2014 Mathematics REU

I will be spending the summer of 2014 working on a research project along with Yifeng Huang, supervised by Professor Jian Song of the Rutgers mathematics department, hosted by the DIMACS REU program.

The project involves complex geometry, specifically solving the Kahler-Ricci soliton equation on the total space of a particular holomorphic vector bundle. The goal is to generalize the paper "On rotationally symmetric Kahler-Ricci solitons" [Li, 2011] to bundle given by the direct sum of n copies of O(-k) for any positive integer k.

Weekly Log

Pre-REU

Familiarized myself with basic properties of complex manifolds, the complexified tangent bundle, its direct sum decomposition into the holomorphic and anti-holomorphic tangent bundles, the induced complex structure on the real tangent bundle, p,q forms. Main example is complex projective space.

Week 1

Read about basic properties of holomorphic line bundles, the transition functions of derived bundles (ex. tensor powers, direct sums, dual bundles) Verified the Fubini-Study metric on complex projective space is Kahler-Einstein.

Week 2

Took notes on connections on smooth vector bundles including the d-bar operator on sections of a holomorphic vector bundle and the chern connection. If L \to M is a holomorphic line bundle and h is a hermitian metric on L, then the 1,1 form \partial \bar{\partial} log h_{\alpha}, where h_{\alpha} is a local expression of the metric is globally well defined, this is called the curvature form of L.

We also computed the local expression of a special class of metrics on the total space of a general n-times direct sum of the line bundle (L,h) over M, as well as its Ricci curvature.

Week 3

Computed the local expression of the Lie derivative term of the KR soliton equation, in the special case that the vector field is a holomorphic function times the vertical radial vector field. It turns out that the equation actually forces this vector field to be a constant multiple of the radial field. Also calculated the curvature of the natural metric on O(-1), and derived the form of the metric on O(-k).

Week 4

Computed the curvature of the natural metric on O(-k) and found that it is indeed -k times the fubini study metric on projective space. Worked to further reduce the KRS equation to an ODE in the smooth function parametrizing the space of metrics. Right now I am waiting to hear back from my advisor to clarify some details in the paper I am working off of which is preventing further progress. In the mean time I am creating a set of comprehensive notes on the topics I've had to learn so far.

Week 5

The derivation of the ODE mentioned previously implied that it was a necessary condition for any metric satisfying the KRS equation. This week I verified that it is also sufficient; any solution of this ODE yields a a Kahler-Ricci soliton. Derived boundary conditions for this ODE based on Calabi's extension theorem. Reduced the equation to a first order linear form by applying the Legendre transform.

Week 6

If F(t) is the solution of the above first order linear ODE, the defining equation for the soliton solution of interest becomes \phi'(t) = F(t)(1).There are two free constants in the solution of the linear ODE. The calabi boundary conditions along with the condition that the soliton be shrinking fix these constants. Based on the zeroes and growth of F, verified that a solution of (1) actually is a complete shrinking soliton.

Week 7

Analyzed the behavior of solutions of (1) as t \to \infty. This corresponds to the behavior of the metric as one moves out to infinity radially along the fibers from the zero section. Found the the metric becomes asymptotic to a conical metric, i.e. one of the form \partial \bar{partial} t^p/p for some p > 0.

Week 8

We began to look back at the equation (1) and studied the possible values of the free parameters of F(t) if the soliton is to be expanding. In this case there is a 1-parameter family of solutions indexed by positive real numbers. We plan to continue this project in the fall where we will begin with studying the asymptotics (as t \to \infty) of expanding solitons and the behavior of the Ricci flow corresponding to these solitons.

Presentation Slides

Introductory Presentation

References & Links

  1. Griffiths, P. , Harris, J. 1978. Principles of Algebraic Geometry. John Wiley and Sons, Inc.
  2. Wells, R.O. 1973. Differential Analysis on Complex manifolds. Prentice-Hall Inc.
  3. Li, C. On Rotationally Symmetric Kahler-Ricci Solitons. arXiv:1004.4049