May 23, 2015
Laboratory of Algebraic Geometry

Workshop on rationally connected varieties:
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10:15-11:15 Jarek Wiesniewski: Symplectic quotients and hyperkaehler manifolds
11:30-12:30 Nikon Kurnosov: Absolutely trianalytic tori in the generalized Kummer variety
13:30-14:30 Justin Sawon: Isotrivial fibrations
14:45-15:45 Misha Verbitsky: Transcendental Hodge algebra
16:00-17:00 Ljudmila Kamenova: On a conjecture of Matsushita


Ljudmila Kamenova (SUNY Stony Brook)
On a conjecture of Matsushita
Matsushita conjectured that the rank of a Lagrangian fibration on a hyperkahler manifold is either zero or maximal, i.e., the fibration is either isotrivial or of maximal variation. This talk is about a joint work with Misha Verbitsky proving the conjecture.
Nikon Kurnosov (HSE)
Absolutely trianalytic tori in the generalized Kummer variety
Let Z \subset M be a closed subset of a hyperk\"ahler manifold. It is called trianalytic, if it is complex analytic with respect to I, J, K, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures (M, I, J, K) containing I. It is known that there no absolutely trianalytic tori in the Hilbert schemes of K3 and O'Grady examples. We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.
Justin Sawon (University of North Carolina)
Isotrivial fibrations
In this talk we present some results about isotrivial elliptic fibrations on K3 surfaces and isotrivial Lagrangian fibrations on higher-dimensional holomorphic symplectic manifolds. For K3 surfaces, we have a complete description of all isotrivial elliptic fibrations. These fibrations induce isotrivial Lagrangian fibrations on the Hilbert schemes of points on K3 surfaces. One can modify these to produce isotrivial Lagrangian fibrations on new symplectic V-manifolds (i.e., holomorphic symplectic orbifolds), but it remains an open problem to find examples that admit symplectic desingularizations.
Misha Verbitsky (HSE)
Transcendental Hodge algebra
Let $M$ be a projective manifold. Transcendental Hodge lattice $H^p_{tr}(M)$ of weight $p$ is the smallest rational Hodge substructure in $H^p(M)$ containing $H^{p,0}(M)$. Transcendental Hodge lattice is a birational invariant of $M$. I will prove that the direct sum $\bigoplus^p H^p_{tr}(M)$ of all transcendental Hodge lattices for $M$ is an algebra, because its orthogonal complement is an ideal, and compute it (using Yu. Zarhin's theorem) explicitly for all hyperkahler manifolds. This result has many geometric consequences; it is used in the proof of Matsushita's conjecture (joint work with Ljudmila Kamenova).
Jaroslaw Wisniewski (Warsaw)
Symplectic quotients and hyperkaehler manifolds

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Laboratory of Algebraic Geometry and its Applications