The log product formula

Herr, Leo

doi:10.2140/ant.2023.17.1281

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The log product formula

Leo Herr

Vol. 17 (2023), No. 7, 1281–1323

DOI: 10.2140/ant.2023.17.1281

Abstract

Let $V, W$ be a pair of smooth varieties. We want to compare curve counts on $V \times W$ with those on $V$ and $W$ . The product formula in Gromov–Witten theory compares the virtual fundamental classes of stable maps to a product ${\bar{M}}_{g, n} (V \times W)$ to the product of stable maps ${\bar{M}}_{g, n} (V) \times {\bar{M}}_{g, n} (W)$ . We prove the analogous theorem for log stable maps to log smooth varieties $V, W$ .

This extends results of Y.P. Lee and F. Qu, who introduced this formula after K. Behrend. We introduce “log normal cones” and “log virtual fundamental classes,” as well as modified versions of standard intersection-theoretic machinery adapted to log geometry.

Keywords

logarithmic geometry, intersection theory, algebraic geometry, algebraic stacks

Mathematical Subject Classification

Primary: 14A21, 14C17, 14N35

Milestones

Received: 18 October 2020

Revised: 5 April 2022

Accepted: 18 August 2022

Published: 30 May 2023

Authors

Leo Herr
Universiteit Leiden Netherlands

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Vol. 17, No. 7, 2023