Short Course

Mohammad Farajzadeh Tehrani, Simons Center for Geometry and Physics
Normal Crossing Divisors and Varieties for Symplectic Topology

Talk 1: Normal Crossing Divisors and Varieties for Symplectic Topology

Talk 2: The Smoothability of Symplectic Normal Crossings Varieties and Possibly

Talk 3: A Multifold Symplectic Cut Construction


In the first talk, I will introduce (in more details) topological notions of symplectic normal crossings divisor and variety. These objects generalize the notion of normal crossings in algebraic geometry. I will introduce the notion of “regularization” which is a generalization of the identification map in the “symplectic neighborhood theorem” for smooth symplectic divisors.
The main result is that every NC divisor/variety, after some perturbation of the symplectic structure, admits a regularization.
After covering the preliminary definitions and results in the first talk, in the second talk,
I will discuss the smoothing problem for normal crossings varieties. I will review the known results in algebraic geometry, state a necessary and sufficient condition for the existence of smoothing in the symplectic category, and go over some examples that appear in applications. The proof is constructional and uses the regularization maps of the first talk. The basic case is known as the symplectic sum construction in the literature.

In the third talk, I will describe a multifold version of the now classical symplectic cut degeneration. It degenerates a symplectic manifold with compatible local Hamiltonian circle actions into several symplectic manifolds with normal crossings symplectic divisors and determines a normal crossings symplectic variety with a natural one-parameter family of smoothings. For symplectic cuts on compact manifolds, the latter provides a one-parameter family of degenerations of the original symplectic manifold to the corresponding normal crossings symplectic variety. Symplectic sum and cut are reverse of each other.

Along the way, I will talk about some tentative applications and the ways one can extend these results to other interesting situations.

Date & Time: 
Sunday, July 24, 2016, 9:00 - 10:30 & 11:00 - 12:30 & 14:00 - 15:30
Lecture Hall 1, Niavaran Building, IPM