General Exam of Mohan Swaminathan
Friday, May 4, 2018
2:00 pm - 4:00 pm
Committee: John Pardon (chair), Gabriele di Cerbo, Tatyana Shcherbyna
Special Topics: Symplectic Topology, Complex Algebraic Geometry
_____________________________________________________________________
Symplectic Topology:
JP: Tell us about Hamiltonian vector fields on a symplectic manifold.
- Defined them and proved that the flow preserves the symplectic form and conserve the Hamiltonian.
JP: Suppose X is a symplectic manifold and R^2 is endowed with the standard symplectic form ds st. Consider the graph of a function f: X x R --> R as a hypersurface
in the symplectic manifold X x R^2 (with product symplectic form) and compute its characteristic foliation.
- Pulled back the symplectic form to X x R, a computation shows that the characteristic foliation is spanned by (X_f,d/dt), where X_f is the vector field associated
to the time dependent Hamiltonian f.
JP: What can you say about the restriction of the symplectic form to this hypersurface?
- I said the symplectic form is pulled back from the quotient of X x R by the characteristic foliation.
JP: This gives another proof that X_f preserves the symplectic form (even for time dependent f). Now, let's move on to holomorphic curves. Define stable maps.
- Defined them for the case of maps from closed Riemann surfaces with marked points.
JP: Suppose Q is a manifold, and L is a Lagrangian submanifold of T^*Q whose projection to Q is a local diffeomorphism. What can you say about L?
- L is locally given by the graph of a closed 1-form.
JP: Suppose we have a pseudoholomorphic map u: C --> X which is an immersion (with dim X = 4). What can we say about the linearized del-bar operator of C?
- Firstly, TC is a subbundle of u^*TX since u is an immersion. We can therefore consider the quotient N_u = (u^*TX)/TC, and we get a Cauchy-Riemann operator on the
normal bundle N_u.
JP: Can you draw a commutative diagram for this?
- The short exact sequence 0 --> TC --> u^*TX --> N_u --> 0 gives a diagram of del-bar operators with 3 rows and 2 columns (with columns being short exact). The
del-bar operator on TC is the natural one, and the one on N_u is induced by the quotient, and the diagram commutes.
JP: Ok, what can you say about the relationship between the kernels and cokernels of these three del-bar operators?
- We can write out an exact sequence of kernels and cokernels as a consequence of the snake lemma.
JP: When is the del-bar operator on the normal bundle surjective?
- There is a numerical criterion: we have surjectivity if deg (K_C \otimes N_u) < 0.
JP: Write this out explicitly in terms of the homology class of the image of u and the number of self-intersections of u.
- After some calculation (using adjunction formula), I got deg N_u = A^2 - 2k, where A is the homology class u_*[C] and k is the number of self-intersections (which I
assumed to be ordinary nodes). I also implicitly assumed that the map u not a multiple cover.
JP: Can you interpret the formula for deg N_u geometrically?
- I got stuck here. It is obviously true for embedded curves. After a lot of prodding, I managed to say that if we push u(C) off itself inside X, then we can consider
this as a small section of the normal bundle. The number of intersections of u(C) with its transverse pushoff are the number of zeros of this section of the normal
bundle along with two extra intersections for each node. This gives A^2 = deg N_u + 2k.
JP: Consider S^2 x R^2 with its standard product symplectic form and complex structure. What are the holomorphic curves in the class [S^2 x pt]?
- Only S^2 x pt since R^2 is aspherical. There is a unique holomorphic sphere in the fibre for every point of R^2.
JP: Are these transversally cut-out?
- Yes, using the numerical criterion we discussed before (since normal bundle is trivial and holomorphic tangent bundle of S^2 has H^1 = 0).
JP: What happens if we perturbed the complex structure J inside some compact subset of S^2 x R^2?
- The immersed spheres would still be cut out transversally, by the numerical criterion.
JP: Ok, now consider a symplectic vector space V. Let L,M be transverse Lagrangian subspaces. Is the space of Lagrnagian subspaces P which are transverse to both L
and M connected? How many connected components?
- The symplectic form is determined by the non-degenerate pairing B : L x M --> R, and P will be realized as the graph of a bijective linear map f:L-->M such that
B(x,f(y)) = B(y,f(x)) for all x,y in L.
JP: Yes, but you can assume that B is diagonal by choosing appropriate bases in L and M.
- Indeed, we may assume B is the identity matrix in this basis, in which case f is required to be an invertible symmetric matrices. Thus, this space is not connected
and its connected components are classified by its signature (dimension of maximal positive/negative definite subspaces).
There were no questions about compactness or gluing, which was a little surprising.
_____________________________________________________________________
Complex Algebraic Geometry
GdC: You mentioned the adjunction formula. Can you elaborate on that?
- Proceeded to prove the adjunction formula for a smooth hypersurface inside a complex manifold but was stopped in between.
GdC: How can you get sections of the canonical bundle of the hypersurface D inside X.
- We can use the Poincare residue map K_X(D) --> K_D. I wrote this out in coordinates.
GdC: What is the kernel of this map?
- Sections of K_X which are holomorphic.
GdC: What does this tell you about the surjectivity of the Poincare residue map on global sections?
- It's surjective if H^1(X,K_X)=0 for e.g. if X = P^n.
GdC: Give me a heuristic to compute the dimension of the moduli space of genus 4 curves.
- I proceeded as in Griffiths--Harris to argue that the canonical curve of a general C of genus 4 is of degree 6 inside P^3. I did the dimension counts to say it lies
on a quadric and a certain linear system of cubics and then computed the dimension using this.
GdC: Does a similar strategy work for genus 5? What about genus 6?
- Genus 5 is ok and is done in Griffiths--Harris. I hadn't thought about genus 6. The canonical curve must be of degree 10 inside P^5, so if I wanted to realize it
as an intersection of 4 hypersurfaces, some of their degrees would have to be 1, which contradicts the non-degeneracy of the canonical curve. Thus, the canonical
curves of genus 6 are never complete intersections.
GdC: State the Abel-Jacobi theorem.
- Stated it.
JP: Is this map injective when C is of genus>=1?
- Yes, since two distinct points p,q are never linearly equivalent except on P^1.
JP: Is it an immersion?
- Yes, since there is no point of C where all the holomorphic differentials vanish.
GdC: Can you state Torelli's theorem?
- I began by talking about the principal polarization of the Jacobian given by the intersection form, the theta divisor, and ample line bundles. I then stated Torelli
as the principally polarized Jacobian determines the curve.
GdC: You mentioned the theta divisor. How is it related to the Abel-Jacobi map?
- It is (a translate of) the image of the Abel-Jacobi map from Sym^{g-1}C (i.e., it is the locus of effective degree g-1 line bundles).
GdC: Using this, what does Torelli's theorem reduce to in the case of genus 2?
- The Abel-Jacobi map C ---> J(C) gives an isomorphism onto the theta divisor. The principal polarization determines the theta divisor upto translation.
GdC: What is the dimension count for hyperelliptic curves and why does this imply the generic curve of genus >=3 is not hyperelliptic?
- Gave the argument from Griffiths--Harris.
GdC: Name a nontrivial rank 2 bundle on P^2.
- After much prodding, I named the tangent bundle of P^2.
GdC: Does TP^2 split into a sum of line bundles?
- At first, I didn't know how to proceed and then I was asked to compute Chern classes. I used the Euler sequence to compute c(TP^2) = 1 + 3H + 3H^2 which is an
irreducible polynomial, and thus we get the result that it doesn't even split topologically.
_____________________________________________________________________
Complex Analysis
TS: How would you compute the integral of cos(kx)/(1+x^2) from -infty to infty as a function of k?
- I observed that the integrand was even and so I could restrict to k>=0. Then I drew the semi-circle countour. They just wanted me to show that the integral on the
semi-circle goes to zero as the radius goes to infinity.
TS: Do you know what a harmonic function is?
- Yes, defined it.
TS: What can you say about an entire harmonic function bounded above by 1?
- Constant.
TS: How do you prove this? Do it in dimension 2 only.
- Let u>=0 be the harmonic function. Take its harmonic conjugate v, set f=u+iv and apply Liouville to e^{-f}.
TS: What if we had an entire function such that |f(z)|\le C log|z| for large z?
- It's then bounded by |z| for large z, thus it's a polynomial of degree <=1, and since it's bounded by log |z|, it's constant.
JP: What conditions can you give for a holomorphic function on the disk to extend continuously to the boundary?
- If the power series was \Sum a_nz^n, then, we could ask for a_n to be absolutely summable.
JP: What if we wanted all derivatives to extend continuously?
- We could require n^k*a_n to be absolutely summable for all k>=0.
JP: What's a condition that's simpler than that?
- I didn't initially realize that they wanted me to say. What they wanted was for me to say a_n must decay faster than the reciprocal of any polynomial.
JP: Can you give an example of this?
- Take a_n = e^{-n}.
JP: Give one which is not of exponential decay.
- Ok, a_n = e^{-sqrt n}.
_____________________________________________________________________
Real Analysis
TS: Talk about Fourier transform.
- Defined it for L^1 functions, and said it is continuous and vanishes at infinity.
TS: Talk briefly about regularity and decay.
- If (1+|x|^n)|f(x)| is in L^1, then f-hat is n times continuously differentiable.
TS: What can we say about f-hat if |f(x)|\le e^{-k|x|}?
- f-hat extends to a holomorphic function on the strip |Im z|