相关论文: Using symmetry to count rational curves
Let X be a smooth complex projective variety, and let Y in X be a smooth very ample hypersurface such that -K_Y is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the…
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 establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these…
Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d-1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we…
In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov-Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This note provides…
We first recall Solomon's relations for Welschinger's invariants counting real curves in real symplectic fourfolds, announced in \cite{Jake2} and established in \cite{RealWDVV}, and the WDVV-style relations for Welschinger's invariants…
Gromov-Witten invariants of a symplectic manifold are a count of holomorphic curves. We describe a formula expressing the GW invariants of a symplectic sum $X# Y$ in terms of the relative GW invariants of $X$ and $Y$. This formula has…
Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more…
In algebraic geometry, Gromov--Witten invariants are enumerative invariants that count the number of complex curves in a smooth projective variety satisfying some incidence conditions. In 2001, A. Givental and Y.P. Lee defined new…
Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as…
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 develop general theory of equivariant quantum cohomology for ample Kahler manifolds and prove the mirror conjecture for projective complete intersections.
We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…
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…
We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge…
We work in the setting of Calabi-Yau mirror symmetry. We establish conditions under which Kontsevich's homological mirror symmetry (which relates the derived Fukaya category to the derived category of coherent sheaves on the mirror) implies…
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 class of perturbations of the Cauchy-Riemann equations for maps from curves to a Calabi-Yau threefold. Our perturbations vanish on components of zero symplectic area. For generic 1-parameter families of perturbations, the…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…
We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed…