Real frontiers of fake planes

TL;DR

This paper explores the classification and examples of fake real planes, complex surfaces with real loci diffeomorphic to , across all Kodaira dimensions, showing their birational diffeomorphism to .

Contribution

It provides elementary descriptions and examples of fake real planes for all Kodaira dimensions, expanding understanding of their structure and classification.

Findings

Examples of fake real planes for all Kodaira dimensions are constructed.

Many fake real planes are shown to be birationally diffeomorphic to .

Classification results are extended with explicit elementary examples.

Abstract

In Dubouloz and Mangolte (Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$, arXiv:1507.01574, 2015) we define and partially classify fake real planes, that is, minimal complex surfaces with conjugation whose real locus is diffeomorphic to the euclidean real plane $\mathbb{R}^{2}$. Classification results are given up to biregular isomorphisms and up to birational diffeomorphisms. In this note, we describe in an elementary way numerous examples of fake real planes and we exhibit examples of such planes of every Kodaira dimension $\kappa\in \{-\infty,0,1,2\}$ which are birationally diffeomorphic to $\mathbb{R}^{2}$.