Quantum periods for 3–dimensional Fano manifolds

Abstract

Abstract. The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors.Our methods are likely to be of independent interest. We rework the Mori-Mukai classification of 3-dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient V /G, where G is a product of groups of the form GLn(C) and V is a representation of G. When G = GL 1 (C) r , this expresses the Fano 3-fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3-fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the Quantum Lefschetz Hyperplane Theorem of Coates-Givental and the Abelian/non-Abelian correspondence of Bertram-Ciocan-Fontanine-Kim-Sabbah. A. IntroductionThe quantum period of a Fano manifold X is a generating function for Gromov-Witten invariants. It is a deformation invariant of X that carries detailed information about quantum cohomology. In this paper we give closed formulas for the quantum periods for all 3-dimensional Fano manifolds. As a consequence we prove a conjecture, made jointly with Golyshev, that identifies Laurent polynomials which correspond under mirror symmetry to each of the 98 deformation families of 3-dimensional Fano manifolds with very ample anticanonical bundle. We also exhibit Laurent polynomial mirrors for the remaining 7 deformation families. Our arguments rely on the classification of 3-dimensional Fano manifolds, due to Iskovskikh and Mori-Mukai: this is a difficult theorem whose proof, even today, requires delicate arguments in explicit birational geometry. On the other hand our mirror Laurent polynomials have a simple combinatorial definition and classification. Given a suitable mirror theorem this classification would give a straightforward, combinatorial, and uniform alternative proof of the classification of 3-dimensional Fano manifolds. The general outlines of such a mirror theorem are beginning to emerge [2,3,38,40,70], as are some promising approaches to proving it [24][25][26][27]42,43].Let X be a Fano manifold, that is, a smooth projective variety such that the anticanonical bundle −K X is ample. The quantum period G X (t) of X, defined in §B below, is a generating function for certain genus-zero Gromov-Witten invariants of X. It satisfies a differential equation:where D = t d dt and the p k are polynomials, called the quantum differential equation for ...

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

“…. Definition 4 (see [CCGK16]). An integral polygon is called of type A n , n ≥ 0, if it is a triangle such that two its edges have integral length 1 and the rest one has integral length n. (In other words, its integral points are 3 vertices and n − 1 points lying on the same edge.)…”

“…The Fano variety X 2−1 is a hypersurface section of type (1, 1) in P 1 × X 1−11 in an anticanonical embedding; in other words, it is a complete intersection of hypersurfaces of types (1, 1) and (0, 6) in P 1 × P (1, 1, 1, 2, 3). The Fano variety X 2−2 is a hypersurface in a certain toric variety, see [CCGK16]. The Fano variety X 2−3 is a hyperplane section of type (1, 1) in P 1 × X 1−12 in an anticanonical embedding; in other words, it is a complete intersection of hypersurfaces of types (1, 1) and (0, 4) in P 1 × P (1, 1, 1, 1, 2).…”

Minimality and Mutation-Equivalence Of polygons

“…Let b = r E q, where 2/j q 1/2, by (11) and (12). Then r 2 E (− jq 2 + jq −1) jd B , and by considering the minimum value achieved by the quadratic in q on the lefthand side of this inequality we obtain…”

“…This gives polygon number (1.1) in Table 4. m P = r E > 3, k = 1, no edge parallel to E, E vertical, l = 0, j 5: Now P has vertices (b − r E , r E ), (b, r E ), (b, r E − jb), and is contained in the rectangle [b − r E , b] × [r E − jb, r E ] where 5 j < 11, by (14), 3 < r E 9, by (15), and 2r E /j b r E /2 by (11) and (12). This gives three minimal polygons: numbers (1.2), (1.3), and (2.6) in Table 4.…”

“…. , x ±1 n ] [5,8,11,12]. Under this correspondence, the regularized quantum period G X of X -a generating function for Gromov-Witten invariantscoincides with the classical period π f of f -a solution of the associated PicardFuchs differential equation -given by…”