Algebraic Geometry Seminar

An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to a qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, we will restrict to the case of log Calabi-Yau pairs (X,\Delta). We prove that if (X,\Delta) admits a polarized endomorphism that preserves the boundary structure, then (X,\Delta) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. This is joint work with Joaquin Moraga and Wern Yeong.

The F-signature of a local singularity is an invariant in positive characteristics that measures the asymptotic properties of the Frobenius map. It can be used to detect regularity, strong F-regularity and other finiteness properties in positive characteristics. Thus, the F-signature seems to play a role analogous to the local volume of KLT singularities over the complex numbers. This talk concerns the behavior of the F-signature under the process of reduction to characteristic p >> 0 of a fixed complex singularity. Motivated by applications to the sizes of local fundamental groups, Carvajal-Rojas, Schwede and Tucker conjectured that the F-signatures remain uniformly bounded away from zero when we reduce a complex KLT singularity to large characteristics . We will present joint works with Yuchen Liu, and with Anna Brosowsky, Izzet Coskun and Kevin Tucker, in which we prove this conjecture in many new cases including for cones over low degree smooth hypersurfaces and most three dimensional KLT singularities. We will present some of the key ideas in the proof, which come from the K-stability theory of Fano varieties.

A lot of moduli spaces have the property that they are locally cut out by zeros of a section of a vector bundle over smooth spaces. These moduli spaces could be highly singular, but possess certain properties called quasi-smoothness. Using this, one can define on the moduli "virtual structure sheaf", which is deformation-invariant. The moduli of stable maps and quasimaps (as well as many other curve counting moduli) satisfy this property, and can be used to define quantum K-theory (or Gromov-Witten theory, Donaldson-Thomas theory etc.). In this talk I will talk about results on moduli of stable maps to a point and to Calabi-Yau threefolds.

Contact manifolds are odd-dimensional analogues of symplectic manifolds. The main aim of this talk is to present some structural results on contact structures on smooth complex projective log varieties. I will generalize a few standard results on rational curves on smooth projective varieties to the logarithmic case. Then, I will use these results to study Mori-type log contractions of contact projective log varieties.

I plan to survey some results on the analogues of Simpson's correspondence for varieties defined over an algebraically closed field of positive characteristic. Special attention will be given to quasi-projective varieties, where our study leads to some interesting problems concerning standard notions of positivity of vector bundles.