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Register a new userCohomologies of harmonic bundles on quasi-compact Kaehler manifolds
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In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds.
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In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmids Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kahler manifolds with carefully controlled asymptotics near the compactifying divisor; such a metric is unique up to some isometry. Such an asymptotic behavior is canonical in some sense.
Let $S$ be a Riemann surface obtained by deleting a finite number of points, called cusps, from a compact Riemann surface. Let $rho: pi_1(S)to Sl(n, mathbb{C})$ be a semisimple linear representation of $pi_1(S)$ which is unipotent near the cusps. We investigate various cohomologies associated to $rho$ of $bar S$ with degenerating coefficients $L_{rho}$ (considered as a local system -- a flat vector bundle, a Higgs bundle, or a $mathcal{D}$-module, depending on the context): the v{C}ech cohomology of $j_*L_{rho}$, the $L^2$-cohomology, the $L^2$-Dolbeault cohomology, and the $L^2$-Higgs cohomology, and the relationships between them. This paper is meant to be a part of the general program of studying cohomologies with degenerating coefficients on quasiprojective varieties and their Kahlerian generalizations. The general aim here is not restricted to the case of curves nor to the one of representations that are unipotent near the divisor. The purpose of this note therefore is to illuminate at this particular case where many of the (analytic and geometric) difficulties of the general case are not present what differences will appear when we consider unipotent harmonic bundles instead of Variations of Hodge Structures where the results are known.
We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kaehler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruchs.
Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO*(2n).
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
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