Prof. G. Kokarev Sommersemester 2015 Complex Geometry Exercise sheet 2 c Problem 1. Let V be a real vector space and V = V ⊗ C be its complexification. Show that: c c (i) a C-linear endomorphism Π: V ! V is the C-linear extension of an R-linear endomorphism V ! V if and only if Π¯ = Π; (ii) any decomposition V c = V 1;0 ⊕ V 0;1 into the direct sum of complex subspaces such that V 1;0 = V 0;1 induces a linear complex structure on V . Problem 2. Let V be a real vector space, dim V = 2n, equipped with a linear complex structure J and a Hermitian scalar product g(·; ·). We assume that V has a natural orientation induced by J. (i) Show that the 2-form !(X; Y ) = g(JX; Y ) is a non-degenerate skew-symmetric form whose c extension to the complexification V = V ⊗ C is an (1; 1)-form. n (ii) Show that ! = n!Vol g, where Vol g is a volume form induced by the scalar product g and the orientation on V . (iii) (Wirtinger inequality) Let W ⊂ V be an oriented subspace of dimension 2m. Show that m ! jW 6 m!Vol gjW and the equality occurs if and only if W is a J-invariant subspace with an orientation induced by J. Remark. For the part (iii) use the following fact from linear algebra: for any skew-symmetric 2-form P 2i−1 2i ! on a Euclidean space E there exists an orthonormal basis basis (ej) such that ! = λiξ ^ ξ , j where (ξ ) is a basis dual to (ej). 1 Problem 3. Check the following properties of the complex projective line CP : (i) it is biholomorphic to a Riemann sphere S2; n 1 n (ii) for any lattice Γ ⊂ C any holomorphic map CP ! C =Γ is constant. n Problem 4. Check the following topological properties of the complex projective space CP : (i) it is compact; k (ii) the subsets ∆ = f(z0 : ::: : zn): zk 6= 0; zj = 0 for j > kg, where k = 0; : : : ; n, are homeomor- k n phic to C and define the cell complex structure on CP ; n n (iii) conclude that CP is simply connected and compute the homology groups Hk(CP ): Remark. The solution to this exercise requires the knowledge of basic topology; the necessary material can be found, for example, in Chapters 1–2 of Algebraic Topology by Allen Hatcher. To be discussed on 8 May 2015.