# Area in real K3-surfaces

### Ilia Itenberg

Sorbonne Université, Paris, France### Grigory Mikhalkin

Université de Genève, Switzerland

## Abstract

For a real K3-surface $X$, one can introduce areas of connected components of the real point set $\mathbb{R}X$ of $X$ using a holomorphic symplectic form of $X$. These areas are defined up to simultaneous multiplication by a positive real number, so the areas of different components can be compared. In particular, it turns out that the area of a non-spherical component of $\mathbb{R}X$ is always greater than the area of any spherical component.

In this paper we explore further comparative restrictions on the area for real K3-surfaces admitting a suitable polarization of degree $2g - 2$ (where $g$ is a positive integer) and such that $\mathbb{R}X$ has one non-spherical component and at least $g$ spherical components. For this purpose we introduce and study the notion of simple Harnack curves in real K3-surfaces, generalizing planar simple Harnack curves from [8].

## Cite this article

Ilia Itenberg, Grigory Mikhalkin, Area in real K3-surfaces. EMS Surv. Math. Sci. 8 (2021), no. 1/2, pp. 217–235

DOI 10.4171/EMSS/48