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Complex Manifolds Complex Manifolds Lecture notes based on the course by Lambertus van Geemen A.A. 2012/2013 Author: Michele Ferrari. For any improvement suggestion, please email me at: [email protected] Contents n 1 Some preliminaries about C 3 2 Basic theory of complex manifolds 6 2.1 Complex charts and atlases . 6 2.2 Holomorphic functions . 8 2.3 The complex tangent space and cotangent space . 10 2.4 Differential forms . 12 2.5 Complex submanifolds . 14 n 2.6 Submanifolds of P ............................... 16 2.6.1 Complete intersections . 18 2 3 The Weierstrass }-function; complex tori and cubics in P 21 3.1 Complex tori . 21 3.2 Elliptic functions . 22 3.3 The Weierstrass }-function . 24 3.4 Tori and cubic curves . 26 3.4.1 Addition law on cubic curves . 28 3.4.2 Isomorphisms between tori . 30 2 Chapter 1 n Some preliminaries about C We assume that the reader has some familiarity with the notion of a holomorphic function in one complex variable. We extend that notion with the following n n Definition 1.1. Let f : C ! C, U ⊆ C open with a 2 U, and let z = (z1; : : : ; zn) be n the coordinates in C . f is holomorphic in a = (a1; : : : ; an) 2 U if f has a convergent power series expansion: +1 X k1 kn f(z) = ak1;:::;kn (z1 − a1) ··· (zn − an) k1;:::;kn=0 This means, in particular, that f is holomorphic in each variable. Moreover, we define OCn (U) := ff : U ! C j f is holomorphicg m A map F = (F1;:::;Fm): U ! C is holomorphic if each Fj is holomorphic. Let f be a holomorphic function. One can write f(z) = g(z)+ih(z), with g; h : U ! R smooth. The condition for f to be holomorphic on U is equivalent to the Cauchy-Riemann conditions: @g @h @g @h (a) = (a) and (a) = − (a) @xj @yj @yj @xj for j = 1; : : : ; n, where zj = xj + iyj. n m Definition 1.2. Let V ⊆ C be open, let F = (F1;:::;Fm): V ! C be a holomorphic map. The complex Jacobian matrix of F is 0 @F1 @F1 1 @z ::: @z 1 n @F B . .. C j JCF := B . C = @ A @zk @Fm ::: @Fm @z1 @zn ~ 2n Now let Fj = Gj +iHj with Gj;Hj : V ! R smooth R-valued functions. Let F : V ! R defined as F~(z) = (G1(z);:::;Gm(z);H1(z);:::;Hm(z)). The real Jacobian matrix of F is @Gj @Gj ! J F := J F~ = @xk @yk R R @Hj @Hj @xk @yk 3 n 4 Chapter 1. Some preliminaries about C Remark. If F : n ! m is holomorphic, then @Gj = @Hj ; @Gj = − @Hj that means C C @xk @yk @yk @xk ! A −B @Gj @Hj JRF = with A = ;B = BA @xk @xk Moreover @Fj 1 @ @ 1 @Gj @Hj @Hj @Gj = − i (Gj + iHj) = + + i − = @zk 2 @xk @yk 2 @xk @yk @xk @yk @Gj @Hj = + i = Ajk + iBjk ) JCF = A + iB: @xk @xk Let M = PQ 2 M ( ), and let J = 0 −Idn . Then Lemma 1.1. RS 2n R Idn 0 ! P −R JMJ −1 = M () P = S; Q = −R () M = RP Proof. ! ! −R −S Q −P JMJ −1 = M () JM = MJ () = PQ S −R Combining the above lemma and the previous remark, we can characterize an holo- n n morphic function F : C ! C (same dimension!) analyzing its real Jacobian matrix: n n −1 Proposition 1.2. A function F : C ! C is holomorphic if and only if J(J F )J = R J F , with J = 0 −Idn . R Idn 0 Remark. n It is worth to notice that J is the matrix representing the multiplication by i from C to itself, as we shall see in the following chapter. Thus, one can also state: a function n n F : C ! C is holomorphic if and only if its real Jacobian matrix is self-conjugate under the conjugation action of the multiplication by i (or simply, its real Jacobian matrix commutes with J). n n Proposition 1.3. Let F : C ! C holomorphic. Then det(JRF ) ≥ 0. Proof. Consider the matrix N defined as ! ! Idn i · Idn −1 1 Idn Idn N = 2 M2n(C);N = Idn −i · Idn 2 −i · Idn i · Idn Notice that ! ! ! −1 1 Idn i · Idn A −B Idn Idn NJRFN = = 2 Idn −i · Idn BA −i · Idn i · Idn ! ! ! ! 1 A + iB −B + iA Id Id A + iB 0 J F 0 = n n = = C 2 A − iB −B − iA −i · Idn i · Idn 0 A − iB 0 JCF 5 Hence −1 −1 det(JRF ) = det(N)det(N )det(JRF ) = det(NJRFN ) = 2 = det(JCF )det(JCF ) = det(JCF )det(JCF ) = jdet(JCF )j ≥ 0 2 Recall that an holomorphic function in one variable is a conformal mapping from R to itself, that is, it preserves orientations of angles. The latter proposition shows that, when dealing with an holomorphic function of several variables, the "orientation preserving" property translates to a really strict condition on the determinant of the real Jacobian of the function. As we will see in the next chapter, this condition is related in some sense with the notion of orientation (it will imply the orientability of complex manifolds, seen as differentiable manifolds). Theorem 1.4 (Maximum principle). Let g : V ! C be holomorphic, V ⊆ C open, connected. Assume there is a v 2 V such that jg(v)j ≥ jg(z)j 8z 2 V (jgj takes its maximum on V ). Then g is constant, so g(z) = g(v) 8z 2 V . This fundamental result about one-variable holomorphic function has many conse- quences in complex analysis; we will only use it once in the following chapter to see that a holomorphic function on a compact complex manifold is nothing but a constant function (see theorem 2.3). There exists also a "holomorphic version" of the Dini theorem (local invertibility of maps with invertible Jacobian): n n Proposition 1.5. Let V ⊆ C open, F : V ! C holomorphic. Assume that JCF has rank n in a 2 V (i.e. JCF (a) has non-zero determinant). Then there is a neighborhood W of a and an holomorphic inverse G : F (W ) ! W such that F ◦ G = idF (W );G ◦ F = idW . 2 Proof. As det(JRF ) has rank 2n, det(JRF ) = jdet(JCF )j 6= 0. So, by the Dini theorem there exist a neighborhood W of a such that it is possible to find an inverse G for the map 2n 2n F regarded as a map from R to R . We are going to show that G is already the map −1 we need, that is, G is holomorphic; or equivalently, J(JRG)J = JRG, J as in lemma 1.1. We know that G ◦ F = idW ) JRG · JRF = Id; moreover, since F is holomorphic −1 −1 −1 −1 −1 −1 −1 −1 J(JRF )J = JRF . Then J(JRG) J = JRG ) J(JRG)J = (J(JRG) J ) = −1 −1 (JRG ) = JRG. n Definition 1.3. A function F is biholomorphic on W ⊆ C if there exists an holomorphic inverse G : F (W ) ! W (as in the previous proposition). Chapter 2 Basic theory of complex manifolds Throughout this chapter, we will assume that the reader has some familiarity with the the- ory of differentiable manifolds, since many times we will refer to (and use) the definitions and the results related to that theory. 2.1 Complex charts and atlases Let X be a topological manifold of dimension 2n, that is, X is a Hausdorff topological space such that each point of X admits an open neighborhood U which is homeomorphic 2n to an open subset V of R . Such an homeomorphism x : U ! V is called coordinate neighborhood. In this course, we do not require X to be second countable (as it happened for differentiable manifolds). Definition 2.1. A local complex chart (U; z) of X is an open subset U ⊆ X and an n 2n homeomorphism z : U ! V := z(U) ⊂ C (≡ R ). −1 Two local complex charts (Uα; zα); (Uβ; zβ) are compatible if the map fβα := zβ ◦ zα : zα(Uα \ Uβ) ! zβ(Uα \ Uβ) is holomorphic. The map fβα is called transition function or coordinate change. (We note that fαβ is holomorphic, too). Definition 2.2. A holomorphic atlas (or complex analytical atlas) of X is a collection A = f(Uα; zα)g of local complex charts, such that X = [αUα and such that all transition functions fαβ are biholomorphic, for each α; β. (In this way, each pair of charts is com- patible). A complex analytic structure on X is a maximal holomorphic atlas A = f(Uα; zα)gα2I . Maximal means: if (U; z) is a local complex chart and (U; z) is compatible with (Uα; zα) 8α 2 I, then (U; z) 2 A. A complex (analytic) manifold is a topological manifold together with a complex analytic structure. 6 2.1. Complex charts and atlases 7 Remark. A holomorphic atlas B = f(Uβ; zβ)gβ2J determines a (unique) maximal atlas A with B ⊂ A and hence it determines a complex manifold. (The atlas is given by A = f(U; z) j (U; z) compatible with (Uβ; zβ) 8β 2 Jg). n+1 n n (C − f0g) Example(The projective space): As usual, we define P (C) = P = ∼ × where u ∼ v , u = tv for some t 2 C = C − f0g. Let n+1 n π : C − f0g ! P (u0; : : : ; un) 7! (u0 : ::: : un) n n −1 be the quotient map.
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