Abelian group actions on algebraic varieties with one fixed
point
(joint work with Amir Assadi and Rebecca Barlow)
Theorem: Let X be a complete algebraic variety over an
algebraically closed field of characteristic p>=0, and let G be a
finite abelian group acting on X. Assume the order of G is l^r, where l
is a prime different from p. If the fixed point set consists of exactly
one point x, then X is singular in X.
Corollary: If X is smooth then G has either no fixed
point or at least two of them.
This is an algebraic analogue of (much deeper) results of Conner-Floyd,
Atiyah-Bott and others in the topological category.
Appeared in: Mathematische Zeitschrift
210 (1992) 129-136
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Last updated: December 20, 2003