On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin

This article presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants; which are the structure constants of the quantum cohomology ring. This c...

For a $\mathbb P^1$-orbifold $\mathscr C$, we prove that its big quantum cohomology is generically semisimple. As a corollary, we verify a conjecture of Dubrovin for orbi-curves. We also show that the small quantum cohomology of $\mathscr C$ is generically semisimple iff $\mathscr C$ is Fano, i.e. it has positive orbifold Euler characteristic.

In their paper "Quantum cohomology of projective bundles over $P^n$" (Trans. Am. Math. Soc. (1998)350:9 3615-3638) Z.Qin and Y.Ruan introduce interesting techniques for the computation of the quantum ring of manifolds which are projectivized bundles over projective spaces; in particular, in the case of splitting bundles they prove under some restrictions the formula of Batyrev about the quantum ring of toric manifolds. Here we prove the formula of Batyrev on the quantum Chern...

In this paper, we propose the formulas that compute all the rational structural constants of the quantum K\"ahler sub-ring of Fano hypersurfaces.

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov-Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincar\'{e} duality. In particular we compute the quantum cohomology of the two exceptional minuscule homogeneous var...

In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which $-K$ is very ample. An argument for the associativity in general is proposed, which also avoids resorting to the symplectic category. The enumerative predictions of Kontsevich and Manin concerning the degree of the rational curve locus in a linear system are reco...

Let $X$ be a smooth projective variety with a semi-simple quantum cohomology. It is known that the blow up $Bl(X)$ of $X$ at one point also has semi-simple quantum cohomology. In particular, the monodromy group of the quantum cohomology of $Bl(X)$ is a reflection group. We found explicit formulas for certain generators of the monodromy group of the quantum cohomology of $Bl(X)$ depending only on the geometry of the exceptional divisor.

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 ne...

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough w.r.t. the degree. As a reward we get a number of surprising enumerative formulas relating lines, conics and twisted cubics on X.

This paper considers Q-Fano 3-folds X with \rho=1. The aim is to determine the maximal Fano index f of X. We prove that f<= 19, and that in case of equality, the Hilbert series of X equals that of weighted projective space PP(3,4,5,7). From the previous version, we restrict the list of all possibilities of X to the case f>8. It will appear in a forthcoming paper.