Higher order operations in Deligne cohomology

Oblatum 10-VII-1993 & 26-IX-1994     

Integral Deligne cohomology for real varieties

We develop an integral version of Deligne cohomology for smooth proper real varieties. For this purpose the role played by singular cohomology in the complex case has to be replaced by the ordinary bigraded Gal(mathbbC/mathbbR){Gal(mathbb{C}/{mathbb{R}})}-equivariant cohomology of Lewis et al. (Bull Am Math Soc (N.S.) 4(2):208–212, 1981), the equivariant counterpart of singular cohomology. The theory is aimed at giving more precise information about the 2-primary components of regulators. We establish basic properties and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2.     

Twisted product and cohomology

LetH be a Hopf algebra,H ₁ be a sub-Hopf algebra ofH, H ₂ be the quotient Hopt algebra ofH modularH ₁. This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex ofH has the same cohomology algebra with a twisted product of the cobar complexes ofH ₁ andH ₂. Supported by National Natural Science Foundation of China     

Chern classes in Deligne cohomology for coherent analytic sheaves

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.     

Chern invariants of some flat bundles in the arithmetic Deligne cohomology

In this note, we investigate the cycle class map between the rational Chow groups and the arithmetic Deligne cohomology, introduced by Green–Griffiths and Asakura–Saito. We show nontriviality of the Chern classes of flat bundles in the arithmetic Deligne Cohomology in some cases and our proofs also indicate that generic flat bundles can be expected to have nontrivial classes. This provides examples of non-zero classes in the arithmetic Deligne cohomology which become zero in the usual rational Deligne cohomology.     

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H ¹(A,T), where T is a maximal compact torus of , the identity component of the group of invariants G ^A . We then prove that the natural map is a bijection, reducing the calculation of H ¹(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.      

Twisted cohomology pairings of knots I; diagrammatic computation

We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every link in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.     

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra A, we define a group homomorphism Hdet : Autdg(A) → k^×, called the homological determinant.As applications, we present a sufficient condition for the invariant DG suba...     

A remark on a theorem by Deligne

We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.

     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.