[Paper Review] The transcendental part of the regulator map for K_1 on a mirror family of K3 surfaces

Pedro Luis del Angel, Stefan Müller–Stach|arXiv (Cornell University)|Aug 28, 2000

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper computes the transcendental part of the regulator map for $K_1$ classes on a mirror family of quartic $K3$ surfaces, showing that the associated normal functions do not satisfy the standard Picard-Fuchs hypergeometric equation. This implies the cycles are indecomposable for generic points in the family, and the inhomogeneous Picard-Fuchs equations are linked to Painlevé VI-type differential equations.

ABSTRACT

We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K_1 of a mirror family of quartic K3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equation are related to Painlev\\'e VI type differential equations.

Motivation & Objective

• To verify H. Esnault's conjecture that certain $K_1$ classes on $K3$ surfaces are detected in the transcendental part of Deligne cohomology via the regulator map.
• To analyze the behavior of normal functions arising from these $K_1$ classes in a one-parameter family of mirror quartic $K3$ surfaces.
• To determine whether these normal functions satisfy the standard Picard-Fuchs hypergeometric differential equation or a more general type.
• To identify the differential equations governing the normal functions and relate them to Painlevé VI-type equations.
• To establish that the cycles are indecomposable for generic points in the family by showing the normal functions are not solutions to the classical Picard-Fuchs equation.

Proposed method

• Constructs a family of mirror quartic $K3$ surfaces parametrized by a base $B$, using a one-parameter deformation of the Fermat quartic.
• Defines a cycle in $CH^2(X_b,1)$ via a rational map to $\mathbb{P}^1$, inducing a normal function $\nu(b)$ in the transcendental part of Deligne cohomology.
• Computes the regulator map $c_{2,1}$ as an Abel-Jacobi integral of holomorphic two-forms over real 2-chains, yielding a multivalued holomorphic function on $B$.
• Derives the inhomogeneous Picard-Fuchs differential equation satisfied by the normal function using the Gauss-Manin connection and monodromy analysis.
• Analyzes the coefficients of the differential operator $\mathcal{D}_{\rm PF}$, showing they are rational functions with poles along degeneracy loci.
• Demonstrates that the resulting inhomogeneous equation is not hypergeometric but instead related to Painlevé VI-type equations via explicit parameterization in terms of $\nu$.

Experimental results

Research questions

• RQ1 Do the normal functions arising from $K_1$ classes on a mirror family of $K3$ surfaces satisfy the standard Picard-Fuchs hypergeometric differential equation?
• RQ2 What is the structure of the differential equation governing the transcendental part of the regulator map for $K_1$ on $K3$ surfaces?
• RQ3 Can the normal functions associated with $K_1$ classes be shown to be non-decomposable via their differential equations?
• RQ4 How are the inhomogeneous Picard-Fuchs equations related to known special functions such as Painlevé transcendents?
• RQ5 What is the monodromy and analytic behavior of the normal function near singular fibers in the family?

Key findings

• The normal function associated with the $K_1$ cycle does not satisfy the standard hypergeometric Picard-Fuchs equation, indicating non-trivial transcendental behavior.
• The inhomogeneous Picard-Fuchs equation governing the normal function is shown to be equivalent to a generalized Painlevé VI-type differential equation.
• The coefficients of the differential operator $\mathcal{D}_{\rm PF}$ are rational functions in the parameter $\nu$, with poles only along degeneracies of the family.
• The normal function is a single-valued meromorphic function on the base $B$, extending holomorphically to the compactified base except at singular fibers.
• The failure of the normal function to satisfy the classical Picard-Fuchs equation implies that the corresponding $K_1$ class is indecomposable for generic points in the family.
• The structure of the differential equation confirms that the cycle lies in the transcendental part of Deligne cohomology and is not in the image of the standard cycle class map from $CH^1(X) \otimes \mathbb{C}^\times$.