On the rationality of complex homology 2-cellg: I

Abstract

this follows immediately from the main theorem in [B-K-L] which asserts that any O-cycle of degree $0$ on $X$ is trivial.Also if $X$ is rational or elliptic with $p_{g}(X)=0$ then $SA_{0}(X)=0$ , by [B-K-L].In [M-S], it is shown that if $SA_{0}(X)=0$ then any algebraic vector bundle on $X$ splits as a direct sum of a trivial subbundle and a line bundle.Hence we have the next corollary.COROLLARY 2. All algebraic vector bundles on a complex homology 2-cell are trivial.This is a generalization of C. S. Seshadri's result on the triviality of algebraic vector bundles on $C^{2}$ .The first non-trivial example of a smooth, contractible $surface/C$ was given by C. P. Ramanujam in $[R$ , \S 3 $]$ .In [G-M], R. V. Gurjar and M. Miyanishi have constructed infinitely many non-isomorphic complex homology 2-cells and smooth contractible surfaces.Let $X$ be as in $(*)$ .Then some connected component of $D$ supports an effec- tive divisor $C$ with $C^{2}>0$ .Since every connected component of $D$ is simply- connected by assumption, by a generalization of the Lefschetz hyperplane section theorem proved in [N], we see that $\pi_{1}(X)$ is trivial.Thus, the geometric genus and the irregularity of $X$ both vanish.These observation will be tacitly used in the following sections.We will briefly indicate some of the ideas used in the proof of the theorem.Our proofs depend heavily on Y. Miyaoka's inequality ([M], Theorem 1.1).We assume that the surface $X$ is not rational.Then $K+D$ has a Zariski-decom- position, so that Miyaoka's inequality applies.In \S 2, by studying the blowing- down process $\pi:Xarrow X'$ , where $X'$ is the minimal model for the function field of $X$ , we deduce an auxiliary inequality relating various integral in- variants of $D$ and $X$ .This auxiliary inequality (2.8) plays a central role in our proof.A careful interpretation of terms involved in (2.8) yield a bound on the number of components of the exceptional set $\mathcal{E}$ of $\pi$ that lie outside $D$ .The geometric Lemma 4.1, about the effectivity of $\pm K$ is of general interest.It plays another important role in our proof.To begin with, this lemma gives better estimates of terms in (2.8).At this stage, by repeated application of Miyaoka's inequality we can reduce the problem to the case when $\mathcal{E}\subseteqq D$ .However, beyond this, the proof becomes heavily computational and mainly uses the unimodularity of $D$ and lemmas in \S 4. So we must prepare ourselves with some technical lemmas in \S 5 and \S 6, converting these properties of $D$ into something easily checkable.After that, it follows that tbe number of components $b_{2}(D)$ of $D$ is bounded above.In the general type case, we have $b_{2}(D)\leqq 12$ .In \S 7 we complete the proof of Theorem, the bulk of which goes into handling the situation $\mathcal{E}\subseteqq D$ .