Parametric h-principle for holomorphic immersions with approximation

TL;DR

This paper establishes a parametric homotopy principle for holomorphic immersions of Stein manifolds into Euclidean space, including approximation on convex sets, using an integration by parts formula for related holomorphic problems.

Contribution

It introduces a new parametric homotopy principle for holomorphic immersions with approximation capabilities, expanding the theoretical framework for complex geometry.

Findings

Proves the parametric homotopy principle for holomorphic immersions.

Develops an integration by parts formula for solving holomorphic vector field problems.

Demonstrates approximation results on holomorphically convex sets.

Abstract

We prove the parametric homotopy principle for holomorphic immersions of Stein manifolds into Euclidian space and the homotopy principle with approximation on holomorphically convex sets. We write an integration by parts like formula for the solution $f$ to the problem $Lf|_\Sigma=g$, where $L$ is a holomorphic vector field, semi-transversal to analytic variety $\Sigma$.