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Failure of the invariant cycle theorem over $\mathbb Z$

Donu Arapura, François Greer, Yilong Zhang

Abstract

We construct a counterexample to both the local and global invariant cycle theorems with integral coefficients. The example is a semistable one-parameter family of elliptic surfaces with $p_g=q=1$ and constant period map. The smooth fibers have the smallest possible discriminant, and are associated with Vinberg's most algebraic K3 surface. Our construction generalizes the Shioda-Inose construction for rational double covers of K3 surfaces.

Failure of the invariant cycle theorem over $\mathbb Z$

Abstract

We construct a counterexample to both the local and global invariant cycle theorems with integral coefficients. The example is a semistable one-parameter family of elliptic surfaces with and constant period map. The smooth fibers have the smallest possible discriminant, and are associated with Vinberg's most algebraic K3 surface. Our construction generalizes the Shioda-Inose construction for rational double covers of K3 surfaces.
Paper Structure (29 sections, 45 theorems, 84 equations, 8 figures, 1 table)

This paper contains 29 sections, 45 theorems, 84 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Failure of local ICT$_{\mathbb{Z}}$
  3. Failure of global ICT$_{\mathbb{Z}}$
  4. The construction
  5. Preliminaries
  6. Transcendental Hodge structures
  7. Elliptic surfaces
  8. Shioda--Inose structure
  9. Quadratic base change and variations of Hodge structure
  10. Euler defect of base change
  11. Defective double cover of elliptic K3
  12. Birational geometry of defective double cover
  13. Hodge theory of defective double cover
  14. Examples of families with constant VHS
  15. Surface degenerations
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $f:\mathcal{X}\to \Delta$ be a flat family of complex projective varieties over a disk. Suppose $\mathcal{X}$ is smooth and $f$ is smooth over the punctured disk $\Delta^*$. Let $X_t$ be a general smooth fiber, then for each $n$, the restriction map surjects onto the invariant part.

Figures (8)

  • Figure 1: Rational elliptic surface with its two projections
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 3 more figures

Theorems & Definitions (89)

  • Theorem 1.1: LICT$_\mathbb{Q}$
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5: GICT$_{\mathbb{Q}}$ Deligne-HodgeII
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 79 more