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Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

Shouhei Ma, Ken Sato

Abstract

In this paper we initiate the study of higher Chow cycles on holomorphic symplectic manifolds. Our concrete central result is construction of explicit indecomposable (2,1)- and (4,1)-cycles on the Fano varieties of lines on cyclic cubic fourfolds. This is the first explicit example of such cycles on holomorphic symplectic manifolds. The proof of indecomposability is done by degeneration to cuspidal cubic fourfolds. Along the way, we develop a method of inducing (p,1)-cycles on Hilbert squares of K3 surfaces. Finally, we study restriction of (2,1)-cycles to Lagrangian subvarieties, and observe the phenomenon that the restricted cycles are always decomposable in the examples in our hand.

Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

Abstract

In this paper we initiate the study of higher Chow cycles on holomorphic symplectic manifolds. Our concrete central result is construction of explicit indecomposable (2,1)- and (4,1)-cycles on the Fano varieties of lines on cyclic cubic fourfolds. This is the first explicit example of such cycles on holomorphic symplectic manifolds. The proof of indecomposability is done by degeneration to cuspidal cubic fourfolds. Along the way, we develop a method of inducing (p,1)-cycles on Hilbert squares of K3 surfaces. Finally, we study restriction of (2,1)-cycles to Lagrangian subvarieties, and observe the phenomenon that the restricted cycles are always decomposable in the examples in our hand.

Paper Structure

This paper contains 23 sections, 21 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Holomorphic symplectic manifolds
  4. Higher Chow cycles
  5. Regulators
  6. Non-symplectic automorphisms
  7. Induced cycles from $K3$ surfaces
  8. $(2, 1)$-cycles
  9. $(4, 1)$-cycles
  10. $(3, 1)$-cycles
  11. Eisenstein $K3$ surfaces and cuspidal cyclic cubic fourfolds
  12. Eisenstein $K3$ surfaces
  13. Cuspidal cyclic cubic fourfolds
  14. $(2, 1)$-cycles
  15. Deformation to smooth cubic fourfolds
  16. ...and 8 more sections

Key Result

Theorem 1.1

When $(F, t)$ is very general, the indecomposable parts of the $(2, 1)$-cycle $\xi_2$ and the $(4, 1)$-cycle $\xi_{4}$ on $F(Y_{F,t})$ are non-torsion. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1:

Theorems & Definitions (46)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 36 more