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An ℓ-th root of a test configuration of exponent ℓ

Toshiki Mabuchi (2016)

Complex Manifolds

Let (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : X → X and η : X → Y, both isomorphic onX X̂ 0, such that [...] whereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor onX such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way that [...] where C1 and C2 are...

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.

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