[Paper Review] Reconstruction of general elliptic K3 surfaces from their Gromov-Hausdorff limits

Kenji Hashimoto, Kazushi Ueda|arXiv (Cornell University)|May 4, 2018

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper proves that a general elliptic K3 surface with a section is uniquely determined by its discriminant—a configuration of 24 points on the projective line—by showing that the decomposition of the discriminant polynomial into a cube and a square is unique up to roots of unity. This uniqueness implies that the surface can be reconstructed from its Gromov–Hausdorff limit as the fiber volume tends to zero, establishing a link between metric limits and algebraic structure in mirror symmetry.

ABSTRACT

We show that a general elliptic K3 surface with a section is determined uniquely by its discriminant, which is a configuration of 24 points on the projective line. It follows that a general elliptic K3 surface with a section can be reconstructed from its Gromov-Hausdorff limit as the volume of the fiber goes to zero.

Motivation & Objective

• To establish the uniqueness of the Weierstrass model decomposition for general elliptic K3 surfaces with a section.
• To show that the discriminant configuration of 24 points on P¹ determines the surface uniquely up to symmetry.
• To connect the Gromov–Hausdorff limit of Kähler–Einstein metrics on elliptic K3 surfaces to the algebraic structure of the surface.
• To provide a geometric reconstruction of the Jacobian of a general elliptic K3 surface from its metric limit.

Proposed method

• Construct an auxiliary elliptic surface X in weighted projective space P(8,12,1,1) from the discriminant polynomial h = f³ + g².
• Use the Picard group structure and lattice theory to analyze the Néron–Tate height pairing and Mordell–Weil group of X.
• Prove that for very general f and g, the Picard number of X is exactly 4, implying that all sections lie in a 4-dimensional lattice.
• Show that the only rational vectors in the Mordell–Weil lattice with self-intersection −8 are the six sections associated with cube and square roots of unity.
• Use scheme-theoretic arguments to extend uniqueness from the very general case to the general case via Zariski openness.
• Relate the Gromov–Hausdorff limit to the conformal structure of the base P¹ minus the discriminant, recovering the complex structure up to conjugation.

Experimental results

Research questions

• RQ1 Can a general elliptic K3 surface with a section be uniquely reconstructed from its Gromov–Hausdorff limit as the fiber volume vanishes?
• RQ2 Is the decomposition of the discriminant polynomial h = f³ + g² into a cube and a square unique for general f and g?
• RQ3 To what extent does the Gromov–Hausdorff limit of Kähler–Einstein metrics on an elliptic K3 surface encode the complex structure of the surface?
• RQ4 How does the Mordell–Weil group of the elliptic surface relate to the Picard lattice and the uniqueness of the Weierstrass model?

Key findings

• The decomposition of h = f³ + g² into a cube and a square is unique for general f and g, up to multiplication by a cube root of unity.
• For very general f and g, the Picard number of the associated elliptic surface X is exactly 4, which forces all sections to lie in a 4-dimensional lattice.
• The Mordell–Weil group of X is isomorphic to the quotient Pic(X)/U, and only six sections have self-intersection −8 in the Mordell–Weil lattice.
• The first projection from the scheme Z of solutions to the equation f³ + g² = ϕ³ + ψ² is generically six-to-one, proving uniqueness outside a Zariski closed set.
• The complex structure of the base P¹ minus the discriminant is reconstructed from the Gromov–Hausdorff limit metric, up to complex conjugation.
• The reconstruction of the Jacobian of a general elliptic K3 surface from its Gromov–Hausdorff limit is possible, establishing a link between metric limits and mirror symmetry.