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Mathematics Colloquium


Date:  Tuesday, April 08, 2014

Title:  Affine Varieties and Potential Theory

Abstract:  A polynomial on $\CC^n$ is an entire function of slowest possible growth. This can be made precise by comparing to a reference strictly plurisubharmonic exhaustion function (just $\|z\|^2 $). Similarly, a complex submanifold $X \subset \CC^N$ is {\em affine algebraic} if it can be defined by the vanishing of polynomials (as opposed to entire functions). The question addressed in this talk is whether there are intrinsic characterizations of which complex manifolds are affine varieties in terms of the plurisubharmonic functions on them. Which compact complex manifolds are {\em projective algebraic varieties} was nicely answered long ago by Kodaira's Embedding Theorem. At present, there are still only partial answers known for the affine case. Some of the conjectures to be discussed were raised by Griffiths, Stoll and the speaker. The results are due to a variety of people, many with connections to UM. We will survey these results up to the present, and give many examples and counter-examples. As a colloquium, pictorial proofs will be emphasized.

Speaker:  Dan Burns
Institution:  University of Michigan


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