Abstract

Given a rational elliptic surface Y with an anti-canonical cycle D, Hein solved the corresponding complex Monge-Ampere equation and proved that the complement Y\D admits complete Ricci-flat metrics. It is natural to ask if the cohomology class of the metric and the asymptotic behavior uniquely determine the Ricci-flat metric. I will give an affirmative answer based on a joint work with T. Collins and A. Jacob. As a consequence, the space of Ricci-flat metrics modulo gauge equivalence is finite dimensional. If time permits, I will explain the application to mirror symmetry.

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