension four and six. It is also proved that the use of genus zero Gromov-Witten invariants in the class of embedded $J$-holomorphic…
We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…
The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working…
We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with…