# Seminars & Events for Algebraic Geometry Seminar

##### Transversality and noncommutative geometry

Birationally commutative graded algebras solve the moduli problem for "point modules" over a graded ring. They have been a fruitful source of counterexamples, examples, and intuition in noncommutative ring theory. We investigate when a large subclass of birationally commutative algebras is noetherian. Formally, these are idealizer subrings of twisted homogeneous coordinate rings. In the process, we give a (purely algebro-geometric) generalization of the Kleiman-Bertini theorem.

##### Homotopy Theoretic methods on Chow varieties

The homotopy theoretic method has been applied to the algebraic cycle theory for a long period of time. In particular, it can be applied to compute topological invariants of Chow varieties. In this talk I will discuss this method in calculating the Euler Characteristic of Chow varieties. The calculation in a direct and simple way (this result has been obtained by Blaine Lawson and Stephen Yau in a different way). This technique also can be applied to Chow varieties with certain group actions and other cases. Furthermore, I will also talk about the application of the method on $l$-adic Euler-Poincare Characteristic of Chow varieties over arbitrary algebraic closed field.

##### Algebraic curves with CM

Is every Abelian variety isogenous with the Jacobian of an algebraic curve? We will study also several other questions in arithmetic geometry and show various implications. We will mention some solutions to these problems.

##### BGG correspondence and the cohomology of compact Kaehler manifolds

The cohomology algebra of the sheaf of holomorphic functions on a compact Kaehler manifold can be naturally viewed as a module over the exterior algebra of a vector space. A well-known result of Bernstein-Gelfand-Gelfand gives a correspondence between such "exterior" modules and linear complexes of modules over the symmetric algebra, i. e. the polynomial ring. I will explain how one can use a modern view on this correspondence, together with the Generic Vanishing theory developed by Green and Lazarsfeld via Hodge-theoretic methods, in order to understand subtle algebraic structures of the cohomology algebra. As a bonus, homological and commutative algebra tools can be applied on the polynomial ring side to obtain new inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kaehler manifolds.

##### Rational Simple Connectedness

Rational simple connectedness is an analog of simple connectedness for complex varieties having important applications: every 2-parameter family of rationally simply connected varieties has a rational section, and a 1-parameter family has so many rational sections that they approximate every power series section to arbitrary order. Unfortunately the condition is quite difficult to verify and is known to hold only for homogeneous spaces and also for some projective hypersurfaces satisfying a list of hypotheses. My new approach for verifying this condition works by studying a canonically defined foliation on the moduli space of rational curves on the variety.

##### Analogue of the Narasimhan-Seshadri theorem in higher dimensions and holonomy

Joint Columbia-Courant-Princeton University Algebraic Geometry Seminar

##### Mirror symetry for del Pezzo surfaces

Joint Columbia-Courant-Princeton University Algebraic Geometry Seminar

##### Smoothing surface singularities via mirror symmetry

We use the Strominger-Yau-Zaslow interpretation of mirror symmetry to describe deformations of surface singularities in terms of counts of holomorphic curves and discs on a mirror surface. In particular we prove Looijenga's conjecture on smoothability of cusp singularities. This is joint work with Mark Gross and Sean Keel, and builds on work of Gross-Siebert and Gross-Pandharipande-Siebert.

##### Rigidity properties of Fano varieties

From the point of view of the Minimal Model Program, Fano varieties constitute the building blocks of uniruled varieties. Important information on the biregular and birational geometry of a Fano variety is encoded, via Mori theory, in certain combinatorial data corresponding to the Neron–Severi space of the variety. It turns out that, even when there is actual variation in moduli, much of such combinatorial data remains unaltered, provided that the singularities are "mild" in an appropriate sense. The talk is based on joint work with C. Hacon.

##### Rational curves on hypersurfaces

This talk is on the geometry of spaces of rational curves on Fano hypersurfaces. I will talk about some of the known results on the dimension, irreducibility, and the Kodaira dimension of these spaces. I will also discuss the problem of bounding the dimension of the cones of non-free rational curves on general hypersurfaces.

##### Rational simple connectedness and Serre's "Conjecture II"

In the early 1960's Serre formulated two conjectures about Galois cohomology. The first was proved by Steinberg shortly thereafter, but the second remains open. I will discuss the proof of Serre's Conjecture II in the "geometric case": every principal homogeneous space for a bundle of simply connected, semisimple groups over a surface has a rational section. Due to the work of many people — Merkurjev and Suslin, E. Bayer and Parimala, Chernousov, Gille—the geometric case further reduces to the "split, geometric case" i.e., the bundle of groups is constant. And this case was proved by de Jong, X. He and myself using "rational simple connectedness." No background in Galois cohomology or rational connectedness will be assumed.

##### Introduction to DB singularities

##### DB pairs and vanishing theorems

##### Smoothing of surface singularities and symplectic 4-manifolds

##### Noncommutaive Hodge structure and applications

In this talk we will look at some classical problems. - rationality - from a new prospective. Our point of view will be based on Homological Mirror Symmetry. Examples will be discussed at the end.