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We prove a version of the Hodge–Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space.

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $\omega \in N^1(X)_\mathbb{R}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker-semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus-flips., Comment: 51 pages; v2: added discussion concerning characteristic of the base field, fixed some typos and inconsistencies; removed "I" from title as second part has a new title; to appear in Geometry & Topology

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli space $M$ of $\mu_{\omega}-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. If $M$ parametrizes only locally free sheaves, it is moreover hyperk\"ahler. Finally, we show that there is an isometry between $v^{\perp}$ and $H^{2}(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective., Comment: 42 pages; major revisions; to appear in Kyoto J. Math

We show that certain moduli spaces of vector bundles over blown-up primary Hopf surfaces admit no compact components. These are the moduli spaces used by Andrei Teleman in his work on the classification of class VII surfaces.

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle., An error in the argumentation has been corrected

Harvey et Lawson ont introduit et calcule le rang de Kahler en relation avec le cone des courants positifs fermes de bidimension (1, 1) pour beaucoup de classes de surfaces complexes compactes. Dans ce travail nous etendons ces calculs a la seule classe de surfaces connues et qui n'avait pas ete consideree par eux, celle des surfaces de Kato. Notre outil principal est la reduction a la dynamique des contractions holomorphes (C 2 ,0) → (C 2 ,0) associees.

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding instanton moduli space. In the general (possibly non-K\"ahlerian) case, the Donaldson-Uhlenbeck compactification is not a complex space, and the set $U^{\mathrm {ss}}$ can be a complicated subset of the base space $U$ that is neither open or closed in the classical topology, nor locally closed in the Zariski topology. This result provides an efficient tool for the explicit description of Donaldson-Uhlenbeck compactifications on arbitrary Gauduchon surfaces.

Etant donnes des corps de nombres K avec s > 0 plopgements reels et 2t > 0 plongements complexes, et des sous groupes "admissibles" U du groupe multiplicatif des entiers inversibles de K, nous construisons et etudions certaines varietes complexes compactes X(K, U). Entre autres, nous montrons que ces varietes ne sont pas kahleriennes, mais admettent des metriques localement conformement kahleriennes lorsque t = 1. En particulier, nous dondons un contre-exemple a une conjecture de I. Vaisman.

Resume On demontre que tout fibre vectoriel holomorphe de rang deux, non-filtrable, sur une surface elliptique non-kahlerienne est une modification elementaire d'une image directe d'un fibre en droites par un revetement double de la surface. Pour citer cet article : M. Aprodu, M. Toma, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On montre que parmi les surfaces compactes complexes de classe VII 0 avec b 2 > 0, les surfaces d'Inoue-Hirzebruch sont caracterisees par le fait qu'elles possedent deux champs de vecteurs tordus. Ce resultat est un pas vers la comprehension des feuilletages sur les surfaces VII 0 .

S. Boucksom, J.-P. Demailly, M. Paun and Th. Peternell proved that the cone of mobile curves ME ( X ) ¯ of a projective complex manifold X is dual to the cone generated by classes of effective divisors and conjectured an extension of this duality in the Kahler set-up. We show that their conjecture implies that ME ( X ) ¯ coincides with the cone of integer classes represented by closed positive smooth ( n − 1 , n − 1 ) -forms. Without assuming the validity of the conjecture we prove that this equality of cones still holds at the level of degree functions.

Nous montrons qu'une surface de la classe VII0 avec b2 > 0 sur laquelle existe un champ de vecteurs non trivial contient exactement b2 courbes rationnelles. Il s'ensuit par un theoreme de I. Nakamura qu'une telle surface se deforme en une surface de Hopf primaire eclatee. Ce resultat contribue a la classification des surfaces complexes compactes avec champs de vecteurs.

Resume Nous montrons qu’une surface minimale de la classe VII0 avec b2 > 0 sur laquelle existe un champ de vecteurs non trivial contient exactement b2 courbes rationnelles. Il s’ensuit par un theoreme de I. Nakamura qu’une telle surface se deforme en une surface de Hopf primaire eclatee. Ce resultat precise la classification des surfaces complexes compactes avec champs de vecteurs.

We study very ample vector bundles on curves. We first give numerical conditions for the existence of non-special such bundles. Then we prove the inequality \[ h^0(\det E)\ge h^0(E) + {\rm rank}(E)-2 \] over curves of genus at least two. We apply this to prove some special cases of a conjecture on scrolls of small codimension.

In this paper we describe logarithmic moduli spaces of pairs (S,D) consisting of minimal surfaces S of class VII with positive second Betti number b_2 together with reduced divisors D of b_2 rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of the complement of D and for the associated holomorphic contraction germ from (C^2,0) to (C^2,0)., Comment: Minor correction of the dimension of the moduli space

It is in general unknown which topological complex vector bundles on a non-algebraic surface admit holomorphic structures. We solve this problem for primary Kodaira surfaces by using results of Kani on curves of genus two with elliptic differentials. Some of the corresponding moduli spaces will be smooth compact and holomorphically symplectic., Comment: 10 pages, latex