# On a mixed Monge-Amp\`ere operator for quasiplurisubharmonic functions with analytic singularities

## TL;DR

This paper develops a method to approximate mixed Monge-Ampère products of quasiplurisubharmonic functions with analytic singularities using smooth functions, linking complex analysis, residue currents, and algebraic geometry.

## Contribution

It introduces a regularization technique for mixed Monge-Ampère products with analytic singularities, extending previous results to the mixed case and connecting to residue current theory.

## Key findings

- Explicit one-parameter limits for regularization of mixed Monge-Ampère products
- Generalization of non-mixed product results to mixed products
- Approximation of Chern and Segre currents by smooth forms

## Abstract

We consider mixed Monge-Amp\`ere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Amp\`ere products of smooth functions, generalizing results of Andersson, B{\l}ocki and the last author in the case of non-mixed Monge-Amp\`ere products. Connections to the theory of residue currents, going back to Coleff-Herrera, Passare and others, play an important role in the proof. As a consequence we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.