Powers of ideals and convergence of Green functions with colliding poles

Alexander Rashkovskii (1) , Pascal J. Thomas (2)

1 Faculty of Science and Technology
2 IMT - Institut de Mathématiques de Toulouse UMR5219

Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green functions to the Green function of the limit ideal; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers of the ideals converge to some ideals, we prove that the Green functions converge, locally uniformly away from the limit pole, to a function which is essentially the upper envelope of the scaled Green functions of the limits of the powers. As examples, we consider ideals generated by hyperplane sections of a holomorphic curve near its singular point. In particular, our result explains recently obtained asymptotics for 3-point models.

Mots clés: pluricomplex potential theory; complex Monge Ampère; ideals of holomorphic functions; Hilbert Samuel multiplicity

Domaines: Variables complexes [math.CV]

Alexander Rashkovskii, Pascal J. Thomas. Powers of ideals and convergence of Green functions with colliding poles. International Mathematics Research Notices, 2014, 2014, pp.1253-1272. ⟨10.1093/imrn/rns251⟩. ⟨hal-00728273⟩