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Cauchy-Riemann meet Monge-Ampère. (English) Zbl 1310.32004
The article is a comprehensive survey on recent developments in that scope of Complex Analysis that is based upon real methods, especially on the \(L^2\)-theory of the Cauchy-Riemann equation with plurisubharmonic weights. \smallskip
There are 10 sections. After a short introduction, the relevant \(L^2\)-estimates for the \(\overline\partial\) operator with plurisubharmonic weight functions are recalled in Section 2. The third section is devoted to the Ohsawa-Takegoshi theorem which has proved as a seminal result with many applications. The simplified proof (due to B.Y. Chen) is given. Section 4 contains beautiful applications of the Ohsawa-Takegoshi theorem, for instance P. H. Hiep’s proof of the strong openness conjecture of Demailly-Kollár. In Section 5 the pluricomplex Green function \(G_D(\cdot,w)\) of a domain \(D\) in \(\mathbb C^n\) is treated with emphasis on its relationship to the Monge-Ampère operator and its behavior under approach of the pole \(w\in D\) to the boundary. The main topic of the next section is the relationship between Kobayashi’s criterion of the Bergman completeness and the behavior of the sublevel sets \(\{G_D(\cdot, w)<-1\}\), when \(w\) tends to a boundary point. The result of Bergman completeness and Bergman exhaustivity of bounded hyperconvex domains is addressed as well as the author’s lower estimate of the Bergman distance on a smoothly bounded domain. The subject of Sections 7 and 8 is the Suita conjecture in one variable and possible generalizations to convex domains in \(\mathbb C^n\). Section 9 is about the Mahler conjecture and the Bourgain-Milman inequality. In this section Nazarov’s proof of this inequality is given, where the problem is reduced to a suitable upper and lower bound of the Bergman kernel of a tube domain \(\text{int\,} (K) + i \mathbb R^n\), for a convex symmetric body \(K \subset \mathbb R^n\). Finally in the last section the question is discussed, whether for a pseudoconvex domain \(\Omega \subset \mathbb C^n\) the function \(t \longmapsto e^{-2nt} \lambda \big(\{G_\Omega (\cdot,w)<t\}\big)\) is nondecreasing on \((-\infty,0)\), \(\lambda\) being the Lebesgue measure. A proof is given for \(n=1\). For \(n \geq 2\) the author presents a relationship between this monotonicity and pluripolar isoperimetric inequalities. \smallskip
A long list of references (119 titles) concludes the article.

MSC:
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32F45 Invariant metrics and pseudodistances in several complex variables
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