We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Graβmannian. It has all the properties asked for by Teixidor.

We study the Kähler-Ricci flow on noncompact Kähler manifolds and provide conditions under which the flow has a long time solution converging to a complete negative Kähler-Einstein metric. We also study the complex parabolic Monge-Ampère equation.

Let f : X → Y be a semistable family of complex abelian varieties over a curve Y of genus g(Y ), and smooth over the complement of s points. If 𝓁 denotes the rank of the nonflat (1, 0) part F^1,0 of the corresponding variation of Hodge structures, the Arakelov inequality says that

2 ∙ deg(F^1,0) ≤ 𝓁 ∙ (2g(Y) - 2 + s).

The family reaches this bound if and only if the Higgs field of the variation of Hodge structures is an isomorphism. The latter is reflected in the existence of special Hodge cycles in the general fibre, and the base of such a family is a Shimura curve.

In particular, for s ≠ 0, such a family must be isogenous to the 𝓁-fold product of a modular family of elliptic curves, and aconstant abelian variety.

For s = 0, if the flat part of the variation of Hodge structures is defined over the rational numbers, one finds the family to be isogenous to the product of several copies of a family h : Z → Y , and a constant abelian variety. In this case, h : Z → Y is obtained from the corestriction of a quaternion algebra A, defined over a totally real numberfield F, and ramified over all infinite places but one.

In case the flat part is not defined over the rational numbers, one still can classify the corresponding variations of Hodge structures.

We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that M#N admits a conformally flat Riemannian metric.

For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves. There exists a so-called theta line bundle on the family of moduli spaces. When the Kodaira-Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves. As an application, the so-called Hitchin connection on the direct image of (powers of) the theta line bundle is derived naturally from the identification theorem. A logarithmic extension to certain singular stable curves is also presented in this paper.