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K3 surfaces with two involutions and low Picard number

Dino Festi, Wim Nijgh, Daniel Platt

TL;DR

The paper investigates K3 surfaces of degree $2d$ that possess commuting holomorphic and anti-holomorphic involutions, establishing minimal Picard numbers and constructing explicit examples with $ρ=2$ for $d=2,3,4$ over $\mathbb{Q}$. It combines three main constructions: double covers of the plane (degree-2 K3s), nodal quartics, and smooth quartics with carefully chosen Picard lattices, plus an analysis of real structures to realize Picard lattices over $\mathbb{R}$. A key outcome is the realization of a nodal-quartic pathway yielding infinitely many $d$-values with $ρ=2$, and a strengthening of Morrison’s theorem by showing every even lattice with rank up to 10 and signature $(1,r-1)$ can be realized by a real K3 with nonempty real locus. The results provide a broad supply of explicit, low-$ρ$ K3 examples suitable for applications in hyperkähler and $G_2$-geometry constructions, including explicit models over $\mathbb{Q}$ and $\mathbb{R}$ and multiple polarizations of degree 2 and 8.

Abstract

Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $ρ$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $ρ\geq 1$ when $d=1$ and $ρ\geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $ρ=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over $\mathbb{Q}$, can be used to realise the minimum $ρ=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $ρ=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $ρ=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$.

K3 surfaces with two involutions and low Picard number

TL;DR

The paper investigates K3 surfaces of degree that possess commuting holomorphic and anti-holomorphic involutions, establishing minimal Picard numbers and constructing explicit examples with for over . It combines three main constructions: double covers of the plane (degree-2 K3s), nodal quartics, and smooth quartics with carefully chosen Picard lattices, plus an analysis of real structures to realize Picard lattices over . A key outcome is the realization of a nodal-quartic pathway yielding infinitely many -values with , and a strengthening of Morrison’s theorem by showing every even lattice with rank up to 10 and signature can be realized by a real K3 with nonempty real locus. The results provide a broad supply of explicit, low- K3 examples suitable for applications in hyperkähler and -geometry constructions, including explicit models over and and multiple polarizations of degree 2 and 8.

Abstract

Let be a complex algebraic K3 surface of degree and with Picard number . Assume that admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, when and when . For , the first example defined over with was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over , can be used to realise the minimum for all . In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum for . We also show that a nodal quartic surface can be used to realise the minimum for infinitely many different values of . Finally, we strengthen a result of Morrison by showing that for any even lattice of rank and signature there exists a K3 surface defined over such that .
Paper Structure (16 sections, 24 theorems, 61 equations, 1 figure)

This paper contains 16 sections, 24 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some background
  3. The Picard lattice of a K3 surface
  4. The second cohomology group and the Hodge decomposition
  5. Reduction maps and upper bounds for the Picard number
  6. The ample cone and polarized K3 surfaces
  7. Moduli spaces of K3 surfaces
  8. Involutions on K3 surfaces
  9. Double covers of the plane
  10. Quartics with a node
  11. Smooth quartics with Picard number equal to 2
  12. A quartic with Picard number 2 and polarizations of degree 2 and 8
  13. Double cover.
  14. Intersection of three quadrics.
  15. An example of a K3 surface of degree 6 with Picard number 2
  16. ...and 1 more sections

Key Result

theorem 1

Let $X$ be a K3 surface of degree $2d$ and with Picard rank $\rho$, admitting a holomorphic and an anti-holomorphic involution which commute. If $d=1$, then $\rho \geq 1$ and there exist examples defined over $\IQ$ with $\rho=1$. If $d>1$, then $\rho \geq 2$ and there exist examples over $\IQ$ with

Figures (1)

  • Figure 1: A double cover of $\IP^2$ branched over the smooth sextic $Z(f)$ is an example of a K3 surface. It admits the holomorphic involution that swaps the sheets of the cover, coloured red and blue in the figure.

Theorems & Definitions (67)

  • theorem 1
  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • theorem 2
  • proof
  • lemma 1
  • ...and 57 more