Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

On the syzygies of flag manifolds

We show that on a complex flag manifold, a very ample line bundle which is a -th power has property in the sense of Green and Lazarsfeld. This is a partial answer to a problem raised by Fulton.

     

A rank-three condition for invariant (1, 2)-symplectic almost Hermitian structures on flag manifolds

This paper considers invariant (1, 2)-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semi-simple Lie group G. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of G, and generalizes the cone-free property for tournaments related to 𝕊l (n,ℂ) case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant (1, 2)-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not B _l , l ≥ 3, G ₂ or F ₄. For B _l and F ₄ a close condition turns out to be sufficient. Received: 28 October 2001     

Nullification and cellularization of classifying spaces of finite groups

In this note we discuss the effect of the -nullification and the -cellularization over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe by means of a covering fibration, and we classify all finite groups for which is -cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of and

     

Dominant K-theory and integrable highest weight representations of Kac-Moody groups

We give a topological interpretation of the highest weight representations of Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we define a version of equivariant K-theory, KG on the category of proper G-CW complexes. We then study Kac-Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable highest weight representations of a Kac-Moody group G of compact type, maps isomorphically onto , where EG is the classifying space of proper G-actions. For the affine case, this agrees very well with recent results of Freed-Hopkins-Teleman. We also explicitly compute for Kac-Moody groups of extended compact type, which includes the Kac-Moody group E₁₀.     

Cohomology of partially ordered sets and local cohomology of section rings

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley-Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster's decomposition of local cohomology of Stanley-Reisner rings.     

On cobordism of manifolds with corners

This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.

     

Hyperbolic metric spaces and the exotic cohomology Novikov conjecture

A geometric version of the Novikov conjecture states that certain cohomology classes of a complete metric space arise from an ideal boundary. We prove this for spaces hyperbolic in the sense of Gromov.     

A generalization of the finiteness problem in local cohomology modules

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H _a ⁱ (N) is a weakly Laskerian R-module for all i < t, then Hom _R (R/a, H _a ^t (M, N)) is weakly Laskerian R-module. Also, we prove that Ext _R ⁱ (R/a, H _a ^t )) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp _R (H _a ⁱ (N)) is a finite set for all i < t, then Ext _R ⁱ (R/a, H _a ^t (N)) is weakly Laskerian R-module for all i = 0, 1.     

Ideals of the cohomology rings of Hilbert schemes and their applications

We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.     

A classification of isometry groups of homogeneous 3‐manifolds

We classify pairs where M is a 3‐dimensional simply connected smooth manifold and G a Lie group acting on M transitively, effectively with compact isotropy group.     

The jumping phenomenon of the dimensions of cohomology groups of tangent sheaf

Let X be a compact complex manifold. Consider a small deformation π: X → B of X, the dimensions of the cohomology groups of tangent sheaf H^q(X_t, TX_t) may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in H^q(X, TX) with parameter t and gets a formula for the obstructions.     

On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds

We will prove that the automorphism groups of the strongly pseudoconvex model domains in almost complex manifolds are isomorphically embedded into the automorphism group of the unit ball.     

The cohomology of the Steenrod algebra and representations of the general linear groups

Let be the algebraic transfer that maps from the coinvariants of certain -representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer . It has been shown that the algebraic transfer is highly nontrivial, more precisely, that is an isomorphism for and that is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree and apply repeatedly at most times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the -representations. As a consequence, every finite -family in the coinvariants has at most nonzero elements. Two applications are exploited.

The first main theorem is that is not an isomorphism for . Furthermore, for every 5$">, there are infinitely many degrees in which is not an isomorphism. We also show that if detects a nonzero element in certain degrees of , then it is not a monomorphism and further, for each \ell$">, is not a monomorphism in infinitely many degrees.

The second main theorem is that the elements of any -family in the cohomology of the Steenrod algebra, except at most its first elements, are either all detected or all not detected by , for every . Applications of this study to the cases and show that does not detect the three families , and , and that does not detect the family .