[Paper Review] An example of compact Kähler manifold with nonnegative quadratic bisectional curvature

Qun Li, Damin Wu+1 more|arXiv (Cornell University)|Oct 8, 2011

Geometry and complex manifolds9 references3 citations

TL;DR

This paper constructs a 7-dimensional compact Kähler C-space with second Betti number 1, demonstrating that its canonical Kähler-Einstein metric has nonnegative quadratic bisectional curvature (QB ≥ 0), yet does not admit any Kähler metric with nonnegative orthogonal bisectional curvature (B⊥ ≥ 0). This provides the first explicit example of a compact Kähler manifold satisfying QB ≥ 0 but not B⊥ ≥ 0, highlighting a strict algebraic and geometric distinction between these curvature conditions in higher dimensions.

ABSTRACT

We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature, which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of positive scalar curvature

Motivation & Objective

• To construct a compact Kähler manifold with nonnegative quadratic bisectional curvature (QB ≥ 0) that does not admit any Kähler metric with nonnegative orthogonal bisectional curvature (B⊥ ≥ 0).
• To provide a concrete counterexample showing that QB ≥ 0 is strictly weaker than B⊥ ≥ 0 in dimension n ≥ 3, despite the algebraic hierarchy.
• To verify the conjecture that certain Kähler C-spaces with b₂ = 1 and non-Hermitian symmetric structure may still satisfy QB ≥ 0.
• To demonstrate via explicit computation that the 7-dimensional Kähler C-space (B₃, α₂) has QB ≥ 0, confirming its role as a test case for curvature conditions in generalized Frankel-Hartshorne theory.
• To explore the differential geometric implications of QB ≥ 0 for the classification of Fano manifolds and Kähler C-spaces with numerically effective tangent bundles.

Proposed method

• The authors analyze the curvature tensor of the 7-dimensional Kähler C-space (B₃, α₂), a homogeneous Kähler manifold with second Betti number 1 and a unique Kähler-Einstein metric up to scaling.
• They compute the quadratic bisectional curvature using the curvature components expressed in terms of matrices A, B, C derived from the curvature tensor components P_{1iar{1j}}, P_{1iar{2j}}, P_{2iar{2j}}.
• The key quantity Φ, representing the quadratic form associated with QB ≥ 0, is decomposed into Φ₁ and Φ₂, with Φ₁ further split into Φ₁′ and Φ₁′′ based on matrix entries.
• Nonnegativity of Φ₂ is established using the inequality ⟨B, B*⟩ ≥ −|B|² and |tr B|² ≤ 3|B|², yielding Φ₂ ≥ (8 − 1 − 6)|B|² = |B|² ≥ 0.
• For Φ₁′′, the authors express Φ₁′′ as a quadratic form (a,c)D^t(a,c), where D is a 6×6 symmetric matrix composed of G and H matrices involving identity, all-ones, and permutation matrices.
• A change of basis via orthogonal matrix T diagonalizes the relevant matrices, allowing verification that G − HG⁻¹H ≥ 0, thus proving Φ₁′′ ≥ 0 and hence Φ ≥ 0, confirming QB ≥ 0.

Experimental results

Research questions

• RQ1 Does there exist a compact Kähler manifold of dimension n ≥ 3 with QB ≥ 0 everywhere but no Kähler metric with B⊥ ≥ 0 everywhere?
• RQ2 Can a Kähler C-space with b₂ = 1 and non-Hermitian symmetric structure still admit a Kähler metric with QB ≥ 0?
• RQ3 Is the condition QB ≥ 0 strictly weaker than B⊥ ≥ 0 in the sense of curvature positivity, both algebraically and geometrically?
• RQ4 Can the conjecture that QB ≥ 0 implies biholomorphy to a Kähler C-space with b₂ = 1 be verified for specific examples?
• RQ5 What is the precise algebraic structure of the quadratic bisectional curvature form on non-symmetric Kähler C-spaces?

Key findings

• The 7-dimensional Kähler C-space (B₃, α₂) admits a canonical Kähler-Einstein metric with nonnegative quadratic bisectional curvature (QB ≥ 0).
• This manifold does not admit any Kähler metric with nonnegative orthogonal bisectional curvature (B⊥ ≥ 0), despite QB ≥ 0.
• The curvature computation confirms that the quadratic form Φ associated with QB ≥ 0 is nonnegative, with Φ₂ ≥ |B|² ≥ 0 and Φ₁′′ ≥ 0 via spectral analysis of the matrix D.
• The matrix G − HG⁻¹H is shown to be nonnegative definite after orthogonal transformation, confirming the nonnegativity of the critical quadratic form Φ₁′′.
• The entire curvature form Φ is decomposed and shown to be nonnegative through component-wise analysis and matrix diagonalization, establishing QB ≥ 0 for the manifold.
• This example confirms that QB ≥ 0 is a strictly weaker condition than B⊥ ≥ 0 in higher dimensions, providing a counterexample to the idea that QB ≥ 0 implies B⊥ ≥ 0.