Workshop on rationally connected varieties:
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talks and abstracts
Ekaterina Amerik (HSE and Orsay)
Characteristic foliation on a smooth hypersurface in a holomorphic
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)
We improve our former result, which established their generic
The proof goes along similar lines, in characteristic zero. This is a
joint work with Mihai Paun.
Determinant pseudo-effectivity of quotients of orbifold cotangent
Olivier Debarre (ENS)
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.
Fano varieties and EPW sextics
(University Roma Tor Vergata)
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 $c_1^2> c_2$.
Failure of smooth specialisation in etale homotopy.
Jason Starr (SUNY Stony Brook)
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.
Rational points of rationally connected and rationally simply
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
with I.Cheltsov and V. Przyjalkowski.
Zhiyu Tian (Caltech)
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.
Hasse principle for quadrics over global function fields
Jaroslaw Wisniewski (Warsaw)
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.
Rational curves and rational homogeneous varieties
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