We classify compact Kähler manifolds of dimension on which acts a lattice of an almost simple real Lie group of rank . This provides a new line in the so-called Zimmer program, and characterizes certain complex tori as compact Kähler manifolds with large automorphisms groups.
Holomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in...
Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant...
Let be a compact Kähler manifold and be a -divisor with simple normal crossing support and coefficients between and . Assuming that is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on having mixed Poincaré and cone singularities according to the coefficients of . As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair .
It is shown that in an elementary extension of a compact complex manifold M, the K-analytic sets (where K is the algebraic closure of the underlying real closed field) agree with the ccm-analytic sets if and only if M is essentially saturated. In particular, this is the case for compact Kähler manifolds.
Let be a compact hyperkähler manifold containing a complex torus as a Lagrangian subvariety. Beauville posed the question whether admits a Lagrangian fibration with fibre . We show that this is indeed the case if is not projective. If is projective we find an almost holomorphic Lagrangian fibration with fibre under additional assumptions on the pair , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian...
Burger et Mozes ont construit des exemples de groupes simples infinis, qui sont des réseaux dans le groupe des automorphismes d’un immeuble cubique. On montre qu’il n’existe pas de morphisme d’un groupe kählérien vers l’un de ces groupes dont le noyau soit finiment engendré. On en déduit que ces groupes ne sont pas kählériens.
We give a model-theoretic interpretation of a result by Campana and Fujiki on the algebraicity of certain spaces of cycles on compact complex spaces. The model-theoretic interpretation is in the language of canonical bases, and says that if b,c are tuples in an elementary extension 𝓐* of the structure 𝓐 of compact complex manifolds, and b is the canonical base of tp(c/b), then tp(b/c) is internal to the sort (ℙ¹)*. The Zilber dichotomy in 𝓐* follows immediately (a type of U-rank 1 is locally...
We show that the moduli space of polarized irreducible symplectic manifolds of -type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman’s characterization of polarized parallel-transport operators of -type.
Without relying on the classification of compact complex surfaces, it is proved that every such surface with even first Betti number admits a Kähler metric and that a real form of the classical Nakai-Moishezon criterion holds on the surface.
In this paper, we continue the study of the possible cohomology rings of compact complex four dimensional irreducible hyperkähler manifolds. In particular, we prove that in the case b 2=7, b 3=0 or 8. The latter was achieved by the Beauville construction.
This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler...
For any compact Kähler manifold and for any equivalence relation generated by a symmetric binary relation with compact analytic graph in , the existence of a meromorphic quotient is known from Inv. Math. 63 (1981). We give here a simplified and detailed proof of the existence of such quotients, following the approach of that paper. These quotients are used in one of the two constructions of the core of given in the previous paper of this fascicule, as well as in many other questions.
We show global existence and convergence results for the pluriclosed flow on manifolds for which certain naturally associated tensor bundles are globally generated.