A Taste of Two-dimensional Complex Algebraic Geometry

These are notes for a talk at a topology seminar at ND. 1. GENERAL FACTS In the sequel, for simplicity we denote the complex projective space CPN by PN , and we denote the projective coordinates by [~z] = [z0, . . . , zN ]. A smooth projective variety is a connected compact complex submanifold of some projective space PN . All the smooth projective varieties are Kähler manifolds, but the converse is not true. An algebraic surface is a smooth projective variety of complex dimension 2. A celebrated theorem of Chow [2, Chap. 1, Sec. 3] states that if X ↪→ PN is a smooth projective variety, then there exist homogeneous polynomials P1, . . . , Pν ∈ C[z0, . . . , zN ] such that X = { [~z] ∈ P ; P1(~z) = · · · = Pν(~z) = 0 } . In other words, all projective varieties are described by a finite collection of homogeneous polynomial equations. If X is a smooth projective variety of (complex) dimension n then the tanget bundle TX has a complex structure and thus we can speak of Chern classes ck(X) := ck(TX), k = 1, . . . , n. We also have an isomorphism of holomorphic vector bundles ΛT ∗X ⊗ C ∼= ⊕