Area in real K3-surfaces

Ilia Itenberg; Grigory Mikhalkin

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arXiv · 2001.06871

Area in real K3-surfaces

Ilia Itenberg·Grigory Mikhalkin

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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.

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Ilia Itenberg, Grigory Mikhalkin. 2020-01-19. Area in real K3-surfaces. https://arxiv.org/abs/2001.06871

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