MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested in the local geometry of holomorphic curves in Calabi-Yau threefolds. The setup and question are then the following:

Consider a $\mathbb{C}^2$ bundle over a compact Riemann surface $\Sigma_g$ of genus $g>1$ endowed with a unitary connection. The first Chern class of the bundle will be $-K$ with $K$ the canonical class of $\Sigma_g$. What are the necessary and sufficient conditions which must be imposed on the unitary connection such that there exists a Ricci-flat metric in a neighborhood of the zero-section which induces the connection on the normal bundle?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.