Generalized bochner theorems and the spectrum of complete manifolds

Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L ² harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L ² harmonic spinors on spin manifolds.     

The cohomology of the complex ofG-invariant forms onG-manifolds

The complex ofG-invariant forms and its cohomology for arbitraryG-manifolds and especially for a certain class ofG-manifolds, which are locally trivial fiber bundles over the orbit space, are considered. The transgression in the differential graded algebra of basic elements for tensor product of two identical Weil algebras of a reductive Lie groupG is calculated. This is used to get two convenient differential graded algebras with the same minimal models as the differential algebra of differential forms on the cross product of two principalG-bundles overG and ofG-invariant forms onG-manifolds of the above class. In particular, for compactG the generalization of the Cartan theorem on the cohomology of a homogeneous space is proved.Partially supported by the grant of the AMS's fSU Aid Fund     

Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

We show that the class of pairs (Γ,H) of a group and a finite index subgroup which verify a conjecture of Moore about projectivity of modules over ZΓ satisfy certain closure properties. We use this, together with a result of Benson and Goodearl, in order to prove that Moore's conjecture is valid for groups which belongs to Kropholler's hierarchy LHF. For finite groups, Moore's conjecture is a consequence of a theorem of Serre, about the vanishing of a certain product in the cohomology ring (the Bockstein elements). Using our result, we construct examples of pairs (Γ,H) which satisfy the conjecture without satisfying the analog of Serre's theorem.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

Bordism-Finiteness and Semi-simple Group Actions

We give bordism-finiteness results for smooth S ³-manifolds. Consider the class of oriented manifolds which admit an S ¹-action with isolated fixed points such that the action extends to an S ³-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontryagin class p ₁, for example p ₁ = 0, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with S ³-action as above.     

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k ₁, ..., k _r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.     

Kawamata–Viehweg vanishing on rational surfaces in positive characteristic

We prove that the Kawamata–Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W ₂(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata–Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.     

Simple singularities and topology of symplectically filling 4-manifold

Topological restrictions of symplectically filling 4-manifolds of links around simple singularities are studied by using the Seiberg-Witten monopole equations. In particular, the intersection form of minimal symplectically filling 4-manifolds of the singularity of type E ₈ is determined. Moreover, for the case of simply elliptic singularities, similar restrictions are obtained. In the proof, a vanishing theorem of the Seiberg-Witten invariant is discussed. Received: June 9, 1998.     

The $${\bar\mu}$$-invariant of Seifert fibered homology spheres and the Dirac operator

We derive a formula for the [`(m)]{\bar\mu}-invariant of a Seifert fibered homology sphere in terms of the η-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.     

Conjugations on 6-manifolds with free integral cohomology

In this article, we show the existence of conjugations on many smooth simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X ⁶ to admit a conjugation with fixed point set M ³ is the obvious one: the existence of a degree-halving ring isomorphism between the \mathbb Z₂{\mathbb Z_2}-cohomologies of X and M. As a consequence certain 6-manifolds, for which Puppe (J Fixed Point Theory Appl 2(1):85–96, 2007) proved the non-existence of non-trivial orientation-preserving finite group actions, do admit many involutions.     

Bauer–Furuta invariants under $${\mathbb{Z}_2}$$ -actions

S. Bauer and M. Furuta defined a stable cohomotopy refinement of the Seiberg–Witten invariants. In this paper, we prove a vanishing theorem of Bauer–Furuta invariants for 4-manifolds with smooth -actions. As an application, we give a constraint on smooth -actions on homotopy K3#K3, and construct a nonsmoothable locally linear -action on K3#K3. We also construct a nonsmoothable locally linear -action on K3.      

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

Barth type vanishing theorem

In this paper we give a vanishing result for cohomology groups of symmetric powers of the co-normal bundle of a non-degenerate smooth subvariety X of projective space, then we use this theorem to give a Barth type vanishing theorem.      

Bounded cohomology and non-uniform perfection of mapping class groups

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective. Oblatum 8-IX-2000 & 20-X-2000?Published online: 29 January 2001     

Polygraph Arrangements

A class of subspace arrangements, Z(n, m), known as polygraph arrangements was exploited by Haiman in order to prove the n! theorem. By showing that their intersection lattices,L (Z(n, m)), are EL-shellable, we determine the cohomology groups of the complements of the arrangements. Moreover, we generalize the shellability results to a class of lattices which deserve to be called Dowling generalizations of L(Z(n, m)). As a consequence, we obtain the cohomology groups of the complements of certain Dowling analogues of polygraph arrangements.     

On SL(3, ℝ)-actions on 4-spheres

We construct a natural, continuous SL(3, ℝ)-action on S ⁴ which is an extension of the SO(3)-action ψ of Uchida. The construction is based on the Kuiper theorem asserting that the quotient space of ℂP(2) by complex conjugation is S ⁴. We also give a new proof of the Kuiper theorem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 99–105, 2005.     

Entropy of Stationary Measures and Bounded Tangential de-Rham Cohomology of Semisimple Lie Group Actions

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and X a compact manifold (or more generally, a Borel space) on which G acts. Assume that ν is a μ -stationary measure on X, where μ is an admissible measure on G, and that the G-action is essentially free. We consider the foliation of K\ X with Riemmanian leaves isometric to the symmetric space K\ G, and the associated tangential bounded de-Rham cohomology, which we show is an invariant of the action. We prove both vanishing and nonvanishing results for bounded tangential cohomology, whose range is dictated by the size of the maximal projective factor G/Q of (X, ν). We give examples showing that the results are often best possible. For the proofs we formulate a bounded tangential version of Stokes’ theorem, and establish a bounded tangential version of Poincaré’s Lemma. These results are made possible by the structure theory of semisimple Lie groups actions with stationary measure developed in Nevo and Zimmer [Ann of Math. 156, 565--594]. The structure theory assert, in particular, that the G-action is orbit equivalent to an action of a uniquely determined parabolic subgroup Q. The existence of Q allows us to establish Stokes’ and Poincaré’s Lemmas, and we show that it is the size of Q (determined by the entropy) which controls the bounded tangential cohomology. Supported by BSF and ISF. Supported by BSF and NSF.     

The Geisser-Levine method revisited and algebraic cycles over a finite field

We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous étale cohomology. Received: 23 November 2000 / Published online: 5 September 2002