Quantum cohomology of flag manifolds

We study two aspects of quantum Schubert calculus: a presentation of the (small) quantum cohomology ring of partial flag manifolds and a quantum Giambelli formula. Our proof gives a relationship between universal Schubert polynomials as defined by Fulton and quantum Schubert polynomials, as defined by Fomin, Gelfand, and Postnikov, and later extended by Ciocan-Fontanine. Intersection theory on hyperquot schemes is an essential element of the proof.     

Invariant almost Hermitian structures on flag manifolds

Let G be a complex semi-simple Lie group and form its maximal flag manifold where P is a minimal parabolic (Borel) subgroup, U a compact real form and T=U∩P a maximal torus of U. We study U-invariant almost Hermitian structures on . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A₂ case.     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K₀ is connected and the action of K₀ on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^∗(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.     

Singular symplectic flops and Ruan cohomology

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient

     

Quantum cohomology of partial flag manifolds

We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties , including the quantum Pieri and quantum Giambelli formulas and the presentation.

     

Stable manifolds under very weak hyperbolicity conditions

We present an argument for proving the existence of local stable and unstable manifolds in a general abstract setting and under very weak hyperbolicity conditions.     

On graded generalized local cohomology

Let be a homogeneous Noetherian ring with local base ring (R₀,m₀) and let M,N be two finitely generated graded R-modules. Let denote the i-th graded generalized local cohomology of N relative to M with support in . We study the vanishing, tameness and asymptotical stability of the homogeneous components of . Received: 22 March 2005; revised: 25 June 2005     

Principal hierarchies of infinite-dimensional Frobenius manifolds: The extended 2D Toda lattice

We define a dispersionless tau-symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞$0/\infty$ respectively, which extends the dispersionless 2D Toda hierarchy of Takasaki and Takebe. Then we construct the deformed flat connection of the infinite-dimensional Frobenius manifold _M0$M_0$ introduced by Carlet, Dubrovin and Mertens (2011) [3] and, by explicitly solving the deformed flatness equations, we prove that the extended 2D Toda hierarchy coincides with principal hierarchy of _M0$M_0$.     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Invariant fibrations of geodesic flows

Let (Σ,g) be a compact C² finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then π₁(Σ) is almost polycyclic. On the other hand, if Σ is a compact, irreducible 3-manifold and π₁(Σ) is infinite polycyclic while π₂(Σ) is trivial, then Σ admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven.     

Moment polytopes for symplectic manifolds with monodromy

A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such “almost-toric 4-manifolds” which admits a Hamiltonian S¹-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.     

Eigenfunctions of transfer operators and cohomology

The eigenfunctions with eigenvalues 1 or −1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue 1 of a transfer operator connected to the nearest integer continued fraction algorithm. This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with coefficients in certain principal series representations.     

Generalized local cohomology and the canonical element conjecture

We study a generalization of the Canonical Element Conjecture. In particular we show that given a nonregular local ring (A,m) and an i>0, there exist finitely generated A-modules M such that the canonical map from to is nonzero. Moreover, we show that even when M has an infinite projective dimension and i>dim(A), studying these maps sheds light on the Canonical Element Conjecture.     

Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures

The Drinfeld-Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of Miura-type transformations.     

Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is two-step. New examples of Einstein solvmanifolds with three-step and four-step nilradicals are also given. This work was partially supported by Grant-in-Aid for Young Scientists (B) 14740049 and 17740039, The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

New examples of Riemannian g.o. manifolds in dimension 7

A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.     

Quantum homogeneous spaces,duality and quantum 2-spheres

For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one is based upon the notion of infinitesimal invariance with respect to certain two-sided coideals in the Hopf algebra dual to the Hopf algebra ofG. These methods are applied to the quantum group SU(2). As two-sided coideals we take the subspaces spanned by twisted primitive elements in the sl(2) quantized universal enveloping algebra. A one-parameter series of mutually non-isomorphic quantum 2-spheres is obtained, together with the spectral decomposition of the corresponding right regular representation of quantum SU(2). The link with the quantum spheres defined by Podle is established.     

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.