Related papers: On semisimplicity of quantum cohomology of $\mathb…
In the present paper the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $P^3$ or the quadric $Q^3$ is explicitely computed. Because of systematic usage of the associativity…
In this short note, we study the semi-simplicity of the quantum cohomology algebra H(V, C) of a Fano complete intersection V, and show that the subspace of H(V, C) generated by the hyperplane sections is semi-simple in the sense of Dubrovin…
We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also…
Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)-quantum cohomology, then the same is true for the blow-up of X at any number of points. This a successful test for…
The paper is dedicated to the study of algebraic manifolds whose quantum cohomology or a part of it is a semisimple Frobenius manifold. Theorem 1.8.1 says, roughly speaking, that the sum of $(p,p)$--cohomology spaces is a maximal Frobenius…
We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized…
This paper is devoted to the study of the quantum cohomology of coadjoint varieties of simple algebraic groups across all Dynkin types. We determine the non-semisimple factors of the small quantum cohomology ring and relate them to…
We show that the big quantum cohomology of the symplectic isotropic Grassmanian $IG(2,6)$ is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture…
We propose an analogue of Dubrovin's conjecture for the case where Fano manifolds have quantum connections of exponential type. It includes the case where the quantum cohomology rings are not necessarily semisimple. The conjecture is…
We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the…
We establish a theory for the existence and regularity of solutions to the cohomological equation over an accessible, partially hyperbolic diffeomorphism. As a by-product of our techniques, we show that for $r>1$, any $C^r$ homogeneous,…
In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semi-simplicity of the quantum homology algebra, and the more general…
In the first section of this note we show that the Theorem 1.8.1 of Bayer--Manin ([BaMa]) can be strengthened in the following way: {\it if the even quantum cohomology of a projective algebraic manifold $V$ is generically semi--simple, then…
We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations.…
We give elementary geometric proofs of the main theorems about the (small) quantum cohomology of partial flag varieties SL(n)/P, including the quantum Pieri and quantum Giambelli formulas and the presentation.
The (small) quantum cohomology ring of a flag manifold F encodes enumerative geometry of rational curves on F. We give a proof of the presentation of the ring and of a quantum Giambelli formula, which is more direct and geometric than the…
We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple crossings symplectic divisor. We also prove…
We consider a prime Fano 6-fold $Y$ of index 3, which is a fine quiver moduli space and a blow down of $\mathrm{Hilb}^3(\mathds{P}^2)$. We calculate the quantum cohomology ring of $Y$ and obtain Quantum Chevalley formulas for the Schubert…
A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into…
The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product…