A higher rank Lefschetz formula

Since the early seventies flows with the Lefschetz property were studied by several authors. In this paper a Lefschetz formula is proved for the geodesic flow of a compact locally symmetric space. The flow is described in terms of actions of split tori of various dimensions and the geometric side of the Lefschetz formula is a sum over closed geodesics which correspond to a given torus. The cohomological side is given in terms of Lie algebra cohomology.     

A remark on Barth’s connectivity theorem

It has been remarked by Hartshorne, that Barth’s theorem for a smooth projective X follows from the strong Lefschetz theorem for the cohomology of X. Using the strong Lefschetz theorem for intersection cohomology, we give an extension of Barth’s theorem to singular X. This naturally raises several questions concerning possible Barth theorems on the level of intersection cohomology.     

Lefschetz decomposition for de Rham cohomology on weakly Lefschetz symplectic manifolds

For a compact symplectic manifold which is s-Lefschetz which is weaker than the hard Lefschetz property, we prove that the Lefschetz decomposition for de Rham cohomology also holds.     

Hard Lefschetz theorem for nonrational polytopes

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a general polytope.     

The Lefschetz coincidence theorem in o-minimal expansions of fields

In this paper we prove the Lefschetz coincidence theorem in o-minimal expansions of fields using the o-minimal singular homology and cohomology.     

Quantum differential systems and construction of rational structures

We consider mirror symmetry (A-side vs B-side, namely singularity side) in the framework of quantum differential systems. We focuse on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau–Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani’s results), depending on numerous combinatorial data which are rearranged after the mirror transformation.     

The Picard–Lefschetz formula for p-adic cohomology

In this paper, we discuss a p-adic analogue of the Picard–Lefschetz formula. For a family with ordinary double points over a complete discrete valuation ring of mixed characteristic (0,p), we construct vanishing cycle modules which measure the difference between the rigid cohomology groups of the special fiber and the de Rham cohomology groups of the generic fiber. Furthermore, the monodromy operators on the de Rham cohomology groups of the generic fiber are described by the canonical generators of the vanishing cycle modules in the same way as in the case of the ℓ-adic (or classical) Picard–Lefschetz formula. For the construction and the proof, we use the logarithmic de Rham–Witt complexes and those weight filtrations investigated by Mokrane (Duke Math. J. 72(2):301–337, 1993).      

Algebraic De Rham cohomology

We announce the development of a theory of algebraic De Rham cohomology and homology for arbitrary schemes over a field of characteristic zero. Over the complex numbers, this theory is equivalent to singular cohomology. Applications include generalizations of theorems of Lefschetz and Barth on the cohomology of projective varieties.     

Density of monodromy actions on non-abelian cohomology

In this paper we study the monodromy action on the first Betti and de Rham non-Abelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski-dense monodromy orbit. In particular, we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.     

Cohomology structure for a Poisson algebra:Ⅱ

For a Poisson algebra, we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor. We show that the(generalized) deformation quantization is equivalent to the formal deformation for Poisson algebras under certain mild conditions. Finally we construct a long exact sequence, and use it to calculate the Poisson cohomology groups via the Yoneda-extension groups of certain quasi-Poisson modules and the Lie algebra cohomology groups.     

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.     

Zeroth Poisson Homology,Foliated Cohomology and Perfect Poisson Manifolds

We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.     

Some consequences of the Riemann hypothesis for varieties over finite fields

We deduce from Deligne's form of the Riemann hypothesis and the hard Lefschetz theoremin l-adic cohomology the corresponding facts for any “reasonable” cohomology theory, in particular for crystalline cohomology, and give some applications to algebraic cycles.     

Local contribution to the Lefschetz fixed point formula

Summary For a class of self-correspondencesC calledweakly hyperbolic, we give a computable formula for the contribution of a fixed point component to the Lefschetz number ofC. The formula applies to Lefschetz numbers of cohomology with coefficients in a constructible complex of sheaves (such as intersection homology).Oblatum 9-XI-1990 & 29-IV-1992In memory of J.L. VerdierPartially supported by NSF grant # DMS8802638 and DMS9001941Partially supported by NSF grant # DMS8803083 and DMS9106522     

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     

Meyer's signature cocycle and hyperelliptic fibrations

We show that the cohomology class represented by Meyer's signature cocycle is of order in the 2-dimensional cohomology group of the hyperelliptic mapping class group of genus . By using the -cochain cobounding the signature cocycle, we extend the local signature for singular fibers of genus 2 fibrations due to Y. Matsumoto [18] to that for singular fibers of hyperelliptic fibrations of arbitrary genus and calculate its values on Lefschetz singular fibers. Finally, we compare our local signature with another local signature which arises from algebraic geometry. Received: 6 August 1998 / in final form: 24 February 1999     

Poisson cohomology in dimension two

It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures onG ² (G=ℝ or ℂ) and recall a result given by Arnold. Then we will compute locally the Poisson cohomology of a particular type of Poisson structure.     

Vanishing theorems for ample vector bundles

Summary. The main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety. In particular, we extend classical theorems of Griffiths and Le Potier to the whole Dolbeault cohomology, prove a variant of an uncorrect conjecture of Sommese, and answer a question of Demailly. As an application, we prove conjectures of Debarre and Kim for branched coverings of grassmannians, and extend a well-known Barth–Lefschetz type theorem for branched covers of projective spaces, due to Lazarsfeld. We also obtain new restriction theorems for certain degeneracy loci. Oblatum 10-IV-1996 & 22-V-1996     

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes' C^∗-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.     

An Example Using Improved Lefschetz Duality

A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S~(4n-1)→ S~(2n)× S~m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.