Workshop on rationally connected varieties:
talks and abstracts

May 18-22, 2015
Laboratory of Algebraic Geometry, Moscow

Home | Venue | Schedule | Program | hyperkähler Saturday

Ekaterina Amerik (HSE and Orsay)
Characteristic foliation on a smooth hypersurface in a holomorphic symplectic manifold

Let X be a projective manifold with a holomorphic symplectic form \sigma and D a smooth hypersurface in X. The characteristic foliation on D is given by the kernel of the restriction of \sigma. Hwang and Viehweg proved that it cannot be algebraic if D is of general type, unless X is a surface. On the other hand, it is always algebraic when D is uniruled. I will explain a joint work with F. Campana where we prove that, unless D is uniruled, the characteristic foliation is algebraic only on products with a surface (up to a finite etale covering); in particular, it is never the case on an irreducible holomorphic symplectic manifold of dimension at least 4. One of the crucial ingredients of the proof is the orbifold semi-positivity theorem by Campana and Paun.

Frederic Campana (Nancy)
Determinant pseudo-effectivity of quotients of orbifold cotangent bundles

We improve our former result, which established their generic semi-positivity. The proof goes along similar lines, in characteristic zero. This is a joint work with Mihai Paun.

Olivier Debarre (ENS)
Fano varieties and EPW sextics

We explore a connection between smooth projective varieties X of dimension n with an ample divisor H such that H^n = 10 and K_X = -(n-2)H and a class of sextic hypersurfaces of dimension 4 considered by Eisenbud, Popescu, and Walter (EPW sextics). This connection makes possible the construction of moduli spaces for these varieties and opens the way to the study of their period maps. This is work in progress in collaboration with Alexander Kuznetsov.

Michael MacQuillan (University Roma Tor Vergata)
Failure of smooth specialisation in etale homotopy.

Appearances not withstanding this is a talk about rational curves because they're the cause of the failure. Similarly, since homotopy groups are constant on the fibres of topological fibrations, a counterexample has to be in positive or mixed characteristic, and the specific one which I'll discuss is bi-disc quotients over Spec Z. The example also has considerable logical implications for studying boundedness of rational curves on surfaces of general type, i.e. it cannot be implied by any theorem in ACF_0. Conversely, and more substantially, this boundedness can be proved uniformly in sufficiently large primes $p$ in ACF_0 provided the surface enjoys $c_1^2> c_2$.

Jason Starr (SUNY Stony Brook)
Rational points of rationally connected and rationally simply connected varieties.

I will give three lectures on the joint work of Graber, Harris and myself and of de Jong, He and myself on existence of rational sections of fibrations by rationally connected and rationally simply connected varieties. I will discuss applications (work of Qi Zhang and Hacon-McKernan on Shokurov's conjecture, work of Debarre-Koll'ar on fundamental groups of rationally connected varieties, and work of Tian-Zong on the Weak Approximation Conjecture of Hassett-Tschinkel). No prerequisites will be required.

Lecture notes: Brauer groups and Galois cohomology of function fields of varieties (2006, 2008).

Constantin Shramov (MIRAN)
Irrational singular quartic double solids

I will speak about irrationality results for nodal quartic double solids that can be obtained using conic bundle structures and intermediate Jacobians. This is a joint work with I.Cheltsov and V. Przyjalkowski.

Zhiyu Tian (Caltech)
Hasse principle for quadrics over global function fields

Given a variety defined over a number field or the function field of a curve defined over a finite field, a natural question is to ask when can we find a rational point. We say that the variety satisfies Hasse principle (or local-global principle) if whenever we can find a rational point everywhere locally, we can find a rational point over this global field. In this talk I will give a new geometric proof (motivated by the study of rationally simply connected varieties of de Jong and Starr) of an old result: Hasse principle for quadrics defined over the function field of a curve defined over a finite field.

Jaroslaw Wisniewski (Warsaw)
Rational curves and rational homogeneous varieties

I will report on works by Munoz, Occhetta, Sola-Conde, Watanabe and myself (in different configurations) which aim at characterizing rational homogeneous varieties in terms of families of rational curves.

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Laboratory of Algebraic Geometry and its Applications