[Paper Review] A remark on compact Kähler manifolds with nef anticanonical bundles and its applications

Junyan Cao|arXiv (Cornell University)|May 19, 2013

Geometry and complex manifolds27 references9 citations

TL;DR

This paper establishes that the Harder-Narasimhan filtration slopes of the tangent bundle on compact Kähler manifolds with nef anticanonical bundles are semi-positive when polarized by $(\omega_X)^{n-1}$. Using this result, it provides a new analytic characterization of rationally connected manifolds and gives a purely analytic proof of the surjectivity and codimension-2 smoothness of the Albanese map, extending prior algebraic results to the Kähler setting.

ABSTRACT

Let $(X, ω_X)$ be a compact Kähler manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the slopes of the Harder-Narasimhan filtration of the tangent bundle with respect to a polarization of the form $ω_X^{n-1}$ are semi-positive. As an application, we give a characterization of rationally connected compact Kähler manifolds with nef anticanonical bundles. As another application, we give a simple proof of the surjectivity of the Albanese map.

Motivation & Objective

• To prove that the slopes of the Harder-Narasimhan filtration of the tangent bundle are semi-positive for compact Kähler manifolds with nef anticanonical bundles under polarization by $(\omega_X)^{n-1}$.
• To provide a new analytic characterization of rationally connected compact Kähler manifolds with nef anticanonical bundles via vanishing of global sections of tensor powers of the cotangent bundle.
• To give a purely analytic proof of the surjectivity and codimension-2 smoothness of the Albanese map for such manifolds, extending results previously known only in the projective case.
• To establish a new result on the non-negativity of the integral of $c_2(T_X)$ against perturbed nef classes, with a structure theorem for equality cases.

Proposed method

• Proves semi-positivity of Harder-Narasimhan slopes via a direct analytic argument using a family of singular metrics and the Harder-Narasimhan filtration with respect to $\omega_X^{n-1}$.
• Applies a Bochner technique on the pullback of the tangent bundle via a resolution $\pi: \widetilde{X} \to X$, using Hermitian-Einstein metrics on stable quotients of the filtration.
• Uses the uniform lower bound on slopes of the filtration quotients to construct a smooth metric $h_\epsilon$ on $\pi^*(T_X)^{\otimes m}$ such that $i\Theta_{h_\epsilon}(\pi^*T_X^{\otimes m}) \wedge (\pi^*\omega + \epsilon \omega_{\widetilde{X}})^{n-1} \geq \frac{m \cdot c}{4} \text{Id}$.
• Employs a contradiction argument based on the exactness of certain currents $T_\epsilon$ and their uniform $L^1$-boundedness to show that certain sections must vanish.
• Utilizes Höring’s idea to relate the non-negativity of $\int_X c_2(T_X) \wedge (c_1(-K_X) + \epsilon \omega_X)^{n-2}$ to the semi-positivity of the Harder-Narasimhan filtration.
• Applies the criterion from [CDP12] to reduce the vanishing of global sections to a slope condition on the Harder-Narasimhan filtration.

Experimental results

Research questions

• RQ1 Are the slopes of the Harder-Narasimhan filtration of the tangent bundle semi-positive for compact Kähler manifolds with nef anticanonical bundles under polarization by $(\omega_X)^{n-1}$?
• RQ2 Can rationally connected compact Kähler manifolds with nef anticanonical bundles be characterized by the vanishing of $H^0(X, (T_X^*)^{\otimes m})$ for all $m \geq 1$?
• RQ3 Is the Albanese map surjective and smooth outside a codimension-2 subvariety for compact Kähler manifolds with nef anticanonical bundles?
• RQ4 Does the second Chern class $c_2(T_X)$ satisfy $\int_X c_2(T_X) \wedge (c_1(-K_X) + \epsilon \omega_X)^{n-2} \geq 0$ for small $\epsilon > 0$?

Key findings

• The slopes of the Harder-Narasimhan filtration of $T_X$ with respect to $\omega_X^{n-1}$ are semi-positive for any compact Kähler manifold with $-K_X$ nef.
• The four conditions (i)–(iv) in Proposition 1.3 are equivalent: vanishing of $H^0(X, (T_X^*)^{\otimes m})$ for all $m \geq 1$, rational connectedness, and strict positivity of the tangent bundle with respect to $\omega_X^{n-1}$.
• The Albanese map is surjective and smooth outside a subvariety of codimension at least 2 for any compact Kähler manifold with $-K_X$ nef.
• For $\epsilon > 0$ small enough, $\int_X c_2(T_X) \wedge (c_1(-K_X) + \epsilon \omega_X)^{n-2} \geq 0$, and equality implies that $X$ is a finite étale cover of a torus or a $\mathbb{P}^1$-fibration over a torus.
• The proof of the vanishing of $H^0(X, (T_X^*)^{\otimes m} \otimes F^{\otimes k})$ for large $m$ relative to $k$ is purely analytic, avoiding algebraic methods used in prior work.
• The result extends the known projective case to the general Kähler setting, including non-algebraic manifolds.