version: December 8, 2015 (revised and corrected) Complex Analytic Geometry by Janusz Adamus Lecture Notes PART I Department of Mathematics The University of Western Ontario c Copyright by Janusz Adamus (2007{2015) 2 Janusz Adamus Contents 1 Preliminaries 4 2 Analytic sets 8 3 Hironaka division 12 4 Rings of germs of holomorphic functions 20 5 Proper projections 24 6 Local representation of analytic sets 31 7 Irreducibility and dimension 35 8 Coherent sheaves 40 9 Complex analytic spaces 48 This document was typeset using AMS-LATEX. Complex Analytic Geometry - Math 9607 3 References [Ab] S. S. Abhyankar, \Local analytic geometry", Vol. XIV Academic Press, New York-London, 1964. [BM] E. Bierstone, P. D. Milman, \The local geometry of analytic mappings", Dottorato di Ricerca in Matematica, ETS Editrice, Pisa, 1988. [Ca] H. Cartan, Id´eauxet modules de fonctions analytiques de variables complexes, Bull. Soc. Math. France 78 (1950), 29{64. [Ch] E. M. Chirka, \Complex analytic sets", 46. Kluwer Academic Publishers Group, Dordrecht, 1989. [Fi] G. Fischer, \Complex Analytic Geometry", Lecture Notes in Mathematics, Vol. 538. Springer, Berlin-New York, 1976. [GR] H. Grauert, R. Remmert, \Analytische Stellenalgebren", Springer, Berlin-New York, 1971. [Har] R. Hartshorne, \Algebraic Geometry", Springer, New York, 1977. [Ku] E. Kunz, \Introduction to Commutative Algebra and Algebraic Geometry", Birkh¨auser, Boston, 1985. [Lo] S.Lojasiewicz, \Introduction to Complex Analytic Geometry", Birkh¨auser,Basel, 1991. [Se] J.-P. Serre, Faisceaux alg´ebriquescoh´erents, Ann. of Math. 61 (1955), 197{278. [Wh] H. Whitney, \Complex analytic varieties", Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1972. 4 Janusz Adamus 1 Preliminaries Some notation: • N = f0; 1; 2;::: g; N+ = N n f0g. k k • Given real r > 0 and k 2 N+, r∆ = f(z1; : : : ; zk) 2 C : jzij < rg. • C0 = f0g. 1.1 Complex manifolds Definition 1.1. Given open subset Ω ⊂ Cn, a continuous function f :Ω ! C is called holomorphic in Ω (or simply, holomorphic) when, for every x = (x1; : : : ; xn) 2 Ω, there is a positive radius r and n a sequence (cν )ν2N ⊂ C, such that X ν n f(z) = cν (z − x) for all z 2 x + r∆ ; n ν2N ν ν1 νn where (z − x) = (z1 − x1) ::: (zn − xn) . m A mapping f = (f1; : : : ; fm):Ω ! C is a holomorphic mapping when all its components are holomorphic functions. Definition 1.2. Let m 2 N.A complex manifold of dimension m is a nonempty Hausdorff space M together with a complex atlas A = f(Uα;'α)gα2A satisfying: (i) Uα nonempty and open in M for all α 2 A m m (ii) 'α : Uα ! C homeomorphism onto a region (open connected subset) in C for all α 2 A S (iii) α2A Uα = M −1 (iv) 'β ◦ 'α : 'α(Uα \ Uβ) ! 'β(Uα \ Uβ) is holomorphic for all α; β 2 A. Two atlases on M are said to be equivalent when their union is an atlas itself. An equivalence class of all equivalent atlases is called a (complex) structure on M.A coordinate chart is any pair (U; ') such that f(U; ')g [ A is an atlas. In particular, the (Uα;'α) are coordinate charts. Remark 1.3. It is sometimes convenient to treat the empty set as a manifold. It is, by definition, of dimension −1. Definition 1.4. Given manifolds M and N and an open Ω ⊂ M, a continuous mapping f :Ω ! N is called holomorphic when, for every pair of charts (U; ') on M and (V; ) on N such that f(U \Ω) ⊂ V , the composite −1 ◦ f ◦ ('jU\Ω) : '(U \ Ω) ! (V ) is holomorphic. We write f 2 O(Ω;N). Remark 1.5. (1) Given a nonempty manifold M and a nonempty open subset Ω in M, Ω is itself a manifold and dim Ω = dim M. We call it an open submanifold of M. The structure is that restricted from M, that is, the equivalence class of the atlas fUα \ Ω;'αjΩgα where A = f(Uα;'α)gα2A is an atlas on M. Complex Analytic Geometry - Math 9607 5 (2) For n 2 N, we regard Cn as a manifold equipped with the canonical complex structure induced by the identity mapping. (3) When N = C, we write O(Ω) for O(Ω; C). Pointwise addition and multiplication of functions define a commutative ring structure on O(Ω). In fact, after identifying C with the constant functions, O(Ω) is a C-algebra. If Ω is conneted, then it follows from the Identity Principle below, that O(Ω) is an integral domain: For f 6= 0 in O(Ω), the set f −1(0) is nowhere dense in Ω, hence fg vanishes on Ω only if f or g does so. Definition 1.6. A subset N of an m-dimensional manifold M is called a submanifold of dimension n ≤ m when, for every point ξ 2 N, there is a coordinate chart (U; ') on M such that ξ 2 U and −1 m N \ U = ' (fx = (x1; : : : ; xm) 2 C : xn+1 = ··· = xm = 0g) : Of greatest interest to us will, in fact, be the submanifolds of Cn. One easily proves the following useful characterisation of such submanifolds: Proposition 1.7. A subset M ⊂ Cn is an m-dimensional submanifold of Cn if and only if M is locally (at every point ξ 2 M) a graph of a holomorphic function; that is, for every ξ 2 M there is n an m-dimensional linear subspace L ⊂ C , open neighbourhoods V of πL(ξ) in L and W of πL? (ξ) in L?, and a holomorphic f : V ! W , such that M \ (V × W ) = Γf ; where Γf denotes the graph of f. Theorem 1.8 (Identity Principle). Let M and N be complex manifolds, and let M be connected. Suppose f; g 2 O(M; N), and there is a nonempty open subset Ω ⊂ M such that fjΩ = gjΩ. Then f = g. Proof. Put D = fξ 2 M : fjW = gjW for some open neighbourhood W of ξg. Then D is open and nonempty. We will show that D is also closed, which in light of connectedness of M will imply D = M. Let ξ 2 D. Then f(ξ) = g(ξ), by continuity of f and g. Let (U; ') and (V; ) be coordinate charts on M and N respectively, such that ξ 2 U, f(ξ) 2 V , and f(U) [ g(U) ⊂ V . Then ◦ f ◦ '−1 and ◦ g ◦ '−1 agree on '(D \ U). Since D \ U is a nonempty open subset of U, it follows from the Identity Principle for holomorphic functions in Cm that ◦ f ◦ '−1 and ◦ g ◦ '−1 agree on '(U). Thus fjU = gjU , and hence ξ 2 D, as required. Definition 1.9. Let M and N be complex manifolds, of dimensions m and n respectively, and let ξ 2 M. For a holomorphic mapping f : M ! N, the rank of f at the point ξ, denoted rkξf, is defined as the rank of the Jacobi matrix @F i ('(ξ)) ; @xj i=1;:::;n j=1;:::;m −1 where F = (F1;:::;Fn) = ◦ fjU ◦ ' : '(U) ! (V ), and (U; '), (V; ) are coordinate charts around ξ and f(ξ) respectively, such that f(U) ⊂ V . The definition is independent of the choice of charts (Exercise). We say that the mapping f : M ! N is of constant rank r when rkxf = r for all x 2 M. We have the following classical result (see, e.g., [Lo,C.4.1] for the proof): Theorem 1.10 (Rank Theorem). Let f : M ! N be a holomorphic map of constant rank r, and let ξ 2 M. Then there exist coordinate charts (U; ') and (V; ) in M and N respectively, such that −1 ξ 2 U, f(U) ⊂ V , and ◦ fjU ◦ ' is a linear map of rank r. Moreover, f(U) is an r-dimensional submanifold of N, and nonempty fibres of fjU are (m − r)-dimensional submanifolds of M. 6 Janusz Adamus 1.2 Set and function germs Definition 1.11. Let X be a topological space, and ξ 2 X. Consider the following equivalence relation on P(X): A ∼ξ B , there is an open nbhd U of ξ st:A \ U = B \ U: The elements of the quotient space P(X)= ∼ξ are called the set germs at ξ. The equivalence class of A is denoted Aξ, and A is a representative of Aξ. Remark 1.12. (1) We say that Aξ ⊂ Bξ when A ⊂ B for some representatives A and B of Aξ and Bξ respectively. (2) Finite set-theoretical operations commute with taking a germ at a point; e.g., Aξ [Bξ = (A[B)ξ, Aξ \ Bξ = (A \ B)ξ. We define the product of germs as Aξ × Bη = (A × B)(ξ,η) (the definition is independent of the representatives chosen, Exercise). (3) Aξ 6= ? , ξ 2 A . (4) Representatives of Xξ are precisely those sets A ⊂ X that satisfy ξ 2 intA. Definition 1.13. Let X be a topological space, ξ 2 X, and let F(X; ξ) be the collection of all complex-valued functions (U; f) with domain U an open neighbourhood of ξ. Consider the following equivalence relation on F(X; ξ): (U; f) ∼ξ (V; g) , there is an open nbhd W of ξ st: fjW = gjW : The elements of the quotient space F(X; ξ)= ∼ξ are the function germs at ξ. The equivalence class of (U; f) is denoted fξ, and (U; f) is a representative of fξ. Remark 1.14. (1) The following are well-defined (i.e., independent of the choice of representatives, Exercise): fξ ±gξ = (f ±g)ξ, fξ ·gξ = (fg)ξ, and fξ=gξ = (f=g)ξ provided g is non-zero in a neighbourhood of ξ.