Logarithmic Cobordism and Donaldson-Thomas Invariants

Abstract

We introduce a logarithmic cobordism $\ddot{I}{‰}^{\text{Log}}$ ring of pairs $(X,D)$ of varieties equipped with a simple normal crossings divisor $D\subset X$, analogous to the algebraic cobordism ring $\ddot{I}{‰}^{\text{LP}}$ of Levine-Pandharipande, and we provide an application to logarithmic DT invariants. We also prove prove that if we impose the relation $(X^{\prime },D^{\prime })=(X,D)$ for $(X^{\prime },D^{\prime })\to (X,D)$ a logarithmic modification, then the new "logarithmic+modification" cobordism ring $\ddot{I}{‰}^{\text{Log+Mod}}$ collapses to the algebraic cobordism ring of Levine-Pandharipande: $\ddot{I}{‰}^{\text{LP}}\cong \ddot{I}{‰}^{\text{Log+Mod}}$

Preliminary version. Comments welcome!