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Seminar by Mohammad Farajzadeh Tehrani (Stony Brook University)

Title: Symplectic normal crossings (sub-)varieties and their smoothings

2017.05.11 | Christine Dilling

Date Fri 26 May
Time 11:15 12:00
Location 1532-122 Aud-G2


Normal crossings (NC) divisors and varieties play a central role in algebraic geometry. They appear in compactification of moduli spaces, Hodge theory, mirror symmetry, etc. After a short introduction to symplectic manifolds, I will introduce symplectic notions of NC divisors and varieties, which generalize the corresponding notions in algebraic geometry. For smooth symplectic divisors, the Symplectic Neighborhood Theorem gives an identification of a neighborhood of the divisor in its normal bundle with a neighborhood of that in the ambient space. I will introduce analogue of such identifications for NC symplectic divisors and varieties, called regularization maps, and show that the space of symplectic structures admitting such regularization maps is an appropriate replacement for the space of all symplectic structures. I will finish with the following application. In complex geometry, d-semistability condition of Friedman is a well-known obstruction for smoothability of NC varieties in a smooth one parameter family. We show that the direct analogue of this condition is the only obstruction for the existence of such a smoothing in the symplectic category. Regularization maps are the essential auxiliary data for the construction of such a smoothing.

This talk is based on the joint works 1603.07661 and 1410.2573 with M. McLean and A. Zinger.

Note: This seminar is aimed at a general audience of mathematicians