Yuchen Liu's general
My committee: Kollár[K](chair), Tian[T], Pausader[P] Advanced topics:
Algebraic Geometry, Complex Geometry.
Time: 1:30pm - 3:10pm, May 7th, 2014.
Real Analysis:
[P]: State the monotone convergence theorem. Give examples that the
theorem does not hold for functions not satisfying your conditions
(non-negative, monotone).
[P]: Define the Fourier transform of a L^1 function. Describe the
image of the Fourier transform(continuous, vanish at infinity). Prove
Riemann-Lebesgue lemma.
[P]: What can you say about the eigenvalues of the Fourier transform?
(roots of unity of order 4) Can you describe the eigenvectors? (I gave
the Gaussian distribution function, but failed to prove that it is an
eigenvector.)
[P]: Compute the Fourier transform of e^(-x^2). (No idea.) State the
residue theorem. Can you use this theorem to compute the integral? (I
messed it up, finally they guide me to draw a long rectangular and use
Cauchy's theorem.)
Complex Analysis:
[P]: Classify domains in the Riemann sphere up to conformal
equivalence. (I did it for simply-connected domains, using Riemann
mapping theorem.)
[P]: If two holomorphic functions have the same real part, what can we
say about them? (Their difference is an imaginary constant.)
[P]: Give me a holomorphic periodic function. (e^z) Give me a doubly
periodic function. (Weierstrass P) Is your function holomorphic? (no)
Is there any doubly periodic holomorphic function? (no, by Liouville
or maximum principle)
Algebra:
[K]: Define Galois extension. Suppose we are working in char 0, is any
deg 2 extension Galois? (yes) deg 3 extension? (no, x^3-2 over Q)
[K]: If all fields are function fields of algebraic curves over C,
what does Galois extension correspond to? (relative Aut group acts
transitively on fibers) When is an field extension not Galois in this
case? (when a point in the target has different branch type) Give an
example that the composition of two Galois extensions is not Galois.
Complex Geometry:
[T]: Define the Chern forms. (connection, curvature, determinant)
Prove that the Chern class is independent of the choice of
connection. (No idea.) Prove that the Chern forms are closed. (I
showed c_1 is closed, but couldn't figure out c_n.)
[T]: When does a complex line bundle have a holomorphic line bundle
structure? (For compact Kaehler manifold, only if c_1 is a
(1,1)-class) Explain the reason. (I compute the unique connection
compatible with hermitian metric and complex strucutre.) What about
the converse (or the if part)? (It's true, I gave the proof of
Lefschetz (1,1) theorem, and mentioned c_1 in H^2(X,Z) determines a
complex line bundle completely.)
[T]: State the Kodaira vanishing theorem. Prove it. (I only remember
the rough idea, to find a harmonic section and use some inequality,
but I couldn't prove it rigorously.)
[T]: Give an example of non-projective compact Kaehler
manifold. (complex torus which does not satisfy Riemann bilinear
relations)
Algebraic Geometry:
[K]: Suppose X is a scheme over a field K, L is a field extension of
K, then what can you say about cohomology of coherent sheaves on X and
on X\times_K L? (Flat base change.)
[K]: State the semi-continuity theorem. Do you know an example that
the dimension of cohomology groups jumps? (I had some rough idea of
Hartshorne's exercise, but I could't figure it out.) Consider an
elliptic curve, what is the cohomology for a divisor? (Riemann-Roch,
compute h^0 when the degree of the divisor is 0) Can you give an
example now? (self-product of an elliptic curve, with a horizontal
divisor minus the diagonal divisor)
[K]: Define a Fano variety. What is your favorite Fano variety?
(projective spaces, del Pezzo surfaces) Is the blow up of a point in
P^3 Fano? (I compute the canonical divisor, and state
Nakai-Moishezon.) Let's look at the intersection of anti-canonical
divisor with curves. (Dividing into two cases, nonexceptional curves
and exceptional curves, always positive.) What about P^3 blowing up
two points? (same idea, except that I forgot to consider the line
through these two points at first.)
[K]: Do you know some properties of rational curves in Fano varieties?
(rational-connected) Can you prove some result? (no idea) It doesn't
matter.
Comments: The professors are very kind during the exam. When you don't
know how to prove something, they will give you hints to lead you to
the proof. Good luck to everyone.