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 ...

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.where x, y, z are the standard co-ordinate functions on C 3 . Write w = mr + w 0 with 0 ≤ w 0 < r. It is known [24,25] that the base of the miniversal qG-deformation 2 of 1 n (1, q) is isomorphic to C m−1 and, choosing co-ordinate functions a 1 , . . . , a m−1 on it, the miniversal qG-family is given explicitly by the equation:xy + (z rm + a 1 z r(m−2) + · · · + a m−1 )z w0 = 0 ⊂ 1 r (1, w 0 a − 1, a) × C m−1 We say that 1 n (1, q) is of class T or is a T -singularity if w 0 = 0, and that it is a primitive T -singularity if w 0 = 0 and m = 1. T -singularities appear in the work of Wahl [28] and Kollár-Shepherd-Barron [25]. We say that 1 n (1, q) is of class R or is a residual singularity if m = 0, that is, if w = w 0 . We say that the singularity

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Abstract. An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.