Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties

Abstract

We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety X, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to X are free as modules over the Novikov ring. In contrast to previous results of this kind, X need not be compact. When X is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

Homological Lagrangian monodromy for some monotone tori

“…For the latter, meanwhile, note that X has an affinisation X aff , given by the spectrum of the ring of global (J 0 -)holomorphic functions on X. This comes with a (J 0 -)holomorphic map π : X → X aff , and in our toric setting this map is projective-see [18,Lemma 4.2] for example. Using this, we can prove the following.…”

Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

“…Let k be an algebraically-closed field and The critical set of W is a 0-dimensional scheme whose ring of functions (the Jacobian ring) is known to be isomorphic to quantum cohomology. For a proof, see [28,Theorem 1.3] for the statement over C and [61] for the statement in arbitrary characteristic; we restrict to algebraically closed fields here only to get an easy correspondence between k-points of the critical scheme and elements of H 1 (L; k × ). Each k-point ξ of Crit(W) therefore gives us (a) a local system ξ on L with non-vanishing Floer cohomology, and (b) a local summand QH(X; k) ξ ⊂ QH(X; k), the local ring at ξ .…”