The Log Product Formula in quantum $K$-theory

TL;DR

This paper establishes a formula relating log Gromov-Witten invariants of product varieties to those of individual factors within a new log K-theory framework, advancing computational tools in algebraic geometry.

Contribution

It introduces a log K-theory and proves a product formula for log Gromov-Witten invariants, expanding the theoretical foundation of quantum K-theory.

Findings

Derived a formula expressing invariants of products in terms of factors

Introduced log virtual fundamental classes in K-theory

Validated functorial properties of the new classes

Abstract

We prove a formula expressing the $K$-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. The proof requires introducing log virtual fundamental classes in $K$-theory and verifying their various functorial properties. We introduce a log version of $K$-theory and prove the formula there as well.