# Geometric Analysis and Topology Seminar

#### On Maslov Classes of Bohr - Sommerfeld Lagrangian Embeddings to Pseudo-Einstein Manifolds

**Speaker:**
Nikolai Tyurin, Dubna & IAS

**Location:**
Warren Weaver Hall 1013

**Date:**
Friday, February 9, 2007, 1 p.m.

**Synopsis:**

Maslov class for a lagrangian immersion into a symplectic vector space plays an essential role in symplectic geometry. This class appears in several fundamental problems forming a rather broad spectrum: from the classical minimality problem for lagrangian submanifolds to the semiclassical quantization due to Maslov. Our interest in generalization of the Maslov class to the case of a compact symplectic manifold is motivated by this last application. Namely ALG(a) - quantization (quantization, based on Algebraic Lagrangian Geometry, proposed by A. Tyurin and A. Gorodentsev) needs a correction for BPU - map to become holomorphic. A standard way to generalize Maslov class is to introduce Maslov index in the general compact symplectic setup. However the index doesn't inherit all the properties of the Maslov class, f.e. the minimality property.

In contrast for a wide variety of lagrangian submanifolds (which are called Bohr - Sommerfeld submanifolds with respect to the anticanonical bundle) one can define an integer valued 1-cohomology class if the ambient manifold is a compact simply connected symplectic manifold with a canonical class proportional to the cohomology class of the symplectic form (we call it pseudo-Einstein manifold following Fukaya).

As a byproduct of this construction one can find an appropriate correction to ALG(a) - quantization, which makes BPU-map holomorphic, in the pseudo Einstein case.