Recent zbMATH articles in MSC 32Jhttps://www.zbmath.org/atom/cc/32J2021-07-10T17:08:46.445117ZWerkzeugTheory of algebraic surfaces. Translated from the Japanese by Kazuhiro Konno. Notes taken by Shigeho Yamashimahttps://www.zbmath.org/1462.140012021-07-10T17:08:46.445117Z"Kodaira, Kunihiko"https://www.zbmath.org/authors/?q=ai:kodaira.kunihiko.1The present review uses heavily excerpts of the excellent foreword to the book by Professor Kazuhiro Konno. As explained in the foreword:
``In the academic year 1967, Kunihiko Kodaira gave a course of lectures at the University of Tokyo on the theory of complex algebraic surfaces. The lecture notes were published in 1968 as Volume 20 in the series of Seminary Notes by the University of Tokyo. This was the copy of the handwritten manuscript in Japanese by Shigeho Yamashima, based on his beautiful notes reflecting faithfully the atmosphere of Kodaira's lectures. The present book is an English translation of that volume with slight modifications, correcting typos, etc. of the Japanese version.
The readers are expected to have only the elementary prerequisites on complex manifolds as background.
The book consists of two parts: Chaps. 1 and 2.
After stating the goal of the lecture in the Introduction, in Chap. 1, basic facts on algebraic surfaces are reviewed, touching upon divisors, linear systems, intersection theory, and the Riemann-Roch theorem. It provides an elegant introduction to the theory of algebraic surfaces covering some classical materials whose modern proofs first appeared in Kodaira's papers. Among others, one can find a concise analytic proof of Gorenstein's theorem for curves on a non-singular surface, which is a detailed explanation of the one given in Appendix I to [\textit{K. Kodaira}, Ann. Math. (2) 71, 111--152 (1960; Zbl 0098.13004)]. Another highlight is the elementary proof of Noether's formula for the arithmetic genus of an algebraic surface. Nowadays, the formula is known and treated as a special case of Hirzebruch's Riemann-Roch theorem. Kodaira's approach is based on the standard fact that, via generic projections, every algebraic surface can be obtained as the normalization of a surface with only ordinary singularities in the projective 3-space. However, unlike the other modern proofs, the argument does not rely on general facts, such as Porteus' formula, which requires a separate treatment. It is self-contained and follows a classical line, using Lefschetz pencils, much more in the style of Noether's original proof. The second part, Chapter 2, discusses the behaviour of the pluricanonical maps of algebraic surfaces of general type, as an application of the general theory provided in Chap. 1. It gives a detailed account of the paper Pluricanonical Systems on Algebraic Surfaces of General Type, [\textit{K. Kodaira}, J. Math. Soc. Japan 20, 170--192 (1968; Zbl 0157.27704)]''.
In the introduction it is explained the the main goal of the lectures is to prove that for minimal surfaces of general type the pluricanonical map \(\phi_{mK}\) is a birational holomorphic map for \(m\geq 6\) and this is done in chapter 2.
Of course subsequent work of many authors among which Kodaira himself and the epochal paper of \textit{E. Bombieri} [Publ. Math., Inst. Hautes Étud. Sci. 42, 171--219 (1972; Zbl 0259.14005)] and the use of new methods like \textit{I. Reider}'s theorem [Ann. Math. (2) 127, No. 2, 309--316 (1988; Zbl 0663.14010)] supplanted this result.
As such one could think that the present book is only interesting from a historical viewpoint and in a sense this is partly true about Chapter 2, although the level of detail of its analytical approach still renders it interesting.
However the clarity and detail of the definitions and proofs of preparatory facts given in Chapter 1 (called ``Fundamentals of algebraic surfaces'') and the analytic proof of Mumford's vanishing theorem given in Chapter 2 make this book very interesting for anyone wanting to learn working hands-on with complex surfaces. The definitions and proofs are extremely detailed and beautifully accompanied by examples and illustrations.
As said above the readers are expected to have only elementary prerequisites on complex manifolds and as such this book (specially Chapter 1) can be very useful for a graduate student or a non-expert starting to work on algebraic geometry.Note on Dolbeault cohomology and Hodge structures up to bimeromorphismshttps://www.zbmath.org/1462.320212021-07-10T17:08:46.445117Z"Angella, Daniele"https://www.zbmath.org/authors/?q=ai:angella.daniele"Suwa, Tatsuo"https://www.zbmath.org/authors/?q=ai:suwa.tatsuo"Tardini, Nicoletta"https://www.zbmath.org/authors/?q=ai:tardini.nicoletta"Tomassini, Adriano"https://www.zbmath.org/authors/?q=ai:tomassini.adrianoWithin the area of non-Kähler geometry, the present work studies complex manifolds which satisfy the so-called \(\partial \bar\partial\)-Lemma (i.e., \(\text{ker}\,\partial\cap\text{ker}\,\bar\partial\cap\text{im}\, d= \text{im}\,\partial\bar\partial\)). The authors are interested in whether the \(\partial \bar\partial\)-Lemma property is a bimeromorphic invariant. As this property is equivalent to the existence of a natural Hodge structure, i.e.,
\[
H^{p,q}_{\bar\partial}(X)\simeq \overline{H^{q,p}_{\bar\partial}(X)}\ \hbox{and}\ H^{h}_{dR}(X)\simeq\bigoplus_{p+q=h}H^{p,q}_{\bar\partial}(X),
\]
the first main result of the present work gives a partial answer to the problem:
Let X be a compact complex manifold and \(Z\subset X\) a submanifold of codimension \(k\). Let \(\tilde X\) denote the blow up of \(X\) along \(Z\). Then, there is an isomorphism
\[
H^{h}_{dR}(X)\oplus\bigoplus_{i=0}^{k-2}H^{h-2i-2}_{dR}(Z)\overset\sim\longrightarrow H^{h}_{dR}(\tilde X).
\]
If \(Z\) has a holomorphically contractible neighbourhood and some other technical conditions are satisfied, there is also a similar isomorphism for the Dolbeault cohomology groups:
\[
H^{p,q}_{\bar\partial}(X)\oplus\bigoplus_{i=0}^{k-2}H^{p-i-1,q-i-1}_{\bar\partial}(Z)\overset\sim\longrightarrow H^{p,q}_{\bar\partial}(\tilde X).
\]
In particular, if \(X\) and \(Z\) admit a Hodge structure, then \(\tilde X\) does.
Explicit computations with Čech cohomology groups make this result interesting despite that the mentioned problem has been solved by \textit{J. Stelzig} with different techniques, see Corollary 28 in [``On the structure of double complexes'', J. London Math. Soc. (to appear); \url{doi: 10.1112/jlms.12453}]: the \(\partial \bar\partial\)-Lemma property is a bimeromorphic invariant if and only if it is invariant under restriction.
Using this general result, the authors obtain the second main result of the present work: There exists a simply-connected compact complex manifold satisfying the \(\partial \bar\partial\)-Lemma which is not Kähler (not even in class \(\mathcal{C}\) of Fujiki).