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Weyl group of type $E_6$ and K3 surfaces

Cédric Bonnafé

TL;DR

This work constructs singular K3 surfaces by intersecting fundamental invariants of the Weyl group $W$ of type ${\mathrm E}_6$ and then studying quotients by its derived subgroup $W'$. The minimal resolution of ${\mathcal X}/W'$ is shown to be a K3 surface; for generic choices yielding ADE singularities, ${\mathcal Y}_{\lambda,\mu}/W'$ is likewise a K3 surface. The paper then analyzes a particularly rich member ${\tilde{\mathcal X}}$ equipped with an elliptic fibration whose singular fibers are of types $E_7+E_6+A_2+2A_1$, and computes the Picard lattice (rank 20, discriminant $-228$) and the transcendental lattice (explicitly given by $200114$). A complete cohomological decomposition of $H^2({\mathcal X},{\mathbb C})$ under $W$ is provided, supported by extensive Magma computations. The results yield a concrete, highly explicit singular K3 surface arising from the interplay between invariant theory, Springer theory, and the geometry of K3 surfaces, with detailed descriptions of elliptic fibrations and lattice structures that are relevant for moduli and arithmetic questions in the field.

Abstract

Adapting methods of previous papers by A. Sarti and the author, we construct K3 surfaces from invariants of the Weyl group of type $\Erm_6$. We study in details one of these surfaces, which turns out to have Picard number $20$: for this example, we describe an elliptic fibration (and its singular fibers), the Picard lattice and the transcendental lattice.

Weyl group of type $E_6$ and K3 surfaces

TL;DR

This work constructs singular K3 surfaces by intersecting fundamental invariants of the Weyl group of type and then studying quotients by its derived subgroup . The minimal resolution of is shown to be a K3 surface; for generic choices yielding ADE singularities, is likewise a K3 surface. The paper then analyzes a particularly rich member equipped with an elliptic fibration whose singular fibers are of types , and computes the Picard lattice (rank 20, discriminant ) and the transcendental lattice (explicitly given by ). A complete cohomological decomposition of under is provided, supported by extensive Magma computations. The results yield a concrete, highly explicit singular K3 surface arising from the interplay between invariant theory, Springer theory, and the geometry of K3 surfaces, with detailed descriptions of elliptic fibrations and lattice structures that are relevant for moduli and arithmetic questions in the field.

Abstract

Adapting methods of previous papers by A. Sarti and the author, we construct K3 surfaces from invariants of the Weyl group of type . We study in details one of these surfaces, which turns out to have Picard number : for this example, we describe an elliptic fibration (and its singular fibers), the Picard lattice and the transcendental lattice.

Paper Structure

This paper contains 39 sections, 30 theorems, 128 equations.

Table of Contents

  1. General notation, preliminaries
  2. Set-up
  3. First properties of ${\boldsymbol{W}}$
  4. Eigenspaces, Springer theory
  5. K3 surfaces
  6. Symplectic form
  7. Singularities
  8. Euler characteristic
  9. Some numerical data for the surface ${\boldsymbol{{\mathcal{X}}/W'}}$
  10. Singularities
  11. Two smooth rational curves
  12. Singular points, equation
  13. Affine charts
  14. The affine chart ${\mathcal{U}}_3$
  15. The affine chart ${\mathcal{U}}_4$
  16. ...and 24 more sections

Key Result

Lemma 1.1

Let $V$ be a finite dimensional vector space, let $f \in {\mathbb{C}}[V]$, let $g \in {\mathrm{G}}{\mathrm{L}}_{\mathbb{C}}(V)$ and let $v \in V \setminus \{0\}$ be such that: Then $f(v)=0$.

Theorems & Definitions (60)

  • Lemma 1.1
  • Lemma 1.2
  • Remark 1.3
  • proof
  • Corollary 1.4
  • Remark 1.5
  • Remark 2.6
  • Theorem 2.12: Springer, Lehrer-Springer
  • Example 2.13
  • Example 2.14
  • ...and 50 more