Punctured log Gromov-Witten theory of log modifications and double ramification cycles with target log variety

Abstract

We expand upon a previous study conducted by the author on the behavior of punctured log Gromov-Witten theory under log étale modifications $\widetilde{X} \rightarrow X$, giving expressions for log Gromov-Witten classes on $\widetilde{X}$ in terms of log Gromov-Witten classes on $X$, facilitating a complete reduction of the punctured log Gromov-Witten theory of any modification $\widetilde{X}$ of an snc log scheme $X$ to the punctured log Gromov-Witten theory of $X$. We apply this result in two settings. First, we prove a log-orbifold correspondence equating the logarithmic invariants of the canonical wall structure of Gross and Siebert with a class of orbifold invariants considered in the relative quantum cohomology ring of Tseng and You, and more generally construct a family of algebra homomorphisms from the intrinsic mirror algebra $R_{(X,D)}$ of Gross and Siebert to appropriate power series rings. Second, we show the punctured log Gromov-Witten classes of split toric bundles are effectively reconstructed in terms of the punctured log Gromov-Witten classes of the base. Additional input for the second application is the introduction and study of double ramification cycles with target log variety, generalizing the double ramification cycles with target variety investigated by Janda-Pandharipande-Pixton-Zvonkine.