[Paper Review] Rational curves on Calabi-Yau threefolds and a conjecture of Oguiso

Simone Diverio|arXiv (Cornell University)|Nov 26, 2013

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper investigates the existence of rational curves on Calabi-Yau threefolds, proposing that such manifolds are non-hyperbolic and contain rational curves under mild Picard number conditions. It combines differential-geometric curvature conditions—particularly non-degenerate negative $k$-jet curvature—with algebro-geometric techniques, proving that a Calabi-Yau threefold with $\rho(X) > 4$ and a non-zero nef non-ample line bundle must contain a rational curve, supporting Oguiso’s conjecture.

ABSTRACT

This short note is an extended abstract of a talk given at the conference "Komplexe Analysis" at the Mathematisches Forschungsinstitut Oberwolfach in September 2012. We explained some recent results about the existence of rational curves on Calabi-Yau threefolds as well as a curvature approach to the non hyperbolicity of such manifolds.

Motivation & Objective

• To address Oguiso’s conjecture on the non-hyperbolicity of Calabi-Yau threefolds by proving the existence of rational curves.
• To connect differential-geometric curvature conditions—specifically non-degenerate negative $k$-jet curvature—with the ampleness of the canonical bundle and the absence of rational curves.
• To establish that Calabi-Yau threefolds with $\rho(X) > 4$ and a non-zero nef non-ample line bundle contain rational curves, advancing the folklore conjecture in algebraic geometry.
• To explore the implications of the Kawamata-Morrison Cone Conjecture for the Kobayashi conjecture in dimension three.
• To bridge differential geometry and algebraic geometry by linking curvature positivity/negativity to birational invariants like Kodaira dimension and nef cones.

Proposed method

• Analyzes the holomorphic sectional curvature and its implications for the Ricci curvature and first Chern class, using averaging and trace arguments to derive contradictions under $c_1(X)_{\mathbb{R}} = 0$.
• Introduces the concept of non-degenerate negative $k$-jet curvature via the compactified jet bundle $X_k$, using singular hermitian metrics on $\mathcal{O}_{X_k}(-1)$ with negative definite curvature along the subbundle $V_k$.
• Applies the Cone Theorem and the log Minimal Model Program (log-MMP) to show that the existence of a non-zero effective non-nef line bundle implies the existence of a rational curve.
• Employs the Kawamata-Morrison Cone Conjecture to deduce that the nef cone is rational polyhedral under automorphism group action, enabling density arguments on rational points in the nef boundary.
• Uses the Euler characteristic computation and Kawamata–Viehweg vanishing to prove effectiveness of $\mathbb{Q}$-divisors $D$ with $c_2(X) \cdot D > 0$, leading to rational curves.
• Reduces the Kobayashi conjecture in dimension three to the non-hyperbolicity of Calabi-Yau threefolds by leveraging the Beauville-Bogomolov decomposition and the absence of rational curves in hyperbolic manifolds.

Experimental results

Research questions

• RQ1 Does a Calabi-Yau threefold with $\rho(X) > 4$ and a non-zero nef non-ample line bundle necessarily contain a rational curve?
• RQ2 Is non-degenerate negative $k$-jet curvature sufficient to imply $c_1(X)_{\mathbb{R}} \neq 0$?
• RQ3 Does the Kawamata-Morrison Cone Conjecture imply the Kobayashi conjecture in dimension three, except possibly for Picard number one Calabi-Yau threefolds?
• RQ4 Can the existence of a non-zero effective non-ample line bundle on a Calabi-Yau threefold be used to guarantee a rational curve via the Cone Theorem?
• RQ5 Is the conjecture that measure hyperbolicity implies general type in threefolds equivalent to the Kobayashi conjecture?

Key findings

• A Calabi-Yau threefold with $\rho(X) > 4$ and a non-zero nef non-ample line bundle contains a rational curve, providing a positive answer to Oguiso’s conjecture under this condition.
• The existence of a rational curve on a Calabi-Yau threefold is guaranteed if there exists a non-zero effective non-ample line bundle, as shown via the Cone Theorem and the log-MMP.
• If the Kawamata-Morrison Cone Conjecture holds, then the Kobayashi conjecture is true in dimension three, except possibly for hyperbolic Calabi-Yau threefolds of Picard number one.
• Non-degenerate negative $k$-jet curvature implies hyperbolicity, and the paper poses the open question of whether it also implies $c_1(X)_{\mathbb{R}} \neq 0$, which would support the ampleness of the canonical bundle.
• The paper shows that negative holomorphic sectional curvature implies negative scalar curvature, and if $c_1(X)_{\mathbb{R}} = 0$, a contradiction arises via Ricci form and $\omega$-Laplacian of a smooth function.
• The absence of rational curves in a Calabi-Yau threefold implies that its automorphism group is finite, and under the Kawamata-Morrison conjecture, the nef cone becomes rational polyhedral, enabling the construction of effective $\mathbb{Q}$-divisors with positive intersection against $c_2(X)$.