[geometry-ml:00821] Seminar at IPMU on 2009/2/26

[Satoshi Kondo]

Dear colleagues:

We announce the following seminar at IPMU
(the Institute for the Physics and Mathematics of the Universe).

Regards,

Satoshi Kondo (IPMU)

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Speaker:	 Misha Verbitsky	(Institute of Theoretical and Experimental Physics)
Title: SYZ conjecture for hyperkaehler manifolds
Date:	 Feb 26, 2009, 13:30 - 15:30
Place: Room 630, 6th floor, Kashiwa Research Complex
Abstract:
A special Lagrangian subvariety of a Calabi-Yau manifold is a
Lagrangian subvariety $S�subset M$ with its Riemannian volume
proportional to the holomorphic volume form of $M$ restricted to $S$.
Such a subvariety is always minimal, and its deformation space is
smooth and identified with $H1(S)$. The SYZ conjecture, due to
Strominger, Yau, Zaslow, is an attempt to explain the Mirror Symmetry
in terms of fibrations by special Lagrangian tori. The only way to
construct such fibrations known so far is the one using hyperkaehler
geometry.

A hyperkaehler manifold $M$ is a Riemannian manifold equipped with an
action by quaternions $I,J,K$ on its tangent bundle, such that $I,J,K$
are parallel with respect to the Levi-Civita connection. Then $(M,I)$
is a holomorphic symplectic Kaehler manifold. Converse is also true,
by Calabi-Yau theorem. It is easy to see that any holomorphic
Lagrangian subvariety of $(M,I)$ is special Lagrangian in $(M,J)$.The
SYZ conjecture predicts that any hyperkaehler manifold can be deformed
to one which admits a holomorphic Lagrangian fibration. This would
follow if one can prove a form of "Abundance Conjecture", which is one
of the standard conjectures in algebraic geometry. I will explain the
proposed strategy of the proof of abundance conjecture for
hyperkaehler manifolds and some partial results obtained so far
(arXiv:0811.0639).
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You can check the location from
http://www.u-tokyo.ac.jp/campusmap/cam03_01_07_e.html

The schedule of the seminar can be checked from
http://db.ipmu.jp/seminar/?mode=seminar_recent