The zero set of a holomorphic 1-form (phi) on a compact complex surface S is studied. The main result gives, under the assumption that (phi) has a one-dimensional zero set with appropriate self-intersection properties, the existence of a holomorphic map f : S (--->) R onto a Riemann surface. The form (phi) is a pullback via f of a holomorphic 1-form on R and the zero set of (phi) is contained in fibers of f. As a direct consequence of this, any divisor D having the same support as D(,(phi)), the divisor associated to (phi), is shown to satisfy D(.)D (LESSTHEQ) 0 In a different direction, the genus of an irreducible component of the zero set of a holomorphic 1-form is proved to be bounded in terms of the Euler number of S. It is also shown that all curves having sufficiently low genus and zero self-intersection must be contained in the zero set of some holomorphic 1-form on S A structure theorem for elliptic surfaces having a non-vanishing holomorphic 1-form is proved and examples are provided / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_23648 |
| Date | January 1983 |
| Contributors | Spurr, Michael Jerome (Author) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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