Weekly Bulletin

We introduce the enumerative geometry of (C*)^2 and connect it with intersection numbers against the higher rank double ramification cycle. A certain all genus generating function of these enumerative/virtual invariants with top lambda class insertion can be explicitly calculated using a refined tropical correspondence theorem (joint work with Patrick Kennedy-Hunt and Qaasim Shafi). This generalizes the refined tropical correspondence theorem of Pierrick Bousseau. We will report on joint work in progress with Dylan Toh on a tropical correspondence theorem in genus 1 without a top lambda class insertion.

I will discuss work in progress with Bae, Feusi and Molcho on the foundations of intersection theory on the universal family of deegeneratin abelian varieties over Mumford's partial compactifications. I will give an overview of intersection theory on an abelian scheme using Fourier transforms and Beauville's weigh decomposition, and then explain how these techinuqes can be extended to the singular fibers. As a consequence, we obtain a formula for the class of the zero section using previous work of Grushevsky-Zakharov, and a collection of relations in the tautological ring

In this first talk, we will talk about a geometric refinement for log Gromov -Witten invariants of P^1-bundles on smooth projective varieties, called correlated Gromov-Witten invariants, introduced in a joint work with T. Blomme. In order to compute them, we proved a correlated refinement of Pixton double-ramification cycle formula with target varieties. We will state the formula and try to give an idea of how it is obtained as an application of the Universal DR formula of Bae-Holmes-Pandharipande-Schmitt-Schwarz.

Abelian surfaces are complex tori whose enumerative invariants seem to satisfy remarkable regularity properties. The computation of their reduced Gromov-Witten invariants in the case of primitive classes has already been well studied with many complete computations by Bryan-Oberdieck-Pandharipande-Yin. A few years ago, G. Oberdieck conjectured a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. This would close the computation of GW invariants for abelian surfaces. In this second talk, we aim to explain how correlated invariants naturally show up in the decomposition formula for abelian surfaces, and how they allow to prove the multiple cover formula conjecture for many instances. This is joint work with F. Carocci.