Cohomological invariants of a variation of flat connection

In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to a $r$-simplex whose points parametrize flat connections on a smooth manifold $X$. These invariants lie in degrees $(2p-r-1)$-cohomology with $C/Z$-coefficients, for $p> r\geq 1$. In turn, this corresponds to a homomorphism on the higher homology groups of the moduli space of flat connections, and taking values in $C/Z$-cohomology of the underlying smooth manifold $X$.Comment: 15 p. Final version, to appear. arXiv admin note: text overlap with arXiv:1310.000

The Chern character of a parabolic bundle, and a parabolic Reznikov theorem in the case of finite order at infinity

In this paper, we obtain an explicit formula for the Chern character of a locally abelian parabolic bundle in terms of its constituent bundles. Several features and variants of parabolic structures are discussed. Parabolic bundles arising from logarithmic connections form an important class of examples. As an application, we consider the situation when the local monodromies are semi-simple and are of finite order at infinity. In this case the parabolic Chern classes of the associated locally abelian parabolic bundle are deduced to be zero in the rational Deligne cohomology in degrees $\geq 2$.Comment: Adds and corrects reference

Regulators of canonical extensions are torsion: the smooth divisor case

In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $>1$) are torsion, of a flat bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion

Murre's conjectures and explicit Chow–Kunneth projectors for varieties with a nef tangent bundle

In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow-Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle, explicit Chow-Kunneth projectors are obtained which satisfy Murre's conjectures, and the motivic Hard Lefschetz theorem is verified

Tautological ring of the moduli space of generalised Parabolic line bundles on a curve

In this article, we consider the tautological ring containing the extended Brill–Noether algebraic classes on the normalization of the compactified Jacobian of a complex nodal projective curve (with one node). This smallest ℚ-subalgebra of algebraic classes under algebraic equivalence, stable under extensions of the maps induced by multiplication maps, Pontrayagin product and Fourier transform, is shown to be generated by pullback of the Brill–Noether classes of the Jacobian of the normalized curve and some natural classes

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