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Stable Bundles, Instantons and C∞-Structures on Algebraic Surfaces

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Complex Manifolds

Abstract

The ties between topology and algebraic geometry go back to the nineteenth century, but specific questions about surfaces being homeomorphic or not, have only come up in the fifties and sixties. Seven asked in 1954 if every algebraic surface homeomorphic to the projective plane, is biregularly equivalent to it. The (positive) answer came three decades later, as a result of Yau’s work on the Calabi conjecture. In 1965 Kodaira constructed examples of elliptic surfaces with the homotopy type of a K3-surface, and he posed the question, whether they are homeomorphic to such a surface. At the time, the only theorem known was a result of J.H.C. Whitehead, stating that two simply-connected surfaces are of the same homotopy type if and only if they have isomorphic (integer-valued) intersections forms. Given the known structure theorems for indefinite forms, it was a simple matter to verify this in Kodaira’s case. The affirmative answer to Kodaira’s question and many similar ones is an immediate consequence of Freedman’s fundamental theorem from 1982, saying in particular that the homeomorphism type of a compact, oriented, simply-connected differentiable 4-fold is completely determined by its intersection form.

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Okonek, C., Van de Ven, A. (1998). Stable Bundles, Instantons and C∞-Structures on Algebraic Surfaces. In: Complex Manifolds. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61299-2_4

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