Logarithmic Donaldson-Thomas theory

TL;DR

This paper develops a logarithmic Donaldson-Thomas theory for threefolds with divisors, extending enumerative geometry techniques to relative settings with expansions, and proposes conjectures on its properties.

Contribution

It introduces a new logarithmic Donaldson-Thomas framework for pairs (X|D), generalizing existing theories and connecting to logarithmic Gromov-Witten theory.

Findings

Constructed moduli spaces with virtual fundamental classes.

Formulated conjectures on rationality, wall-crossing, and evaluation.

Specializes to known relative ideal sheaf theories.

Abstract

Let $X$ be a smooth threefold with a simple normal crossings divisor $D$. We construct the Donaldson-Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on $X$ relative to $D$. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li-Wu theory of relative ideal sheaves when the divisor is smooth, and is parallel to recent work on logarithmic Gromov-Witten theory with expansions.