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The Kähler Ricci flow on Fano manifolds (I)

Xiuxiong Chen, Bing Wang (2012)

Journal of the European Mathematical Society

We study the evolution of pluri-anticanonical line bundles K M - ν along the Kähler Ricci flow on a Fano manifold M . Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of M . For example, the Kähler Ricci flow on M converges when M is a Fano surface satisfying c 1 2 ( M ) = 1 or c 1 2 ( M ) = 3 . Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups....

The Kuranishi space of complex parallelisable nilmanifolds

Sönke Rollenske (2011)

Journal of the European Mathematical Society

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several examples and also analyse when small deformations remain complex parallelisable.

The L 2 ∂ ¯ -Cauchy problem on weakly q -pseudoconvex domains in Stein manifolds

Sayed Saber (2015)

Czechoslovak Mathematical Journal

Let X be a Stein manifold of complex dimension n ≥ 2 and Ω ⋐ X be a relatively compact domain with C 2 smooth boundary in X . Assume that Ω is a weakly q -pseudoconvex domain in X . The purpose of this paper is to establish sufficient conditions for the closed range of ∂ ¯ on Ω . Moreover, we study the ∂ ¯ -problem on Ω . Specifically, we use the modified weight function method to study the weighted ∂ ¯ -problem with exact support in Ω . Our method relies on the L 2 -estimates by Hörmander (1965) and by Kohn (1973).

The Levi problem for cohomology classes

Mihnea Coltoiu (1984)

Annales de l'institut Fourier

In this paper we extend to complex spaces and coherent analytic sheaves some results of Andreotti and Norguet concerning the extension of cohomology classes.

The Levi problem for domains spread over locally convex spaces with a finite dimensional Schauder decomposition

Martin Schottenloher (1976)

Annales de l'institut Fourier

It is proved that the Levi problem for certain locally convex Hausdorff spaces E over C with a finite dimensional Schauder decomposition (for example for Fréchet or Silva spaces with a Schauder basis) the Levi problem has a solution, i.e. every pseudoconvex domain spread over E is a domain of existence of an analytic function. It is also shown that a pseudoconvex domain spread over a Fréchet space or a Silva space with a finite dimensional Schauder decomposition is holomorphically convex and satisfies...

The Lie group of real analytic diffeomorphisms is not real analytic

Rafael Dahmen, Alexander Schmeding (2015)

Studia Mathematica

We construct an infinite-dimensional real analytic manifold structure on the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is defined to be real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known, the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove that this group is regular in the sense of Milnor. ...

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