MS Seminar (Mathematics - String Theory)

Speaker: Misha Verbitsky (Institute of Theoretical and Experimental Physics)
Title: SYZ conjecture for hyperkaehler manifolds
Date: Thu, Feb 26, 2009, 13:30 - 15:30
Place: Room630
Related File: 48.pdf
Abstract: (Abstract in PDF is available.)

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
$H^1(S)$. 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).