Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold

Generalizing the Harder–Narasimhan filtration of a vector bundle it is shown that a principal G-bundle over a compact K?hler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where , this reduction is the Harder–Narasimhan filtration of the vector bundle associated to by the standard representation of . The reduction of in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and the canonical reduction, then the first condition says that the principal L-bundle obtained by extending the structure group of the P-bundle using the natural projection of P to L is semistable. Denoting by the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in , the associated vector bundle is of positive degree; here is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to for is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here. Received: 10 November 1999 / Revised version: 31 October 2001 / Published online: 26 April 2002     

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001     

Bogomolov inequality for Higgs parabolic bundles

We show that over some smooth projective varieties every semistable Higgs logarithmic vector bundle is semistable in the ordinary sense, hence satisfies Bogomolov inequality. More generaly, we prove that semistable Higgs parabolic vector bundles of rank two over smooth projective varieties of dimension ≥ 2 satisfy the “parabolic” 'Bogomolov inequality Received: 1 March 1999 / Revised version: 11 June 1999     

Multiplicity of holomorphic functions

The main result of this paper concerns an analytic version of Birkhoff's ergodic theorem. It allows us to define asymptotic multiplicities given any algebraically stable rational map . This is the key for our study of the singularities of the Green current T associated to f. We characterize the points where the Lelong number of T is strictly positive. Received: 30 July 1999     

Invariant metrics of positive Ricci curvature on principal bundles

Let be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let be a principal bundle over with compact connected structure group . If the fundamental group of is finite, we show that admits a invariant metric with positive Ricci curvature so that is a Riemannian submersion. Received 14 January 1997     

Very ample line bundles on quasi-abelian varieties

Received January 27, 1999; in final form May 31, 1999 / Published online October 30, 2000     

Globalizations, fiber bundles, and envelopes of holomorphy

We describe a relationship between globalizations of local holomorphic actions on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for particular Stein Riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms. We then derive some consequences on the existence of Stein globalizations. Received August 11, 1998; in final form March 8, 1999     

On signed normal-Poisson approximations

For lattice distributions a convolution of two signed Poisson measures proves to be an approximation comparable with the normal law. It enables to get rid of cumbersome summands in asymptotic expansions and to obtain estimates for all Borel sets. Asymptotics can be constructed two-ways: by adding summands to the leading term or by adding summands in its exponent. The choice of approximations is confirmed by the Ibragimov-type necessary and sufficient conditions. Received: 20 November 1996 / Revised version: 5 December 1997     

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction. Received: 19 February 1998 / Revised version: 28 December 1998     

On isomorphisms of blowing-ups of complete ideals¶of a rational surface singularity

Given a rational surface singularity (S,P), we relate S-isomorphism classes of normal blowing-ups of S and properties of semifactorization for complete ideals in . This relation is understood via the natural cell decomposition of the characteristic cone P˜(X/S), where X is a desingularization of S. Received: 1 April 1998 / Revised version: 2 July 1998     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

Oka's principle for holomorphic submersions with sprays

We classify, up to a local isometry, all non-K?hler almost K?hler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost K?hler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. Received: 1 October 2001 / Published online: 17 June 2002