I am reading through a proof in W. Ding and G. Tian's 1992 paper on the generalised Futaki invariant. To provide context, we are looking for obstructions to the existence of Kähler--Einstein metrics with positive scalar curvature. The Futaki invariant provides an example of such an obstruction. My confusion is not in the Kähler geometry, but in some standard Riemannian Geometry/Geometric Analysis.

Suppose that $X$ is a compact $\mathbb{Q}$-Fano variety, i.e., there is an ample line bundle $L \longrightarrow X$ which restricts to the pluri-anticanonical line bundle $K_X^{-m}$ over the regular part of $X$ for some $m$. Let $\omega$ be an admissible Kähler metric on $X$ which represents $\frac{1}{m} c_1(L)$, where $c_1(L)$ denotes the first Chern class of $L$. Note that by $\omega$ being admissible, we mean that it is given by the pullback of the Fubini-Study metric on $\mathbb{CP}^n$, i.e., $$\omega = \frac{\alpha}{m} \phi_m^{\ast} \left( \omega_{\text{FS}} + \frac{\sqrt{-1}}{2\pi} \partial \overline{\partial} \psi \right),$$ $\psi \in \mathscr{C}^{\infty}(\mathbb{CP}^n, \mathbb{R})$ and $\phi_m$ is the Kodaira embedding furnished from the global sections of $L \to X$.

Now let $\pi : \widetilde{X} \longrightarrow X$ be a smooth resolution of $X$, with $\pi$ given simply by projection. It is clear that the support of the cohomology class $\text{Ric}(\widetilde{\omega}) - \omega$ is contained the exceptional divisors of $\widetilde{X}$. Here $\widetilde{\omega}$ denotes a Kähler metric on $\widetilde{X}$ such that $\pi^{\ast} \omega \leq \omega$. Let $E_1, ..., E_{\ell}$ be the exceptional divisors, then $$\text{Ric}(\tilde{\omega}) - \pi^{\ast} \omega = \sum_{k=1}^{\ell} \alpha_k C_1([E_k]),$$ where $C_1([E_k])$ denotes the Poincaré duals to $E_k$ in $\widetilde{X}$.

For each $k$, let $\| \cdot \|_k$ denote a Hermitian metric on the line bundle $[E_k]$ and $S_k$ a section of $[E_k]$ whose zero locus is exactly $E_k$. Then in the sense of distributions, $$\text{Ric}(\tilde{\omega}) - \pi^{\ast} \omega = \frac{\sqrt{-1}}{2\pi} \partial \overline{\partial} \left( - \sum_{k=1}^{\ell} \alpha_k \log \| S_k \|_k^2 + \varphi \right),$$ where $\varphi \in \mathscr{C}^{\infty}(\widetilde{X}, \mathbb{R})$.

Continuing the proof, we obtain a function $f$ of the form $$f = - \sum_{k=1}^{\ell} \alpha_k \log \| S_k \|_k^2 + \varphi + \log \left( \frac{\widetilde{\omega}^n}{\pi^{\ast} \omega^n} \right) + \text{constant}$$ on the regular part of $X$.

Question: How does one show that $f \in L^p(X, \omega^n)$, i.e., $$\int_X \left| f \right|^p \omega^n < \infty,$$ where $p > 1$?

Of course, $\varphi$ is smooth, and $X$ is compact, so this is no concern. I cannot control the logarithm terms however, is there something I am missing that is blindingly obvious?


In polar coordinates

$ \int_{|z|<1} |\log(|z|)|^p = 2 \pi \int_0^1 (-\log r)^p r dr < +\infty .$


In a compact space it is enough to check the intergrability condition locally, only the terms $\log \|S\|$ might give trouble at points on $E = \{z=0\}$ (in local coordinates), so $ \log \|S\| = H + \log|z| $ with $H$ a smooth function (the log of the norm of a trivilalizing section). Fubini together with the above computation imply that $ \log \|S\|$ is in $L^p$; therefore also is $f$..unless I missunderstood the question.


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