Year: 2025
Type: article
Abstract: We associate a ring R R to a log Calabi-Yau pair ( X , D ) (X,D) or a degeneration of Calabi-Yau manifolds X → B X\to B . The vector space underlying R R is determined by the tropicalization of ( X , D ) (X,D) or X → B X\to B , while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D D is maximally degenerate, then we propose that Spec R \operatorname {Spec} R is the mirror to X ∖ D X\setminus D , while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj R \operatorname {Proj} R . The main result in this paper is that R R as defined is an associative, commutative ring with unit, with associativity the most difficult part.
DOI: 10.1090/jams/1066
Authors: Mark Gross, Bernd Siebert
Institutions: University of Cambridge, The University of Texas at Austin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416873134