Homework Assignments for Math 708 - Spring 2024

• Assignment 1   (Due Tuesday, February 20)
  Huybrechts' book Section 1.1: Read this section.
  • Solve the following problems on page 23: 4 (i.e., 1.1.4), 6 (see hint below), 8, 9, 11, 15, 18, 20 (see hint below).
  • Hint for 1.1.6: First choose a coordinate z_i for which the hypotheses of WPT are satisfied. Next read the proof of WPT and determine the degree of g accordingly. Next, factor f_w(z_i) as a product of linear polynomials in z_i (use the quadratic formula). Use degree considerations to conclude that g(z_1,z_2) should vanish along the zero locus of the product of a subset of these factors. So g would be the monic polynomial in z_i, which ``is'' the product of this subset of linear factors.
  • Hint for 1.1.20: Analyse the proof of Hartog's theorem and note that the proof used only the first two coordinates.
  • Solution to Section 1.1 graded problems 6, and 11, 15, 20
• Assignment 2   (Due Thursday, March 7)
  • Huybrechts' book Section 1.2: Read this section up to Definition 1.2.18.
  • Solve the following problems on page 40: 1.2.1 (see hint), 1.2.5 (use the circle action by f_t:=cos(t)id+i sin(t)I discussed in class to verify that the type of the form is (2,0)).
  • Huybrechts' book Section 1.3: Read this section.
  • Solve the following problems on page 50: 1.3.2, 1.3.3, 1.3.5.
  • Section 2.1: 2.1.2, 2.1.3, 2.1.4 (Generalize the statement of 2.1.4 replacing the projective space by any simply connected compact complex manifold and prove it in this generality).
  • Solution to Section 1.2 problems 1,5
  • Solution to Section 1.3 problems 3,5
  • Solution to Section 2.1 problem 4
• Assignment 3   (Due Thursday, March 28)
  • Section 2.1 page 65: 2.1.12.
  • Section 2.2 page 75: 2.2.3, 2.2.5, 2.2.6, 2.2.7, 2.2.11, 2.2.13.
  • Section 2.4 page 96: 2.4.6, 2.4.10 (Suggestion: for a coordinate free proof generalize the proof given in class for the computation of the tangent bundle to projective space).
  • Solution to problems 2.1.12, 2.2.3, 2.2.5, 2.2.13, 2.4.10
• Assignment 4   (Due Thursday, April 11)
  • Homework 4
  • Solution to the HW4 problems,
• Assignment 5   (Due Tuesday, April 30)
  • Homework 5
  • Solution to problems 2, 2.3.4, 2.3.7, 2.3.10,
• Assignment 6   (Due Wednesday, May 15)
  Do five of the following problems (though all are highly recommended)
  • Section 3.1 page 123: 3.1.5, 3.1.4 (see hint), 3.1.6 (see hint).
    • Hint for 3.1.4: Use 3.1.5 and the case n=1 computed in the text to show that the cohomology class of w_FS is Poincare dual to a hyperplane in P^n and then use the fact that Poincare Duality conjugates the cup product pairing on cohomology to the intersection pairing on homology.
    • Hint for 3.1.6: Once w_FS is pulled back to C^{n+1} minus the origin the coordinates z_i become functions. Show first that one can replace in the formula for w_i over U_i in Example 3.1.9 (i) the quotient |z_j/z_i|^2 by |z_j|^2 without changing the two form.
  • Section 3.2 page 130: 3.2.5, 3.2.6, 3.2.7, 3.2.8, 3.2.11.
  • Section 3.3 page 142: 3.3.4, 3.3.5 (use both the Hard Lefschetz and the Poincare Duality theorems), 3.3.6 (see hint below).
  • Hint for 3.3.6: Choose one of the following two ways to solve the problem:
    • Using the periods of the weight 1 Hodge structure of a compact complex torus.
    • Using the weight 2 Hodge structure of a 2-dimensional compact complex torus.