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.
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.
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.
Took notes on connections on smooth vector bundles, the d-bar operator on sections of a holomorphic vector bundle, and the chern connection on holomorphic vector bundles. 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.