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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 ...

Abstract. We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class AF to a Fano manifold F . We say that F satisfies Gamma Conjecture I if AF equals the Gamma class ΓF . When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by ΓF Ch(Ei) for an exceptional collection {Ei} in the derived category of coherent sheaves D b coh (F ). Gamma Conjecture II refines part (3) of Dubrovin's conjecture [18]. We prove Gamma Conjectures for projective spaces and Grassmannians.

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.Comment: Revised version, accepted to Advances in Mathematic