Homework 6 - Topics in Geometry - Spring 2009 | MATH 6490, Assignments of Geometry

Download Homework 6 - Topics in Geometry - Spring 2009 | MATH 6490 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 6 Exercise 1. Let M be a Riemann surface and OM the structure sheaf on M . Show that for all points x ∈ M the stalk OM,x is isomorphic to the subring of the ring C[[Z]] consisting of all those power series in the variable Z which have a positive radius of convergence. Exercise 2. Let Ω be a lattice in C and consider the canonical projection π : C → C/Ω. Show that π is a morphism of complex manifolds. Show that π induces an isomorphism at the level of stalks of the corresponding structure sheaves. Does this mean that π is an isomorphism? Why or why not? Exercise 3. Let µn be the group of n-th roots of unity generated by ζ = e2πi/n. Consider the action of ζ (and hence of µn) on C by z 7→ ζz. Show how you can make C/µn into a manifold. (See Example 17 on p.42 of Harder’s book.) Exercise 4. (Skyscraper sheaves.) Let X be a topological space, P be a point in X, and A be an abelian group. Define the skyscraper sheaf SP on X determined by the point P and the group A as: SP (U) = A if P ∈ U and SP (U) = 0 if P /∈ U , for any open set U . Determine the stalks of SP . (Note: It is not assumed that X is Hausdorff, or for that matter X need not even be a T1-space.) ∗ ∗ ∗ ∗ ∗ ∗ ∗