BibTeX
@ARTICLE{Przyjalkowski_minimalgromov–witten,
author = {Victor Przyjalkowski},
title = {Minimal Gromov–Witten ring},
journal = {Izv. Math},
year = {}
}
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Abstract
Abstract. We build the abstract theory of Gromov–Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal natural (with respect to Gromov– Witten theory) class of varieties). In particular, we consider “the minimal Gromov– Witten ring”, i. e. a commutative algebra with generators and relations of the form used in the Gromov–Witten theory of Fano variety (of unspecified dimension). Gromov–Witten theory of any quantum minimal variety is a homomorphism of this ring to C. We prove the Abstract Reconstruction Theorem which states the particular isomorphism of this ring with a free commutative ring generated by “prime twopointed invariants”. We also find the solutions of the differential equations of type DN for a Fano variety of dimension N in terms of generating series of one-pointed Gromov–Witten invariants. Consider a smooth Fano variety V of dimension N. Let H ∗ H (V, Q) ⊂ H ∗ (V, Q) be a subspace multiplicatively generated by the anticanonical class H ∈ H2 (V, Q). It is tautologically closed with respect to the multiplication in cohomology. The Gromov–Witten theory (more precisely, the set of Gromov– Witten invariants of genus 0) enables one to “deform ” the cohomology ring, that is, to define quantum
Keyphrases
gromov witten theory minimal gromov witten fano variety gromov witten invariant one-pointed gromov witten invariant differential equation quantum minimal fano variety smooth fano variety commutative algebra anticanonical class h2 cohomology ring unspecified dimension abstract reconstruction theorem quantum minimal variety minimal gromov witten ring prime twopointed invariant free commutative ring particular isomorphism type dn abstract theory
