2012/03/31 by Tom Coates, Hiroshi Iritani · 11 citations
Mathematics · #Absolute convergence #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cohomology #Convergence (economics) #Flag (linear algebra) #Formal power series #Power (physics) #Power series #Series (stratigraphy) #math.AG #math.SG
paper · pdf · doi:10.1307/mmj/1441116660
published in The Michigan Mathematical Journal 64(3) (Institute of Mathematical Statistics) · 38 pages, 1 figure, v2: corrected several errors
arxiv created 2014/11/25 · openalex publication_date 2015/09/01 · arxiv updated 2015/10/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let X be a smooth projective variety. The Gromov-Witten potentials of X are generating functions for the Gromov-Witten invariants of X: they are formal power series, sometimes in infinitely many variables, with Taylor coefficients given by Gromov-Witten invariants of X. It is natural to ask whether these formal power series converge. In this paper we describe and analyze various notions of convergence for Gromov-Witten potentials. Using results of Givental and Teleman, we show that if the quantum cohomology of X is analytic and generically semisimple, then the genus-g Gromov-Witten potential of X converges for all g. We deduce convergence results for the all-genus Gromov-Witten potentials of compact toric varieties, complete flag varieties, and certain noncompact toric varieties.