Enumeration of Genus-Three Plane Curves with a Fixed Complex Structure

Enumeration of Genus-Two Curves with a Fixed Complex Structure in P^2 and P^3

We express the genus-two fixed-complex-structure enumerative invariants of P^2 and P^3 in terms of the genus-zero enumerative invariants. The approach is to relate each genus-two fixed-complex-structure enumerative invariant to the corresponding symplectic invariant.

Enumeration of real curves in CP^{2n-1} and a WDVV relation for real Gromov-Witten invariants

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 invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all c…

射影平面爆破中的曲线热带计数

Tropical enumeration of curves in blowups of the projective plane

提出一种利用tropical曲线和相对Gromov-Witten不变量递归计算射影平面所有爆破的Gromov-Witten不变量的方法，该递归公式不同于已有在零亏格和无爆破情形下的公式。

Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane

We study the growth of the genus zero Gromov-Witten invariants GW_{nD} of the projective plane P^2_k blown up at k points (where D is a class in the second homology group of P^2_k). We prove that, under some natural restrictions on D, the sequence log GW_{nD} is equivalent to lambda n log n, where lambda = D.c_1(P^2_k).

Enumerative Geometry of Calabi-Yau 5-Folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov-Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi-Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise. Two main examples are considered: the local Calabi-Yau P^2 with balanced normal bundle 3O(-1) and the compact Calabi-Y…