We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the previously determined analytic continuation of the fundamental solution of the QDE of the Hilbert scheme. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves.
The equivariant orbifold Gromov-Witten theory of the symmetric product of the plane is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture.
We give a classification of birational transformations on smooth projective surfaces
which have a Zariski-dense set of noncritical periodic points. In particular, we show
that if the first dynamical degree is greater than one, the union of all noncritical periodic
orbits is Zariski-dense.
In this paper, we explore the connections between the Minimal Model Program and the
theory of Berkovich spaces. Let k be a field of characteristic zero and letX be a smooth and projective
k((t))-variety with semi-ample canonical divisor. We prove that the essential skeleton of X coincides
with the skeleton of any minimal dlt-model and that it is a strong deformation retract of the Berkovich
analytification of X. As an application, we show that the essential skeleton of a Calabi-Yau variety
over k((t)) is a pseudo-manifold.