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BOUNDNESS OF INTERSECTION NUMBERS FOR ACTIONS BY TWO-DIMENSIONAL BIHOLOMORPHISMS

Published online by Cambridge University Press:  22 January 2021

Javier Ribón*
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Campus do Gragoatá, Rua Marcos Valdemar de Freitas Reis s/n, 24210-201 Niterói, Rio de Janeiro, Brazil (jribon@id.uff.br)

Abstract

We say that a group G of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(\phi (V), W)$ takes only finitely many values as a function of G for any choice of analytic sets V and W of complementary dimension. In dimension $2$ we show that G satisfies the uniform intersection property if and only if it is finitely determined – that is, if there exists a natural number k such that different elements of G have different Taylor expansions of degree k at the origin. We also prove that G is finitely determined if and only if the action of G on the space of germs of analytic curves has discrete orbits.

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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