Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n.1 in Cn+1. Tanaka introduced a spec- tral sequence E(p,q) r (X) with E(p,q) 1 (X) being the Kohn-Rossi cohomology group and E(k,0) 2 (X) being the holomorphic De Rham cohomology denoted by Hk h(X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated sin- gularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has re- mained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dimX = 3 and H2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.

Let f : X .仺 Y be a holomorphic map of complex manifolds, which is proper, K丯ahler, and surjective with connected fibers, and which is smooth over Y \ Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X. In our previous paper, we discussed the Nakano semi-positivity of Rqf(KX/Y . E) for q . 0 with respect to the so-called Hodge metric, when the map f is smooth. Here we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(Rqf(KX/Y . E)) 乬over Y \ Z乭 as a singular Hermitian metric with semi-positive curvature 乬over Y 乭. As a particular consequence, if Y is projective, Rqf(KX/Y . E) is weakly positive over Y \ Z in the sense of Viehweg.

On a complete noncompact K¨ahler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by m2 if the Ricci curvature is bounded from below by −2(m+1). Then we show that if this upper bound is achieved then either the manifold is connected at infinity or it has two ends and in this case it is diffeomorphic to the product of the real line with a compact manifold and we determine the metric.

We show that the first Johnson subgroup of the mapping class group of a surface  of genus greater than 1 acts ergodically on the moduli space of representations of π1() in SU2. Our proof relies on a local description of the latter space around the trivial representation and on the Taylor expansion of trace functions.

We introduce a gauge-theoretic integer valued lift of the Rohlin invariant of a smooth 4-manifold X with the homology of S1×S3. The invariant has two terms: one is a count of solutions to the Seiberg–Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of X. Each term is metric (and perturbation) dependent, and we show that these dependencies cancel as the metric and perturbation vary in a generic 1-parameter family.