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Projectivity of Kähler manifolds – Kodaira’s problem

Daniel Huybrechts (2005/2006)

Séminaire Bourbaki

Every compact Kähler surface is deformation equivalent to a projective surface. In particular, topologically Kähler surfaces and projective surfaces cannot be distinguished. Kodaira had asked whether this continues to hold in higher dimensions. We explain the construction of a series of counter-examples due to C. Voisin, which yields compact Kähler manifolds of dimension at least four whose rational homotopy type is not realized by any projective manifold.

Pull-back of currents by meromorphic maps

Tuyen Trung Truong (2013)

Bulletin de la Société Mathématique de France

Let  X and Y be compact Kähler manifolds, and let  f : X → Y be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator f ♯ for currents of bidegrees ( p , p ) of finite order on  Y (and thus foranycurrent, since Y is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can...

Pulling back cohomology classes and dynamical degrees of monomial maps

Jan-Li Lin (2012)

Bulletin de la Société Mathématique de France

We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.

The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold

Jonathan Pridham (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this paper is to study the pro-algebraic fundamental group of a compact Kähler manifold. Following work by Simpson, the structure of this group’s pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.’s result on the de Rham fundamental group, and Goldman and Millson’s result on deforming representations of Kähler groups, and can...

Three-manifolds and Kähler groups

D. Kotschick (2012)

Annales de l’institut Fourier

We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kähler and the fundamental group of a closed three-manifold is finite. We also prove that a group that is both the fundamental group of a closed three-manifold and of a non-Kähler compact complex surface is ℤ or ℤ ⊕ ℤ 2 .

Towards a Mori theory on compact Kähler threefolds III

Thomas Peternell (2001)

Bulletin de la Société Mathématique de France

Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold (i.e. K X is nef) is good,i.e.its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of K X is generated by global sections); unless X is simple. “Simple“ means that there is no compact subvariety through the very general point of X and X not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities whose canonical...

Which 3-manifold groups are Kähler groups?

Alexandru Dimca, Alexander Suciu (2009)

Journal of the European Mathematical Society

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then G must be finite—and thus belongs to the well-known list of finite subgroups of O ( 4 ) , acting freely on S 3 .

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