Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.     

Hamiltonian Gromov-Witten invariants

In this paper we introduce invariants of semi-free Hamiltonian actions of S¹ on compact symplectic manifolds using the space of solutions to certain gauge theoretical equations. These equations generalise both the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants.     

Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and G⊂Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology of any smooth symplectic resolution X?V/G (multiplicative McKay correspondence). We prove further that if is an irreducible Weyl group and , then no smooth symplectic resolution of V/G exists unless G is of types .     

On homotopy Poisson actions and reduction of symplectic Q-manifolds

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.     

Witten's nonabelian localization for noncompact Hamiltonian spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303-368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.     

Equivariant deformations of meromorphic actions on compact complex manifolds

Meromorphicity is the most basic property for holomorphic -actions on compact complex manifolds. We prove that the meromorphicity of -actions on compact complex manifolds are not necessarily preserved by small deformations, if the complex dimension of complex manifolds is greater than two. In contrast, we also show that the meromorphicity of -actions on compact complex surface depends only on the topology (the first Betti number) of the surface. We construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold, so called a twistor space. Received January 25, 2000 / Published online October 30, 2000     

On some examples of duality groups

In [1, 3] it was shown: Theorem A. If G is the fundamental group of a finite graph of λ-dimensional duality groups with |G _o(e) : G _e | < ∞ and |G _τ(e) : G _e | < ∞ for every edge e of the corresponding G-tree, then G is an (λ + 1)-dimensional duality group. Here we use the methods of Brown and Geoghegan in [3] to obtain examples of duality groups under weaker conditions than those of Theorem A. Received: 5 June 2007     

On a relation between certain cohomological invariants

Let G be a group, the supremum of the projective lengths of the injective ZG-modules and the supremum of the injective lengths of the projective ZG-modules. The invariants and were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203-223] in connection with the existence of complete cohomological functors. If is finite then [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203-223] and , where is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422-457]. Note that if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ, Arch. Math. 89 (1) (2007) 24-32] that if is finite then G admits a finite dimensional model for , the classifying space for proper actions.We conjecture that for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type .     

Algebraic deformations of compact Kähler surfaces

A short proof is given of Kodaira's result that every compact kähler surface is a deformation of an algebraic surface under the extra assumption that all infinitesimal deformations are unobstructed.     

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.     

Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.     

On Nonprojective Block Components of Lefschetz Characters for Sporadic Geometries

This work examines the possible projectivity of 2-modular block parts of nonprojective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M ₁₂, J ₂, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co ₃, Ru, O′N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a nonprincipal block part contains a nonprojective summand. The case of O′N is additionally found to have a nonprojective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters.     

Hermitian matrices and cohomology of Kähler varieties

We give some upper bounds on the dimension of the kernel of the cup product map $H^{1}(X,\mathbb{C}) \otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})We give some upper bounds on the dimension of the kernel of the cup product map , where X is a compact K?hler variety without Albanese fibrations.     

Stable bundles with torsion Chern classes on non-Kählerian elliptic surfaces

If π:X→B is a non-Kählerian elliptic surface with generic fibreF, the moduli space of stable holomorphic vector bundles with torsion Chern classes onX has an induced fibred structure with base Pic^o(F) and the moduli space of stable parabolic bundles onB ^orb as fibre. This is specific to the non-Kähler case.     

Non-autonomous second order Hamiltonian systems

We study the existence of periodic solutions for a second order non-autonomous dynamical system containing variable kinetic energy terms. Our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other. We study sublinear problems and the existence of non-constant solutions.     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

On the Schur multiplier of special p-groups

Let G be a special p-group minimally generated by $d\ge 3$ elements and having derived subgroup of order $p^{\frac{1}{2}d(d?1)}$. Berkovich asked to find the Schur multiplier and covering groups of such groups G Berkovich and Janko (2011) [1]. We try to give an answer to this question in this article.