Logarithmic enumerative geometry for curves and sheaves

Abstract

We propose a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas correspondence, with descendants, and study its behavior under simple normal crossings degenerations. The formulation of the logarithmic correspondence requires a matching of tangency conditions with relative insertions. This is achieved via a version of the Nakajima basis for the cohomology of the Hilbert schemes of points on a logarithmic surface. Next, we establish a strong form of the degeneration formula in logarithmic DT theory - the numerical DT invariants of the general fiber of a degeneration are determined by the numerical DT invariants attached to strata of the special fiber. The GW version of this result, which we prove in all target dimensions, strengthens currently known formulas. A key role is played by a certain exotic class of insertions, introduced here, that impose non-local incidence conditions coupled across multiple boundary strata of the target geometry. Finally, we prove compatibility of the new logarithmic GW/DT correspondence with degenerations. In particular, the logarithmic conjecture for all strata of the special fiber of a degeneration implies the traditional GW/DT conjecture on the general fiber. Compatibility is a strong constraint, and can be used to calculate logarithmic DT invariants. Several examples are included to illustrate the nature and utility of the formula.