Why do we always need the Schwarz lemma when bounding the trace of a Kähler metric?

My undergraduate thesis topic is Kähler geometry. The general direction is something like the Calabi-Yau theorem or more adventurously some singular Calabi-Yau theorem, but this is not certain yet. One thing that I am noticing a lot of in my reading of Kähler geometry is that if we have two Kähler metrics $\omega$, $\eta$, then to get a bound of the form $$\text{tr}_{\omega}(\eta) \leq C$$ we need to use the Schwarz lemma -- Essentially, we apply the maximum principle to some term like $$\log \text{tr}_{\omega}(\eta) - A \varphi,$$ where $\omega = \eta + dd^c \varphi$ and $A>0$ is large. This requires an assumption on the (Ricci/bisectional/holomorphic sectional) curvatures of $\omega$, $\eta$ (depending on which Laplacian one computes with).

I feel that I understand how to use the Schwarz lemma to get these estimates, but I want to ask why we have to use it (if we have to?).

This is prompted by studying singular metrics, for examples cone and cusp metrics: To formulate my question, let $D$ be a divisor in a compact Kähler manifold $M$, and for simplicity, assume that $D$ has simple normal crossings. A cone Kähler metric is a Kähler metric which is smooth on $M - D$ and is quasi-isometric to $$\frac{i}{2} \sum_{j=1}^k | z_j |^{2(1-\beta_j)} dz_j \wedge d\overline{z}_j + \frac{i}{2} \sum_{j \geq k+1} dz_j \wedge d\overline{z}_j.$$

A cusp Kähler metric is a smooth Kähler metric on $M-D$ which is quasi-isometric to $$\frac{i}{2} \sum_{j=1}^k | z_j |^{-2}| \log | z_i |^2 |^2 dz_j \wedge d\overline{z}_j + \frac{i}{2} \sum_{j \geq k+1} dz_j \wedge d\overline{z}_j.$$

From these descriptions, can one not see immediately that if $\omega$ is cusp and $\eta$ is cone, then $$\text{tr}_{\omega}(\eta) \leq C | z_i|^2 | \log | z_i |^2|^2,$$ which would give $$\text{tr}_{\omega}(\eta) \leq C \prod_j | \sigma_j |^2 | \log | \sigma_j |^2 |^2,$$ if $\sigma_j$ are the defining sections for the divisor $D$?

What initially came to my mind is a coordinate dependence problem, but this seems to contradict the fact that many calculations of this type involve normal coordinate calculations.

Sorry if this question is silly.

Solution 1:

There are various ways of achieving the estimate, as pointed out by Henri in his comment on MO. For the convenience of readers, let me summarize some of the approaches. For the moment, I'll restrict considerations to the case of two smooth Kähler metrics $\omega,\eta$ on a compact Kähler manifold.

Chern-Lu inequality: Suppose $\text{Ric}(\omega) \geq - C_1 \omega - C_2 \eta$ and $\text{HBC}(\eta) \leq C_3$, for constants $C_1, C_2, C_3 \in \mathbb{R}$. Then $$\Delta_{\omega}(\log \text{tr}_{\omega}(\eta)) \geq - C_1 - (C_2 + 2C_3) \text{tr}_{\omega}(\eta).$$ In particular, if $\omega = \eta + \sqrt{-1} \partial \overline{\partial} \varphi$, then $$\Delta_{\omega}(\log \text{tr}_{\omega}(\eta) - (C_2 + 2C_3 + 1) \varphi) \geq - C_1 - (C_2 + 2C_3 + 1)n + \text{tr}_{\omega}(\eta).$$

Aubin-Yau: Suppose $\text{Ric}(\eta) \leq C_1 \omega + C_2 \eta$ and that $\text{HBC}(\omega) \geq - C_3$, for constants $C_1, C_2, C_3 \in \mathbb{R}$. Then $$\Delta_{\eta}(\log \text{tr}_{\omega}(\eta)) \geq - \frac{n(C_1 + C_3)}{\text{tr}_{\omega}(\eta)} - C_2 - C_3 \text{tr}_{\eta}(\omega).$$

In particular, if $\omega = \eta + \sqrt{-1} \partial \overline{\partial} \varphi$, then $$\Delta_{\eta}(\log \text{tr}_{\omega}(\eta) - (2C_3 + C_1 +1)\varphi) \geq - (C_2 + n(2C_3 + C_1 + 1)) + \text{tr}_{\eta}(\omega).$$

Royden's refinement: The bounds on the holomorphic bisectional curvature (HBC) were refined to bounds on the holomorphic sectional curvature by Royden.

Siu-Yau inequality: Let $\omega_{\varphi} = \omega + \sqrt{-1} \partial \overline{\partial} \varphi$. Then $$\Delta_{\omega_{\varphi}} (\log \text{tr}_{\omega}(\omega_{\varphi})) \geq \frac{1}{\text{tr}_{\omega}(\omega_{\varphi})} \left( - \text{tr}_{\omega}(\text{Ric}(\omega_{\varphi}) + \text{tr}_{\omega_{\varphi}} \left( \text{tr}_{\omega_{\varphi}^{-1}} \left( \sqrt{-1} \widetilde{\Theta}_{\omega} \right) \right) \right),$$ where, in the last term, we take the trace $\text{tr}_{\omega_{\varphi}}$ of the covariant part of the curvature tensor, and then take the trace $\text{tr}_{\omega_{\varphi}^{-1}}$ of the covariant part.

There are also backwards Chern-Lu and backwards Aubin-Yau inequalities which are treated in section 7 of Yanir Rubinstein's amazing notes Smooth and Singular Kähler-Einstein metrics.