A note on morphic cohomology and algebraic cycles

We give examples of smooth projective complex varieties of dimension d ≥ 4 and primes ℓ such that the morphic cohomology group is infinite, and is not finitely generated as a rational vector space. In particular, for these examples the semi-topological K-group has infinite dimension.     

Hodge structures on cohomology algebras and geometry

We study restrictions on cohomology algebras of compact Kähler manifolds, imposed by the presence of a polarized Hodge structure on cohomology groups, compatible with the cup-product, but not depending on the h ^p,q numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the formality condition, the Lefschetz property and having commutative or trivial π ₁, but not having the cohomology algebra of a compact Kaehler manifold. We also prove a stability theorem for these restrictions : if a compact Kähler manifold is homeomorphic to a product X × Y, with one summand satisfying b ₁ = 0, then the cohomology algebra of each summand carries a polarized Hodge structure.     

Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps $$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map $$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with \(H^n \left( {X,\mathcal{O}_X } \right) \ne 0\) , wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.     

Variations of mixed Hodge structure, Higgs fields, and quantum cohomology

Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ^* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds. Received: 14 February 2000     

Products in Hochschild cohomology and Grothendieck rings of group crossed products

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.     

Hodge cohomology of some foliated boundary and foliated cusp metrics

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of L² harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required.     

Cyclic cohomology and the family Lefschetz theorem

Any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra. We prove that such cocycle furnishes additive maps from the vertically equivariant K-theory to the scalars. This enables to associate to any closed current on the base of the fibration, a Lefschetz formula for fiber-preserving isometries. Using geometric operators on the base, we deduce the integrality of some characteristic numbers. Received: 28 June 2001 / Published online: 1 February 2002     

The crossed coproduct theorem and Galois cohomology

We give a cohomological interpretation of the Brauer group of a coalgebra in terms of Galois coextensions and Galois cohomology. There is a crossed coproduct structure theorem, and the co-version of the classical splitting theorem holds for the Brauer group of an irreducible coreflexive coalgebra but it does not hold in general.     

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.     

The Grothendieck duality theorem via Bousfield's techniques and Brown representability

Grothendieck proved that if is a proper morphism of nice schemes, then has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown's representability theorem. 1It follows almost as immediately, by ``smashing' arguments, that the adjoint is given by tensor product with a dualising complex. Verdier's base change theorem is an easy consequence.

     

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let M be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E be an r.i. space on (0,∞). Let E(M) be the associated symmetric space of measurable operators. Then to any bounded linear map T from E(M) into a Hilbert space H corresponds a positive norm one functional f∈E(2)^∗(M) such that

     

A separation theorem and Serre duality for the Dolbeault cohomology

LetX be a complex manifold with finitely many ends such that each end is eitherq-concave or (n−q)-convex. If , then we prove thatH ^pn−q (X) is Hausdorff for allp. This is not true in general if (Rossi’s example withn=2 andq=1). If all ends areq-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for ). Moreover the result was already known in the case when theq-concave ends can be ‘filled in’ (again also for ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, . This research was partially supported by TMR Research Network ERBFMRXCT 98063.     

Bounds and conjectures for the signless Laplacian index of graphs

Using the AutoGraphiX system, we obtain conjectures of the form l(n)?q₁⊕i(G)?u(n) where q₁ denotes the signless Laplacian index of graph is one the four operations is another invariant chosen among minimum, average and maximum degree, average distance, diameter, radius, girth, proximity, remoteness, vertex, edge and algebraic connectivities, independence number, domination number, clique number, chromatic number and matching number, Randi? index, l(n) and u(n) are best possible lower and upper bounds function of the order n of G. Algebraic conjectures are obtained in 120 cases out of 152 and structural conjectures in 12 of the remaining cases. These conjectures are known, immediate or proved in this paper, except for 17 of them, which remain open.     

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck’s conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher’s bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.