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.
Siegel varieties are locally symmetric varieties. They are important and interesting in algebraic geometry and number theory.
We construct a canonical Hodge bundle on a Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle; we obtain that the canonical Bergman metric on a Siegel variety is same as the induced Hodge metric and we describe asymptotic behavior of this unique K\"ahler-Einstein metric explicitly; depending on these properties and the uniformitarian of K\"ahler-Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal compactification.
We apply these results of Hodge bundles, to study dimension of Siegel cusp modular forms and general type for Siegel varieties