A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring

By Y. Siu. 2006

57 citations recorded by the source corpus.

On August 5, 2005 in the American Mathematical Society Summer Institute on Algebraic Geometry in Seattle and later in several conferences I gave lectures on my analytic proof of the finite generation of the canonical ring for the case of general type. After my lectures many people asked me for a copy of the slides which I used for my lectures. Since my slides were quite sketchy because of the time limitation for the lectures, I promised to post later on a preprint server my detailed notes from which my slides were extracted. Here are my detailed notes giving the techniques and the proof.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7c3202007437ccafccf2a56cf394e0d25dde16fb

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Citations in the corpus

Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, methodology): “We start now the actual proof of 3.1 and denote by the numerical dimension of the divisor KX +B. We proceed as in [28], [15], [3], [ 35 ].”
Existence of minimal models for varieties of log general type (2006, unknown): “ with a Kahler-Einstein metric. Note that (4) follows from (2) and Theorem D of [4]. Note that Siu has announced a proof of ﬁnite generation for varieties of general type, using analytic methods, see [36]. Corollary 1.1.2. Let (X,∆) be a projective kawamata log terminal pair, where KX+∆ is Q-Cartier”
Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds (2009, methodology): “…the canonical ring for a compact complex manifold of general type, dynamic multiplier ideal sheaves are used (together with the notion of deviation from sufficient ampleness instead of minimum centers of log canonical singularities as deviation from freeness) [Siu2006, Siu2007, Siu2008, Siu2008a].”
Monge–Ampère equations in big cohomology classes (2008, methodology): “’ theorem we will thus have Z X−A d[vdcu∧Θ] = 0, and the desired formulae will follow by (1.5). The proof of (1.6) will again follow rather closely Sibony’s proof of the SkodaEl Mir extension theorem [41, 18]. Note that (1.6) is a local equality, and that it clearly holds on X−A. We will thus focus ”
Canonical measures and Kähler-Ricci flow (2008, background): “Recently, the finite generation of canonical rings on algebr aic varieties of general type was proved independently by [BiCaHaMc] and [ Si2 ].”
Finite generation of canonical ring by analytic method (2008, methodology): “minimum singularity with curvature current Θϕ. We have the modified restriction Θϕ − γ[Y ] defined in Appendix A.11, which we denote by ΘY . We assume that the current ΘY is of the special form (i.e., in the second case of the dichotomy according to the terminology of [ 7 ]), which means that there ”
A variational approach to complex Monge-Ampère equations (2009, unknown): “ been the subject of intensive studies in the past decade, in connection with the search for canonical models and metrics of complex algebraic varieties (see e.g. [Ko l98], [Tian], [Che00], [Don05a], [Siu06], [BCHM06], [EGZ09], [ST08]). Many of these results ultimately relied on the seminal work of ”