Volume 2010, Issue 649

We give a version in characteristic p > 0 of Mumford's theorem characterizing a smooth complex germ of surface ( X, x ) by the triviality of the topological fundamental group of U = X \{ x }.

Let S = S g, p be a compact, orientable surface of genus g with p punctures and such that d ( S ) ≔ 3 g – 3 + p > 0. The mapping class group Mod S acts properly discontinuously on the Teichmüller space of marked hyperbolic structures on S . The resulting quotient ℳ( S ) is the moduli space of isometry classes of hyperbolic surfaces. We provide a version of precise reduction theory for finite index subgroups of Mod S , i.e., a description of exact fundamental domains (for the thin part of ). As an application we show that the asymptotic cone of the moduli space ℳ( S ) endowed with any metric in the bi-Lipschitz class of the Teichmüller metric is bi-Lipschitz equivalent to the Euclidean cone over the finite simplicial (orbi-) complex , where of S is the complex of curves of S . We also show that if d ( S ) ≧ 2, then ℳ( S ) does not admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the bi-Lipschitz class of the Teichmüller metric. These two applications confirm conjectures of B. Farb.

Under assumptions about complete intersection, we prove that Coleff–Herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the Coleff–Herrera product of the currents.

We show that for every smooth generic projective hypersurface , there exists a proper subvariety such that codim X Y ≧ 2 and for every non constant holomorphic entire map ƒ : ℂ → X one has ƒ (ℂ) ⊂ Y , provided deg X ≧ 2 n 5 . In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in .

We construct a new 20-dimensional family of projective hyper-Kähler fourfolds and prove that they are deformation-equivalent to the second punctual Hilbert scheme of a K3 surface of genus 12.

In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck–Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon–Ngô. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.

We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F -manifold is canonical. We then apply this construction to integrable systems of Hitchin type.

Consider genus g curves that admit degree d covers of an elliptic curve. Varying a branch point, we get a one-parameter family W of simply branched covers. Varying the target elliptic curve, we get another one-parameter family Y of covers that have a unique branch point. We investigate the geometry of W and Y by using admissible covers to study their slopes, genera and components. The results can be applied to study slopes of effective divisors on the moduli space of stable genus g curves.

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.