[Paper Review] Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle

Ryosuke Nomura|arXiv (Cornell University)|Feb 15, 2018

Geometry and complex manifolds4 references3 citations

TL;DR

This paper proves the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle by establishing an $L^2$-norm estimate of the scalar curvature along the Kähler-Ricci flow. The key innovation lies in controlling the $L^2$-norm of scalar curvature via a Dirichlet energy functional $E(t)$, which enables the extension of the inequality beyond projective varieties to the broader Kähler setting, confirming the inequality under semi-positivity without requiring bigness.

ABSTRACT

In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.

Motivation & Objective

• To extend the Miyaoka-Yau inequality to compact Kähler manifolds with semi-positive canonical bundle, beyond the previously known projective or big canonical bundle settings.
• To establish a uniform $L^2$-bound on the scalar curvature along the Kähler-Ricci flow under semi-positivity and non-bigness of the canonical bundle.
• To overcome the lack of uniform scalar curvature bounds in non-big cases by introducing a novel energy functional $E(t)$ related to the Dirichlet norm of the log-volume ratio.
• To prove that the Kähler-Ricci flow satisfies the necessary cohomological and curvature decay conditions to imply the Miyaoka-Yau inequality.

Proposed method

• Define the functional $E(t) = \int_X \sqrt{-1} \partial f_t \wedge \overline{\partial}f_t \wedge \omega_t^{n-1}$, where $f_t = \log(\omega_t^n / \Omega)$, to track the $L^2$-norm of the gradient of the log-volume ratio.
• Derive the time derivative $\frac{d}{dt}E(t)$ and relate it to the $L^2$-norm of the scalar curvature $R(\omega_t)$, using curvature decomposition and Ricci curvature estimates.
• Use the assumption that $K_X$ is semi-positive and not big to control the decay rate of the Ricci curvature term via the numerical dimension $\nu < n$, leading to exponential decay $e^{-(n-\nu)t}$.
• Apply the maximum principle to obtain a uniform lower bound $R(\omega_t) \geq -C$, ensuring integrability of the scalar curvature term.
• Establish that $\int_0^\infty \int_X R(\omega_t)^2 \omega_t^n \, dt < \infty$ via integration of the differential inequality $\frac{d}{dt}E(t) \leq -\frac{2}{n}\int_X R^2 \omega_t^n + C'e^{-(n-\nu)t}$.
• Leverage the long-time existence of the Kähler-Ricci flow and cohomological convergence $[\omega_t] \to 2\pi c_1(K_X)$ to verify the required limit conditions for the Miyaoka-Yau inequality.

Experimental results

Research questions

• RQ1 Does the Miyaoka-Yau inequality hold for compact Kähler manifolds with semi-positive but not big canonical bundle?
• RQ2 Can the $L^2$-norm of the scalar curvature be uniformly bounded along the Kähler-Ricci flow in the absence of bigness of $K_X$?
• RQ3 Is the Dirichlet energy $E(t)$ of the log-volume ratio sufficient to control the scalar curvature $L^2$-norm via its time derivative?
• RQ4 Can the cohomological limit $[\omega_t] \to 2\pi c_1(K_X)$ and the decay of $|\mathrm{Ric}(\omega_t) + \omega_t|_{\omega_t}^2$ be achieved under semi-positivity alone?
• RQ5 Does the absence of bigness in $K_X$ prevent the application of existing scalar curvature estimates, and if so, can a new functional $E(t)$ compensate?

Key findings

• The Miyaoka-Yau inequality holds for all compact Kähler manifolds with semi-positive canonical bundle, extending previous results that required bigness or projectivity.
• The $L^2$-norm of the scalar curvature satisfies $\int_0^\infty \int_X R(\omega_t)^2 \omega_t^n \, dt < \infty$ under the assumption that $K_X$ is semi-positive and not big.
• The functional $E(t)$, defined as the Dirichlet energy of $f_t = \log(\omega_t^n / \Omega)$, provides a crucial control mechanism for the scalar curvature via its time derivative.
• The decay rate of the Ricci curvature term $|\mathrm{Ric}(\omega_t) + \omega_t|_{\omega_t}^2$ is controlled by $e^{-(n-\nu)t}$, where $\nu$ is the numerical dimension of $K_X$, and $\nu < n$ due to non-bigness.
• The proof establishes that the Kähler-Ricci flow satisfies the two required limit conditions: $[\omega_t] \to 2\pi c_1(K_X)$ and $\int_0^\infty \int_X |\mathrm{Ric}(\omega_t) + \omega_t|_{\omega_t}^2 \omega_t^n \, dt < \infty$, which imply the Miyaoka-Yau inequality.
• The result confirms that the Miyaoka-Yau inequality is valid in the broader Kähler category under semi-positivity, even without assuming projectivity or bigness.