Logarithmic Cobordism and Donaldson-Thomas Invariants

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