Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero
Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero
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DOI：
10.1155/imrn/2006/37385
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发表时间：
2006-12
影响因子：
1
通讯作者：
Tian-Jun Li
中科院分区：
文献类型：
--
作者：
Tian-Jun Li
定义了非极小流形的任意极小模型的科代拉维数。文[12]证明了，若ω是复曲面（M，J）上的Kahler型，则κ（M，ω）与（M，J）的通常的全纯科代拉维数一致.在[12]中还表明，κ = 0的极小辛4−流形恰好是具有挠标准类的流形，因此可以被视为辛Calabi-Yau曲面。已知的具有挠正则类的辛4−流形的例子是（全纯）科代拉维数为零的Kahler曲面或T2上的T2 −丛（[10]，[12]）。它们都有小的贝蒂数和欧拉数：B+ ≤ 3，B ≤ 19，b1 ≤ 4;欧拉数在0到24之间。[12]这是他们唯一的猜测。在本文中，我们证明了它是真的理性同调。
The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.