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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.

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space., Comment: minor changes to the exposition based on referee's comments; final version to appear in Geometry & Topology; 95 pages

We compute the Dolbeault cohomology of geodesically convex domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology., Comment: To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

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 consider a notion of stability for sheaves, which we call multi-Gieseker stability that depends on several ample polarisations $L_1, \dots, L_N$ and on an additional parameter $\sigma \in \mathbb{Q}_{\geq 0}^N\setminus\{0\}$. The set of semi stable sheaves admits a projective moduli space $\mathcal M_{\sigma}$. We prove that given a finite collection of parameters $\sigma$, there exists a sheaf- and representation-theoretically defined master space $Y$ such that each corresponding moduli space is obtained from $Y$ as a Geometric Invariant Theory (GIT) quotient. In particular, any two such spaces are related by a finite number of "Thaddeus-flips". As a corollary, we deduce that any two Gieseker-moduli space of sheaves (with respect to different polarisations $L_1$ and $L_2$) are related via a GIT-master space. This confirms an old expectation and generalises results from the surface case to arbitrary dimension., Comment: 18 pages

We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that generalises the classical notion of Gieseker-stability to allow for several polarisations. As such we are able to prove that on smooth threefolds certain moduli spaces of Gieseker-stable sheaves are related by a finite number of Thaddeus-flips (that is flips arising for Variation of Geometric Invariant Theory) whose intermediate spaces are themselves moduli spaces of sheaves., Comment: 40 pages, reorganised and new introduction and new title. Main results unchanged

International audience; We study the following question: Let (X,g) be a compact Gauduchon surface, (E,h) be a differentiable rank r vector bundle on X⁠, D be a fixed holomorphic structure on D≔det(E) and a be the Chern connection of the pair (D,det(h))⁠. Does the complex space structure on MaASD(E)* induced by the Kobayashi–Hitchin correspondence extend to a complex space structure on the Donaldson–Uhlenbeck compactification M¯aASD(E)? Our results answer this question in detail for the moduli spaces of SU(2)-instantons with c2=1 on general (possibly unknown) class VII surfaces.

We extend Langton's valuative criterion for families of coherent algebraic sheaves to a complex analytic set-up. As a consequence we derive a set of sufficient conditions for the compactness of a moduli space of semistable sheaves over a compact complex manifold. This applies also to some cases appearing in complex projective geometry not covered by previous results., v2: added section with criterion for separatedness

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 show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kähler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterion expressed via the Hilbert polynomial to the Kähler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kähler manifold.

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.

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama moduli spaces with respect to two different chosen polarisations are related via Thaddeus-flips through other "multi-Gieseker"-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson-Uhlenbeck moduli space., 21 pages; v2: minor changes as requested by referee, added Remark 4.7 concerning Compactified Jacobians; to appear in the proceedings of VBAC 2014 "Algebraic Varieties: Bundles, Topology, Physics"

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.

We give an affirmative answer to the following conjecture of Ma. Kato: Let $S$ be a compact complex surface in Kodaira's class $VII 0$ which contains a strictly positive number of rational curves being exactly equal to the second Betti number of $S$. Then $S$ admits a global spherical shell.

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 prove boundedness of rationally-connected threefolds in $\mathbb P^6$ under some extra-assumptions.

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.

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Furthermore, we describe the geometry of the newly construced moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces., v1: 41 pages, the threefold case, general case pending; v2: 51 pages, new features: generalisation of results to base manifolds of arbitrary dimension, identification of equivalence relation represented by the moduli space, comparison with Gieseker moduli spaces; v3: 50 pages, small corrections, updated references; a slightly shortened version will appear in "Algebraic Geometry"

The Kahler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on Kahler manifolds as a measure of “kahlerianity”. Here we give a partial classification of compact complex surfaces of Kahler rank 1. These are either elliptic surfaces, or Hopf surfaces, or they admit a holomorphic foliation of a very special type. As a consequence we give an affirmative answer to the question raised by Harvey and Lawson whether the Kahler rank is a bimeromorphic invariant.

We prove that any two-dimensional moduli space of stable 2-vector bundles, in the non-filtrable range, on a primary Kodaira surface is a primary Kodaira surface. If a universal bundle exists, then the two surfaces are homeomorphic up to unramified covers., Advances in Mathematics, accepted

It is shown that if $X$ is an Inoue surface of type $S_M$ then the irreducible components of the Douady space of $X^n$ are compact, for all $n>0$. This gives an example of an essentially saturated compact complex manifold (in the sense of model theory) that is not of Kaehler-type. Among the known compact complex surfaces without curves, it is shown that these are the only examples., 10 pages

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

This paper is concerned with a sufficient criterion to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov-McQuillan and using recent works of Langer and Graber-Harris-Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic p. We apply the result to show that Q-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration., Minor simplification. A few typos and an obvious issue in the main statement are fixed

We give an affirmative answer to the following conjecture of Ma, Kato: Let $S$ be a compact complex surface in Kodaira's class $\rm {VII_{0}}$ which contains a strictly positive number of rational curves being exactly equal to the second Betti number of $S$. Then $S$ admits a global spherical shell.

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces., LaTeX, 6 pages, to appear in Comptes Rendus

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

We consider Poncelet pairs $(S,C)$, where $S$ is a smooth conic and $C$ is a degree$-c$ plane curve having the Poncelet property with respect to $S$. We prove that for $c>4$ the projection $(S,C)\mapsto C$ is generically one-to-one and use this to describe a birational model of the variety of Poncelet curves for $c$ odd., revised TeX-version, latex2e