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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 × 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 M¯g,n(V × W) to the product of stable maps M¯g,n(V ) ×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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