Columbia University, New York, NY 10027.
Proc Natl Acad Sci U S A. 1986 Sep;83(18):6665-6. doi: 10.1073/pnas.83.18.6665.
In the classification problem of algebraic surfaces of general type, an important conjecture states that for simply connected such surfaces Chern numbers satisfy the inequality c(1) (2) </= 2c(2) (or equivalently, the index tau </= 0). We disprove this conjecture by computing fundamental groups of Galois coverings corresponding to generic CP(2) projections of projective embeddings of CP(1) x CP(1) related to linear systems [unk]al(1) + bl(2)[unk], a >/= 3, b >/= 2. Also, we proved the existence of simply connected minimal surfaces of general type with zero index (e.g., c(1) (2) = 2c(2)). Previously, it was conjectured that these are exactly the surfaces uniformizable in the polydisk. So this conjecture is also disproved.
在一般型代数曲面的分类问题中，有一个重要的猜想表明，对于单连通的这种曲面，陈数满足不等式 c(1) (2) </= 2c(2)（或者等价地，指标 tau </= 0）。我们通过计算与线性系统 [unk]al(1) + bl(2)[unk]（a >/= 3，b >/= 2）相关的 CP(1) x CP(1) 的投影到 CP(2) 上的射影嵌入的 Galois 覆盖的基本群，来否定这个猜想。此外，我们还证明了存在具有零指标的单连通一般型极小曲面（例如，c(1) (2) = 2c(2)）。此前，有人猜测这些曲面恰好是可以在多圆盘上均匀化的曲面。因此，这个猜想也被否定了。
