The Topology of Algebraic Varieties

Igor R. Shafarevich
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Igor R. Shafarevich: Steklov Mathematical Institute of the Russian Academy of Sciences, Algebra Section

Chapter Chapter 7 in Basic Algebraic Geometry 2, 2013, pp 115-148 from Springer

Abstract: Abstract A variety over $\mathbb{C}$ has a Euclidean topology defining an underlying topological space which, for a nonsingular variety, is a real differentiable manifold, that is orientable and of twice the complex dimension. The variety inherits the usual topological invariants such as fundamental group and cohomology. In this context, complete implies compact. A set of less elementary questions centres around the connectedness of the topological space underlying an irreducible variety. The chapter includes a sketch of the local and relative versions of this problem, that includes the Zariski connectedness theorem. The topology of algebraic curves leads to the famous picture of a compact Riemann surfaces as a sphere with g handles and Euler characteristic 2−2g. The chapter also discusses the geometry of the nested ovals of real algebraic plane curves and the possible complex conjugation maps.

Keywords: Topological Space; Algebraic Variety; Euler Characteristic; Betti Number; Algebraic Curve (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-3-642-38010-5_3

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