# Seminars & Events for Algebraic Geometry Seminar

##### A variety with non-finitely generated automorphism group

If X is a projective variety, then Aut(X)/Aut^0(X) is a countable group, but little is known about what groups can occur. I will construct a projective variety for which this group is not finitely generated, and discuss how the construction can adapted to give an example of a complex projective variety with infinitely many non-isomorphic real forms.

##### On boundedness of some algebraic fiber spaces

In this talk I will describe few examples where it is possible to obtain boundedness of the total space of a fibration from boundedness of the bases. The techniques presented will apply to certain Mori fiber spaces and to Calabi-Yau varieties with an elliptic fibration. If time permits, I will discuss some open problems.

##### Singularities on general fibres

I shall describe a canonical bundle formula which is useful to study the failure of generic smoothness, and give an outline of its proof. This is joint work with Zsolt Patakfalvi.

##### The Hodge decomposition for some non-Kahler threefolds with trivial canonical bundle.

We show that the \partial\bar{\partial}-lemma holds for the non-Kahler compact complex manifolds of dimension three with trivial canonical bundle constructed by Clemens as deformations of Calabi-Yau threefolds contracted along smooth rational curves with normal bundle of type (-1, -1), at least on an open dense set in moduli. The proof uses the mixed Hodge structure on the singular fibers and an analysis of the variation of the Hodge filtration for the smooth fibers.

##### The semi-continuity problem of normalized volume of singularities

Motivated by work in differential geometry, Chi Li introduced the normalized volume of a klt singularity as the minimum normalized volume of all valuations centered at the singularity. This invariant carries some interesting geometric/topological information of the singularity. In this talk, we show that in a Q-Gorenstein flat family of klt singularities, normalized volumes only jump down at

(possibly) countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using the valuative characterization of K-semistability developed by Li, Xu and the speaker, we show that K-semistability is a very generic property in a Q-Gorenstein flat family of Q-Fano varieties.

##### Transcendence of period maps

Period domains D can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient D/\Gamma by an arithmetic group. In the very special case when D/\Gamma is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization D\rightarrow D/\Gamma is a crucial component of the modern approach to the André-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps X\rightarrow D/\Gamma which says that atypical algebraic relations between X and D are governed by Hodge loci. We will also discuss some geometric and arithmetic applications. This is joint work with J. Tsimerman.

##### Dominating varieties by liftable ones.

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.