一类量子Koszul代数的${\mathbb{Z}}_{2}$-Galois 覆盖的 Hochschild上同调

考虑一类量子Koszul代数的 ${\mathbb{Z}}_{2}$-Galois覆盖$\Lambda_{\q}$, 并计算 这类代数的各阶Hochschild上同调群的维数, 进而利用道路的语言, 刻画了 Hochschild上同调环的cup积. 作为应用, 给出了这类代数的Hochschild上同调环模掉幂零理想的 代数结构.     

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.     

The cohomology of the moduli space of controllable linear systems

In this paper we explicitly describe, by generators and relations, the cohomology ring of the manifold _n,m (F) of controllable linear systems having m inputs and state-space dimension n. It is shown that the cohomology ring of _n,m (F) is isomorphic to the invariant cohomology ring of a product of projective spaces. Estimates for the cup length of the cohomology ring are obtained.     

Functors and Computations in Floer Homology with Applications, I

This paper is concerned with Floer cohomology of manifolds with contact type boundary. In this case, there is no conjecture on this ring, as opposed to the compact case, where it is isomorphic to the usual cohomology (with the quantum product). We construct two mappings in Floer cohomology and prove some functorial properties of these two mappings. The first one is a map from the Floer cohomology of M to the relative cohomology of M modulo its boundary. The other is associated to a codimension zero embedding, and may be considered as a cohomological transfer. These maps are used to define some properties of symplectic manifolds with contact type boundary. These are algebraic versions of the Weinstein conjecture, asserting existence of closed characteristics on . This is proved for many cases, Euclidean space and subcritical Stein manifolds, vector bundles, products, cotangent bundles. It is also proved that the above property implies some restrictions on Lagrangian embeddings, and also yields in certain cases, existence results for holomorphic curves bounded by the Lagrange submanifold. The last section is devoted to applications of this existence result, to real forms of Stein manifolds and obstructions to polynomial convexity in Stein manifolds. Many of our applications rely on the computation of the Floer cohomology of a cotangent bundle, that is the subject of Part II. Submitted: December 1997, revised version: February 1999.     

Quantum Cohomology of Grassmannians

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.     

Giambelli formulae for the equivariant quantum cohomology of the Grassmannian

We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.

     

On Ring Structures Under Strong Cup Products

Can we construct a strong s-stage cohomology ring on inverse systems? The following statements show the answer to this question: After defining a strong cup product, we construct a ring structure with right identity under strong cup product and obtain a strong cup product preserving homomorphism induced by a rigid system map on the level of strong s-stage cochain groups. On the other hand, we show that the coboundary formula for strong s-stage cochains does not hold.     

Quantum cohomology of the Hilbert scheme of points in the plane

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ². The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.     

Quantum cohomology of projective bundles over

In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space . Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.

     

Quantum Pieri rules for isotropic Grassmannians

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.     

Quantum D-modules and equivariant Floer theory for free loop spaces

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry'. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the notion of ``abstract quantum D-module' which generalizes the D-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology of toric complete intersections rigorously as a D-module, using Givental's model. This is shown to satisfy the axioms of abstract quantum D-module. By Givental's mirror theorem [Giv3], it follows that equivariant Floer cohomology defined here is isomorphic to the quantum cohomology D-module.     

The quantum euler class and the quantum cohomology of the Grassmannians

The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

Products in Hochschild cohomology and Grothendieck rings of group crossed products

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.     

The Quantum Cohomology Ring of Flag Varieties

We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263-277). We also give a geometric proof of the quantum Monk formula.

     

Poisson (co)homology of truncated polynomial algebras in two variables

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection. To cite this article: S. Launois, L. Richard, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Combinatorial Quantum Cohomology Ring of <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">B</Emphasis>

A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which—for G = SL_n( )—was proved previously in the paper [5].     

Degree bounds in quantum Schubert calculus

Fulton and Woodward have recently identified the smallest degree of that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result, and provide an alternative characterization of this smallest degree in terms of the rim hook formula for the quantum product.