Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties

TL;DR

This paper extends the theory of disc functionals and extremal functions to singular complex varieties, providing new formulas and demonstrating plurisubharmonicity in more general settings.

Contribution

It generalizes classical results from complex manifolds to locally irreducible complex spaces, introducing new formulas for extremal functions on singular varieties.

Findings

Plurisubharmonicity of envelopes on singular spaces established

New formulas for Siciak-Zaharyuta extremal functions derived

Extension of classical disc functional results to singular varieties

Abstract

We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in C^n by Lempert and by Larusson and Sigurdsson.