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 $(X',D') = (X,D)$ for $(X',D')\rightarrow (X,D)$ a logarithmic modification, then the new "logarithmic+modification" cobordism ring $\omega^{\text{Log+Mod}}$ collapses to the algebraic cobordism ring of Levine-Pandharipande: $\omega^{\text{LP}}\cong \omega^{\text{Log+Mod}}$