ON A CLASS OF FOLIATED NON-KÄHLERIAN COMPACT COMPLEX SURFACES
ON A CLASS OF FOLIATED NON-KÄHLERIAN COMPACT COMPLEX SURFACES
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一类叶状非卡勒致密复杂曲面
DOI：
10.2748/tmj/1318338951
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Brunella
中科院分区：
文献类型：
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作者：
M. Brunella
最近的结果的启发非凯勒紧致复杂的表面与小的第二贝蒂数，我们分类上的全纯叶理（奇点）存在。导论.根据科代拉，一个紧致连通复曲面S属于VII 0类，如果它是极小的，并且它的第一个Betti数b1（S）等于1。这仍然是一个开放的和基本的问题，以获得这些曲面的分类，这不是凯勒，因此相当难以捉摸。让我们简短地回顾一下关于这个问题的一些进展[Nak]，[DOT]，[Tel]，以便放置和激励我们的结果。由于第二Betti数在下面的讨论中的重要作用，我们可以方便地用VII 0，n ∈ N表示VII 0曲面类S，其中b2（S）= n，并设置VII 0 = n>0 VIIn 0。VII 0级表面已在科代拉、井上、博戈莫洛夫、Li-Yau-Zheng和Teleman的一系列作品中完全分类。现在让我们把注意力集中在VII 0类表面上。大约在1977年，Kato [Kat]发现了大量的VII 0曲面，现在称为Kato曲面。具有全局球壳的曲面）。他们是，在某种意义上说，推广的经典霍普夫曲面（属于第七类），并有相当数量的论文一直致力于他们，使加藤表面今天可能被认为是“众所周知的”表面。到目前为止，还没有发现VII 0曲面的其他例子，事实上，一些作者大胆地推测，每个VII 0曲面都应该是加藤曲面。这方面的一个重要结果已由中村[Na 1]，[Na 2]在某些特殊情况下证明，然后由Dloussky-Oeljeklaus-Toma [DOT]在一般情况下证明：如果S是类VIIn 0（n > 0）的曲面并包含n条有理曲线，则S是Kato曲面（通过构造，匡威也成立）。这一结果激发了在VII 0曲面上寻找有理曲线的研究。近年来，Teleman开发了一个通用的战略，寻找这些合理的曲线，使用规范理论的方法[Tel]。到目前为止，他的策略已经成功地为第二Betti数的小值：它是在[Te 1]和[Te 2]中证明，每一个曲面类VII 0或VII 20包含至少一个有理曲线。我们将在第1节中对这一惊人的结果作更详细的说明。2010年数学学科分类。小学32 J15;中学37 F75。
Motivated by recent results on non-Kählerian compact complex surfaces with small second Betti number, we classify those on which a holomorphic foliation (with singularities) exists. Introduction. According to Kodaira, a compact connected complex surface S belongs to the class VII0 if it is minimal and its first Betti number b1(S) is equal to 1. It is still an open and fundamental problem to get a classification of these surfaces, which are not Kählerian and hence rather elusive. Let us shortly recall some advances about this problem [Nak], [DOT], [Tel], in order to place and to motivate our result. Because of the important rôle of the second Betti number in the following discussion, it is convenient to denote by VIIn0, n ∈ N , the class of VII0 surfaces S with b2(S) = n, and to set VII0 = ⊔ n>0 VII n 0. Surfaces of class VII0 have been completely classified in a series of works by Kodaira, Inoue, Bogomolov, Li-Yau-Zheng and Teleman. Let us henceforth concentrate our attention to surfaces of class VII0 . Around 1977, Kato [Kat] discovered a large collection of VII0 surfaces, nowadays called Kato surfaces (a.k.a. surfaces with a global spherical shell). They are, in some sense, generalizations of the classical Hopf surfaces (which belong to class VII0), and a significant number of papers has been dedicated to them, so that Kato surfaces may be today considered as “well known” surfaces. No other examples of VII0 surfaces have been discovered so far, and indeed some authors courageously conjecture that every VII0 surface should be a Kato surface. An important result in that direction has been proved by Nakamura [Na1], [Na2], in some particular cases, and then Dloussky-Oeljeklaus-Toma [DOT], in the general case: if S is a surface of class VIIn0 (n > 0) and contains n rational curves, then S is a Kato surface (the converse also being true, by construction). That result motivates the search for rational curves on VII0 surfaces. In recent years, Teleman developed a general strategy for finding those rational curves, using methods of gauge theory [Tel]. Up to now, his strategy has been successfull for small values of the second Betti number: it is proved in [Te1] and [Te2] that every surface of class VII0 or VII 2 0 contains at least one rational curve. We shall give in Section 1 more details on this spectacular result. 2010 Mathematics Subject Classification. Primary 32J15; Secondary 37F75.