This paper in several complex variables, complex analysis, and partial differential equations, by a well-known top expert, surveys two major topics.
The first one is about applications of the analytic techniques of Nadel multiplier ideal sheaves and weighted twisted \(L^2\)-estimates for the \({\overline\partial}\)-equation to such important questions of algebraic geometry as the Fujita conjecture, effective Matsusaka big theorem, invariance of plurigenera, finite generation of the canonical ring, and the minimal model program.
The second major topic is the application of complex geometry to questions of analysis, namely, the effective generation of Kohn’s multiplier ideal for subelliptic estimates, at least for the case of ‘special domains’.
Correspondingly, the paper has two parts and numerous subdivisions. The following listing of the first two of the three levels of headings may give an idea of the paper’s contents: 1.Application of analysis to algebraic geometry, 1.1.Algebraic geometric problems reduced to \(L^2\) estimate for domain (spread over \({\mathbb C}^n\)), 1.2.Multiplier ideal sheaves and effective problems in algebraic geometry, 1.3.Background of finite generation of canonical ring, 1.4.Deformational invariance of plurigenera and the two ingredients for its proof, 1.5.Two tower argument for invariance of plurigenera, 1.6.Convergence argument with the most singular metric, 1.7.Maximally regular metric for the canonical line bundle, 1.8.Finite generation of canonical ring and Skoda’s estimate for ideal generation, 1.9.Stable vanishing order, 2.Application of algebraic geometry to analysis, 2.1.Regularity, subelliptic estimates, and Kohn’s multiplier ideals, 2.2.Finite type and subellipticity, 2.3.Algebraic formulation for special domains, 2.4.Interpretation as Frobenius theorem over Artinian subschemes. 2.5.Sums of squares and Kohn’s counterexample.
Part 1 details the author’s two-tower argument for the invariance of plurigenera and its attendant topics. Here the two main ingredients are the effective global generation of Nadel ideal sheaves, and a version of the Ohsawa-Takegoshi extension theorem for holomorphic sections with \(L^2\)-bounds with respect to possibly singular metrics. §1.8.explains how to bring Skoda’s theorem on ideal generation into play for showing that rings of sections of line bundles over complex projective algebraic manifolds may be finitely generated. §1.9.reduces the question of finite generation of canonical rings to the rationality of stable vanishing orders and properties of Lelong numbers. For the rationality of stable vanishing orders the author proposes to study a degenerate Monge-Ampère equation and some auxiliary linear equations with rational coefficients.
Part 2 explains Kohn’s multipliers and their ideals in the context of finite type domains. The author proposes to give an explicit formula for the subelliptic gain \(\varepsilon\) in terms of the dimension and a quantity closely related to the type for a class of special domains of the form \(\text{Re\;}w+\sum_{j=1}^N| h_j(z_1,\dots,z_n)| ^2<0\), where the \(h_j\) are holomorphic near the origin. He also points out that the finite type condition, nonexistence of holomorphic curves in the boundary, and differential conditions in Kohn’s process of generating \(1\) are analogous to the classical Frobenius theorem on integrable plane fields. Here the role of the plane field is played by the plane field of the complex directions in the boundary, and points of the boundary are to be counted with multiplicity as some Artinian subschemes. The differential formulation of the Frobenius integrability condition should now be replaced with Kohn’s process of wedge products and exterior differentiation.
The paper covers a large amount of material and describes the author’s program to prove the finite generation of canonical rings, and to use algebro geometric techniques to put Kohn’s ideal generation on effective grounds.
For further developments on canonical rings see arXiv:math.AG/0610740, and arXiv:0704.1940v1 [math.AG] by the author. For more on Kohn ideals see arXiv:0706.4113v1 [math.CV] by the author, and arXiv:0711.0429 by Andreea C.Nicoara.