Logarithmic Cobordism and Donaldson-Thomas Invariants

TL;DR

The paper introduces a logarithmic cobordism framework $\omega^{\text{Log}}$ for pairs $(X,D)$ with simple normal crossings divisors, extending Levine-Pandharipande's theory to logarithmic settings. It applies this framework to logarithmic Donaldson-Thomas invariants, connecting DT theory to logarithmic geometry via relative and modified cobordism structures. A central result shows that imposing logarithmic modification relations collapses $\omega^{\text{Log+Mod}}$ onto the classical algebraic cobordism $\omega^{\text{LP}}$, i.e. $\omega^{\text{LP}} \cong \omega^{\text{Log+Mod}}$. This provides a cobordism-theoretic lens for studying logarithmic DT invariants and their computational harness.

Abstract

We introduce a logarithmic cobordism $ω^{\text{Log}}$ ring of pairs $(X,D)$ of varieties equipped with a simple normal crossings divisor $D\subset X$, analogous to the algebraic cobordism ring $ω^{\text{LP}}$ 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 $ω^{\text{Log+Mod}}$ collapses to the algebraic cobordism ring of Levine-Pandharipande: $ω^{\text{LP}}\cong ω^{\text{Log+Mod}}$

Theorems & Definitions (1)

• Theorem 1.1: Levine and Pandharipande LP, Li Li06 Let $X$ be a smooth projective threefold, then we have an equality of generating functions $\text{Z}(X) = M(q)^{\int_X c_3(T_X\otimes K_X)}$ where $M(q)$ is the MacMahon function $M(q) = \prod_n \frac{1}{(1-q^n)^n}$ A key technique in the study of Donaldson-Thomas theory is the use degenerations, which was introduced by Li-Wu in Wu. The set up is as follows: we're given a smooth projective 4-fold $\mathcal{X}$ together with a flat morphism $\pi: \mathcal{X}\rightarrow \mathbb{P}^1$ where a general fiber is a smooth threefold $X$ and the central fiber $\pi^{-1}(0) = A\bigcup_D B$ is the union of two smooth divisors $A,B$ in $\mathcal{X}$ that meet transversally along a smooth divisor $A\cap B = D$. The degeneration $\pi$ also induces a degeneration of DT moduli spaces, and in particular we get a degeneration \text{Hilb}^n(X) \rightsquigarrow \bigcup_{a+b = n}\text{Hilb}^a(A,D)\times \text{Hilb}^b(B,D) where $\text{Hilb}^a(A,D)$ is the Hilbert scheme of $n$ points of $A$ relative to the smooth divisor $D$. The relative Hilbert schemes also carry a zero dimensional virtual fundamental class, and for a pair of 3-fold and smooth divisor $(Y,E)$ we can form the relative generating function \text{Z}(Y,E) = 1 +\sum_{n\geq1} \deg [\text{Hilb}^n(Y,E)]^{\text{vir}} q^n The degeneration formula of Wu applied to our degeneration $\pi:\mathcal{X}\rightarrow \mathbb{P}^1$ tells us that $\text{Z}(X) = \text{Z}(A,D)\cdot \text{Z}(B,D)$ With the introduction of the the technique of degenerations in DT theory, the following conjecture for the relative zero dimensional DT generating function was made in MNOP02 and was proven by Levine-Pandharipande: