Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance of Semipositively Twisted Plurigenera for Manifolds Not Necessarily of General Type

Hyperbolicity of generic high-degree hypersurfaces in complex projective space

“…Its relevancy was realized for the first time in [22] and [23] partly because of our seemingly completely unrelated earlier work on the deformational invariance of the plurigenera [20,21].…”

Strong openness of multiplier ideal sheaves and optimal L 2 extension

“…The optimal L 2 extension theorem (Theorem 1.4 [21]) gives unified optimal estimate versions of various well-known L 2 extension theorems in [39,35,10,52,44,4,38,16], etc. Some interesting relations between the optimal L 2 extension and some questions are found, so that the questions are solved in [21] by using optimal L 2 extension (Theorem 1.4), such as Suita's conjecture (see [57] [45]), L-conjecture (see [59]), extended Suita conjecture (see [59]), and an open question posed by Ohsawa (see [42]), etc.…”

“…Let X be a complex manifold with dimension n and ϕ be a plurisubharmonic function on X (see [31,48]). The multiplier ideal sheaf I(ϕ) is defined to be the sheaf of germs of holomorphic functions f such that |f | 2 e −2ϕ is locally integrable (see [37,58,50,52,11,53,12]). The basic properties of multiplier ideal sheaves include: I(ϕ) is a coherent analytic and integrally closed sheaf and satisfies the Nadel vanishing theorem.…”

Finite generation of canonical ring by analytic method

“…An adaptation of the argument here for use in a proof of the abundance conjecture would require an analytic argument of controlling the estimates in passing to limit, which is analogous to the situation of extending the proof of the deformational invariance of the plurigenera for the case of general type [Siu 1998] to the general algebraic case without the general type assumption [Siu 2002]. …”

“…Unfortunately the argument of absorption of small ample line bundle discussed in (A.12) is used here, making it impossible to use directly the argument for a proof of the abundance conjecture. To adapt the argument for use in a proof of the abundance conjecture, we encounter the situation similar to adapting the proof of the deformational invariance of the plurigenera for the case of general type [Siu 1998] to the general algebraic case without the general type assumption [Siu 2002], requiring an analytic argument of controlling the estimates in passing to limit.…”