The Smoothability of Normal Crossings Symplectic Varieties

Abstract

Our previous paper introduces topological notions of normal crossings symplectic divisor and variety and establishes that they are equivalent, in a suitable sense, to the desired geometric notions. Friedman's d-semistability condition is well-known to be an obstruction to the smoothability of a normal crossings variety in a one-parameter family with a smooth total space in the algebraic geometry category. We show that the direct analogue of this condition is the only obstruction to such smoothability in the symplectic topology category. Every smooth fiber of the families of smoothings we describe provides a multifold analogue of the now classical (two-fold) symplectic sum construction; we thus establish an old suggestion of Gromov in a strong~form.