MATH 514, COMPLEX ALGEBRAIC GEOMETRY, FALL 2022, LECTURE AND HOMEWORK PLANS
[V]=Voisin, [GH]=Griffiths and Harris, [GF] = Grauert and Fritzsche
- Week 1, August 22, 24, 26.
Reading for the week. [GH] Pages 2-12; [V] Chapter 1; alternative and more detailed versions can be found in [GF] Chapter 1 and 3.1-3.3.
Homework for Wednesday, September 7: 1: Prove that the divergence theorem in R3 is a special case of Stokes's Theorem. 2: Use the proof of the Weierstrass division theorem to write f(z,w) = zw +w2 ew + ew z4 + sin(z) near the origin in the form q(z,w)(w2+z4)+r(z,w), where q and r are holomorphic and r is a polynomial in w of degree less than two. 3,4,5: [V] P37 1-3.
- Week 2, August 29, 31, September 2
Reading for the week. [V] Pages 38-53; [GH] 14-18.
Additional Homework for Wednesday, September 7: [V] Page 62 3.
Homework for Monday, September 19: 1,2. [V] P62 1,2.
- Week 3, September 7 and 9. Wednesday, September 7 is a problem session
Reading for the week. [V] Pages 46-61; [GH] 17-18.
Homework for Monday, September 19: 3. Let M be a complex manifold, let E be a holomorphic vector bundle on M, and let s be a continuous section of E, so that s is a continuous map from M to E. Let {U_j} be an open cover of M and let \pi^{-1}(U_j) \to U_j \times C^r be trivializations, so that the restriction of s to U_j can be identified with a continuous map s_{U_j}: U_j \to C^r. Prove that s is holomorphic if and only s_{U_j} is holomorphic for all j.
- Week 4, September 12, 14, 16.
Reading for the week. [V] Pages 53-69; [GH] 23-31
Additional Homework for Monday, September 19: 4. Give an example of a non-integrable distribution on R3.
Homework for Monday, October 3:1-2. [V] P82 1,2 3. Suppose that X and Y are Kahler manifolds. Prove that the product XxY is also a Kahler manifold.
- Week 5, September 19, 21, 23. Monday, September 19 is a problem session
Reading for the week. [V] Pages 68-78, [GH] Pages 27-34
Additional Homework for Monday, October 3:4. Let S be the universal line bundle on Pn, the line subbundle of the rank n+1 trivial bundle described in the text. Let h be restriction to S of the standard Hermitian metric on the trivial bundle. Compute the Chern form of this metric. How is it related to the Kahler form of the Fubini-Study metric?
- Week 6, September 26, 28, 30.
Reading for the week. [V] Pages 76-94, [GH] Pages 34-38
Homework for Monday, October 17:1-2. [V] P113 1,2
3 Prove that any compact Riemann surface is Kahler. 4 Let X be a complex manifold, and let OX* be the sheaf which associates to U the multiplicative group of nowhere vanishing holomorphic functions on U. For any f in OX(U), put E(f) = exp(2π if). Show that E defines a surjective homomorphism OX → OX* of sheaves of abelian groups.
- Week 7, October 3,5,7. Monday, October 3 is a problem session.
Reading for the week. [V] Pages 76-102, [GH] Pages 34-38
Additional Homework for Monday, October 17: 5 Let X be a topological space, U an open subset, {V_i} an open cover of U, F a sheaf on X, and Ff the associated sheaf. Given a collection of sections s_i of F on V_i which agree on the intersections, show that these determine a section s of Ff on U. Given another section t of Ff on U similarly determined by a cover {W_j} and compatible sections t_j of F on W_j, give necessary and sufficient conditions for s and t to be equal as sections of Ff.
- Week 8, October 10,12,14.
Reading for the week. [V] Pages 91-113, [GH] Pages 38-49
Homework for Monday, October 31: 1. Let U={U0,U1} be the standard open cover of P1. Compute the Cech cohomology groups Hj(U,O(n)) for j=0,1 and all integers n.
- Week 9, October 17,19,21.
Reading for the week. [V] Pages 102-113, [GH] Pages 38-49
Additional homework for Monday, October 31: 2. Use cohomology to compute the set of isomorphism classes of locally free sheaves of rank 1 on P1. 3. Find all holomorphic vector bundles G on P1 which fit into a short exact sequence 0 -> O(-2) -> G -> O -> 0.
- Week 10, October 24,26,28.
Reading for the week. [V] Pages 117-136
Still more homework for Monday, October 31: 4. Let F be a sheaf on X and pick a in H1(X,F). Use the discussion on [V] P 111 applied to two arbitrary choices of injectives I,I' containing F, we get a Cech cocycle b for an open cover U={Ui} , and another Cech cocycle b' for an open cover U'={U'j}. Without assuming [V] Theorem 4.44, prove that U and U' have a common refinement V so that the two Cech cocycles for the covering V induced by b and b' differ by a coboundary.
Homework for Monday, November 14: 1-2. [V] P 136 1-2.
- Week 11, October 31, Nov 2,4
Reading for the week. [V] Pages 129-148 ([GH] Pages 111-121 for another approach)
Additional Homework for Monday, November 14: 3. Let X be a complex manifold with a Hermitian metric. Compute the symbol of the d-bar Laplacian acting on (p,q)-forms. 4-5. [V] P 154 1-2.
- Week 12, November 7,9,11. Class on Monday, November 7 will be covered by Pierre Albin by Zoom. The Zoom link has been emailed to all registered students.
Reading for the week. [V] Pages 139-141, 144-154, 157-159
- Week 13, November 14,16,18. Monday, November 14 is a problem session.
Reading for the week. [GH] Pages 12-14, 20-22, 53-65.
Homework for Monday, December 5. 1. Let M=Cn/L be a complex torus, where L is a lattice (the free abelian group generated by 2n vectors which are linearly independent over R. a) Calculate the Hodge numbers hp,q(M). b) For n=2, what is the signature of the intersection form on H2(M,R)? 2. Let C be a compact Riemann surface of genus g. a) Find the Hodge numbers of C. b) For any Kahler class, describe the primitive cohomology and Lefschetz decomposition of C. 3. Let Ci be a Riemann surface of genus gi for i=1,2. Let S = C1 x C2 a) Find the Hodge numbers of S. b) What is the signature of the intersection form on H2(X,R)?
- Week 14, November 28,30, December 2.
Reading for the week. [V] 150-154, 156-168..
Additional homework for Monday, December 5. 4-5. [V] P 182, 1-2.
- Week 15, December 5,7. Class on Monday, December 5 will begin with a problem session, which may not take up the entire class period as we have not covered all of the required background material for problems 4 and 5.
Reading for the week. [V] Prop 6.17 (P 144) ("d d-bar lemma"); [V] 161-168.