Skoda division of line bundle sections and pseudo-division

Abstract

We first present a Skoda type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. It is derived from Varolin's theorem as a corollary. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger Geometric Effective Nullstellensatz such as 'vanishing order 1 division', which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseudo-division and show that it can replace the usual division in the finite generation lemma. We also give a vanishing order 1 pseudo-division result when the line bundle is ample.2010 Mathematics Subject Classification. 32J25(primary) and 14E30(secondary).

Boundary points, Minimal $L^{2}$ integrals and Concavity property

“…I(ψ) = I + (ψ) := ∪ ǫ>0 I((1 + ǫ)ψ)) is an important feature of multiplier ideal sheaves and "opened the door to new types of approximation techniques" (see [38]) (see e.g. [27,35,4,5,15,6,46,30,3,47,48,16,36,7]). The strong openness property was conjectured by Demailly [10], and proved by Guan-Zhou [27] (the 2-dimensional case was proved by Jonsson-Mustat ¸ȃ [33]).…”

Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity

“…I(ψ) = I + (ψ) := ∪ ǫ>0 I((1 + ǫ)ψ)) is an important feature and has been widely used in the study of several complex variables, complex algebraic geometry and complex differential geometry (see e.g. [38,46,8,9,21,10,58,41,6,59,60,22,47,11]), where ψ is a plurisubharmonic function on a complex manifold M (see [12]) and multiplier ideal sheaf I(ψ) is the sheaf of germs of holomorphic functions f such that |f | 2 e −ψ is locally integrable (see e.g. [56,49,51,15,16,14,17,48,52,53,13,42]).…”

Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity II -- on Stein manifolds

“…I(ψ) = I + (ψ) := ∪ ǫ>0 I((1+ǫ)ψ)) has opened the door to new types of approximations, which has been widely used in the study of several complex variables, complex algebraic geometry and complex differential geometry (see e.g. [39,47,8,9,22,10,59,42,7,60,61,23,48,11]), where ψ is a plurisubharmonic function on a complex manifold M (see [12]) and the multiplier ideal sheaf I(ψ) is the sheaf of germs of holomorphic functions f such that |f | 2 e −ψ is locally integrable (see e.g. [57,50,52,15,16,14,17,49,53,54,13,43]).…”