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Sequence of families of lattice polarized $K3$ surfaces, modular forms and degrees of complex reflection groups

Atsuhira Nagano

Abstract

We introduce a sequence of families of lattice polarized $K3$ surfaces. This sequence is closely related to complex reflection groups of exceptional type. Namely, we obtain modular forms coming from the inverse correspondences of the period mappings attached to our sequence. We study a non-trivial relation between our modular forms and invariants of complex reflection groups. Especially, we consider a family concerned with the Shepherd-Todd group of No.34 based on arithmetic properties of lattices and algebro-geometric properties of the period mappings.

Sequence of families of lattice polarized $K3$ surfaces, modular forms and degrees of complex reflection groups

Abstract

We introduce a sequence of families of lattice polarized surfaces. This sequence is closely related to complex reflection groups of exceptional type. Namely, we obtain modular forms coming from the inverse correspondences of the period mappings attached to our sequence. We study a non-trivial relation between our modular forms and invariants of complex reflection groups. Especially, we consider a family concerned with the Shepherd-Todd group of No.34 based on arithmetic properties of lattices and algebro-geometric properties of the period mappings.

Paper Structure

This paper contains 15 sections, 23 theorems, 91 equations, 2 figures, 4 tables.

Table of Contents

  1. Arithmetic arrangement of hyperplanes and Looijenga compactification
  2. Lattice ${\bf A}$ and arrangement $\mathcal{H}$
  3. Family $\mathfrak{F}_0$ of $K3$ surfaces with Picard number $15$
  4. Singular fibres
  5. Local period mapping
  6. Double covering $K_a$ of $S_a$
  7. Moduli space of ${\bf M}$-polarized $K3$ surfaces
  8. Sequence of families of $K3$ surfaces and complex reflection groups
  9. Transcendental lattices for subfamilies of $\mathfrak{F}_0$
  10. Transcendental lattices for subfamilies of $\mathfrak{G}_0$ of Kummer-like surfaces
  11. Relation between modular forms and invariants of complex reflection groups via theta functions
  12. Meromorphic modular forms
  13. Meromorphic modular forms of character ${\rm id}$
  14. Meromorphic modular forms of character ${\rm det}$
  15. Transcendental lattice for family $\mathfrak{G}_0$ of Kummer-like surfaces with Picard number $15$

Key Result

Theorem 1.1

(L Corollary 7.5) Suppose that every $\pi_\sigma (\mathscr{D})$ in (DSigmaH) is not $(n-1)$-dimensional. Then, the algebra where $\mathscr{L}$ is the natural automorphic bundle over $\mathscr{D}$, is finitely generated with positive degree generators. Its Proj gives the Looijenga compactification $\widehat{X^\circ}^{\bf L}$ of (LCompact). The boundary $\widehat{X^\circ}^{\bf L}-X^\circ$ is the st

Figures (2)

  • Figure 1: Singular fibres for $\pi_a: S_a\rightarrow \mathbb{P}^1(\mathbb{C})$
  • Figure 2: Basis of $E_8(-1)^{\oplus 2}$

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Remark 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Remark 2.1
  • ...and 31 more