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Quantum periods for 3-dimensional Fano manifolds. (English) Zbl 1348.14105

Let \(X\) be a Fano threefold. That is, \(X\) is a threefold with ample anticanonical divisor \(-K_X\). Mirror symmetry conjectures that \(X\) has a mirror which is a Landau-Ginzburg model \(f : (\mathbb{C}^\times)^3 \to \mathbb{C}\) for \(f\) a Laurent polynomial. There are different ways to describe this mirror relation (in terms of special Lagrangian torus fibrations as in the SYZ conjecture, in terms of the Gross-Siebert program, etc.). The approach taken in the paper under review is from an enumerative point of view. To \(f\) is associated its period \(\pi_f\) and the mirror symmetry conjecture states that \(\pi_f\) equals the quantum period of \(X\). The latter is a certain generating function of genus 0 Gromov-Witten invariants of \(X\).
With this enumerative mirror statement in mind, the correct equivalence relation to consider on the symplectic side is that of deformation equivalence. Indeed, Gromov-Witten invariants are deformation-invariant and hence so are quantum periods. Similarly, on the Landau-Ginzburg model side two Laurent polynomials are considered equivalent if they induce the same period.
The present paper under review presents the following three major results:
\((1)\) To each of the 105 deformation families of Fano threefolds, the construction of a key variety (an element of the family) that is well suited for calculations of Gromov-Witten invariants.
\((2)\) Through these key varieties, the calculation of the resulting 105 quantum periods.
\((3)\) The observation that the quantum periods of Fano threefolds \(X\) with very ample \(-K_X\) correspond to the Minkowski periods of manifold type. Up to equivalence, there are 98 classes of each.
More precisely:
\((1)\) The authors provide a reformulation of the Mori-Mukai classification of Fano threefolds, with different models. The key varieties that the authors exhibit are constructed inside a GIT quotient given by a representation of a product of general linear groups. This is proven by considering each deformation class individually.
\((2)\) As these key varieties lie in suitable GIT quotients, this puts at disposition a vast array of techniques that are available to calculate Gromov-Witten invariants. These include Givental’s mirror theorem for toric complete intersections, the quantum Lefschetz theorem of Coates and Givental, as well as the abelian/non-abelian correspondence of Bertram, Ciocan-Fontanine, Kim and Sabbah. Using these methods, all of the 105 quantum periods are calculated.
\((3)\) This proves the corresponding conjecture made by the authors and Golyshev in a previous paper. It points to a novel approach to treat the classification of Fano manifolds. Namely, it suggests a two-step approach:
\((a)\) Prove a mirror statement relating each deformation-family of Fano manifolds to an equivalence class of Laurent polynomials.
\((b)\) Classify the Laurent polynomials that appear, up to equivalence.
The advantage of this approach is that compared to Fano manifolds, Laurent polynomials are much simpler (and more efficient) to classify as they are given by combinatorial data. The present implementation of this program in the case of Fano threefolds with very ample anticanonical divisor shows that it is a promising approach.
The paper is structured in two main parts. Sections A to H introduce the problems and cover the background. Then, sections 1 to 105 treat each Fano threefold deformation class separately, calculating the quantum periods and pointing out the Minkowski periods of manifold type when available. It ends with section 106, which describes the case of a Fano manifold of dimension 66 which does not admit a description in terms of GIT quotients as above, and the Appendix, which points to an online supplement containing all the mirror Laurent polynomials (in dimension 3).
The paper is overall well-written and includes a good amount of introductory material. It will be of interest to researchers working on mirror symmetry and Gromov-Witten invariants, as well as to algebraic geometers interested in birational geometry and classification problems.

MSC:

14J45 Fano varieties
14J33 Mirror symmetry (algebro-geometric aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J30 \(3\)-folds
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14J10 Families, moduli, classification: algebraic theory
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