## Geometry Seminar - Abstracts

### Talk

Thursday 10 November, 14:00 - 15:00 in HG03.085

**Nikolai Tyurin** (MPIM Bonn)

*Special Bohr - Sommerfeld geometry*

### Abstract

The main interest is the Lagrangian geometry of compact algebraic
varieties. Every 1 - connected algebraic variety $X$ can be regarded as
a real symplectic manifold if we choose an appropriate principal
polarization, an ample line bundle. The particular choice of the
corresponding Kahler form $\omega$ regarded as symplectic form is not
important since all the forms from the same cohomology class are
equivalent thus all the corresponding "Lagrangian geometries" are
equivalent as well. The term "Lagrangian geometry" means that we study
lagrangian submanifolds on $(X, \omega)$ so such real $S \subset X$
that $dim_{\mathbb{R}}S = dim_{\mathbb{C}} X$ and $\omega|_S \equiv 0$.
The main aim is to construct certain finite dimensional moduli space of
lagrangian submanifolds (or cycles = submanifolds with singularities of
prescribed types) for algebraic varieties.

As examples: SpLag geometry proposed by N. Hitchin for Calabi - Yau varieties;
ALAG moduli space of Bohr - Sommerfeld lagrangian cycles constructed
by A. Tyurin and A. Gorodentsev.

Our way: Special Bohr - Sommerfeld submanifolds = certain mixed of SpLag - and
BS- approaches, applicable for much more wider situation
than CY, in general for any compact 1 - connected algebraic variety.

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