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An obstruction to homogeneous manifolds being Kähler

Bruce Gilligan (2005)

Annales de l’institut Fourier

Let G be a connected complex Lie group, H a closed, complex subgroup of G and X : = G / H . Let R be the radical and S a maximal semisimple subgroup of G . Attempts to construct examples of noncompact manifolds X homogeneous under a nontrivial semidirect product G = S ⋉ R with a not necessarily G -invariant Kähler metric motivated this paper. The S -orbit S / S ∩ H in X is Kähler. Thus S ∩ H is an algebraic subgroup of S [4]. The Kähler assumption on X ought to imply the S -action on the base Y of any homogeneous fibration X → Y is algebraic...

Analytic inversion of adjunction: L 2 extension theorems with gain

Jeffery D. McNeal, Dror Varolin (2007)

Annales de l’institut Fourier

We establish new results on weighted L 2 -extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold.

Asymptotic behaviour of numerical invariants of algebraic varieties

F. L. Zak (2012)

Journal of the European Mathematical Society

We show that if the degree of a nonsingular projective variety is high enough, maximization of any of the most important numerical invariants, such as class, Betti number, and any of the Chern or middle Hodge numbers, leads to the same class of extremal varieties. Moreover, asymptotically (say, for varieties whose total Betti number is big enough) the ratio of any two of these invariants tends to a well-defined constant.

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