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4-dimensional anti-Kähler manifolds and Weyl curvature

Jaeman Kim (2006)

Czechoslovak Mathematical Journal

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

A class of non-algebraic threefolds

Matei Toma (1989)

Annales de l'institut Fourier

Let X be a compact complex nonsingular surface without curves, and E a holomorphic vector bundle of rank 2 on X . It turns out that the associated projective bundle P E has no divisors if and only if E is “strongly” irreducible. Using the results concerning irreducible bundles of [Banica-Le Potier, J. Crelle, 378 (1987), 1-31] and [Elencwajg- Forster, Annales Inst. Fourier, 32-4 (1982), 25-51] we give a proof of existence for bundles which are strongly irreducible.

A compactification of ( * ) 4 with no non-constant meromorphic functions

Jun-Muk Hwang, Dror Varolin (2002)

Annales de l’institut Fourier

For each 2-dimensional complex torus T , we construct a compact complex manifold X ( T ) with a 2 -action, which compactifies ( * ) 4 such that the quotient of ( * ) 4 by the 2 -action is biholomorphic to T . For a general T , we show that X ( T ) has no non-constant meromorphic functions.

A converse to the Andreotti-Grauert theorem

Jean-Pierre Demailly (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e. of asymptotic 0 -cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert...

A note on Berezin-Toeplitz quantization of the Laplace operator

Alberto Della Vedova (2015)

Complex Manifolds

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

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