Correction: Logarithmic Bergman Kernel and Conditional Expectation of Gaussian Holomorphic Fields

1 Correction to: The Journal of Geometric Analysis (2021) 31:8520–8538 https://doi.org/10.1007/s12220-020-00602-z

In this article, the statement and proof of Theorem 3.3 in Logarithmic Bergman kernel and Conditional expectation of Gaussian holomorphic fields has been corrected. Please find below the corrected Theorem 3.3.

The set-up for Theorem 3.3 is summarized below:

Let (M, L) be a polarized Kähler manifold of dimension m. We endow L with a Hermitian metric h with positive curvature. And we use $\omega =\frac{i}{2}\Theta _h$ as the Kähler form. By abuse of notation, we still use h to denote the induced metric on the kth power $L^k$. Then we have a Hermitian inner product on $H^0(M,L^k)$, defined by

$$\begin{aligned} \langle s_1,s_2\rangle =\int _M h(s_1,s_2)\frac{\omega ^m}{m!} \end{aligned}$$

Let V be a smooth subvariety of M, and we denote by ${\mathcal {H}}_{k,V}$ the subspace of $H^0(M,L^k)$ consisting of sections that vanish along V. And ${\mathcal {H}}_{k,V}^{\perp }$ is the orthogonal complement. Let $n=\dim V$. The original statement of Theorem 3.3 is

Theorem 1.1

The restriction map $R: {\mathcal {H}}_{k,V}^{\perp }\rightarrow H^0(V,L^k)$ has norm satisfying

$$\begin{aligned} \parallel R\parallel ^2=O\left( \frac{1}{k^{m-n}}\right) \end{aligned}$$

However both the proof and the application of Theorem 3.3 is for the following statement

Theorem 1.2

The inverse of the restriction map $R: {\mathcal {H}}_{k,V}^{\perp }\rightarrow H^0(V,L^k)$ has operator norm satisfying

$$\begin{aligned} \parallel R^{-1}\parallel ^2=O\left( \frac{1}{k^{m-n}}\right) \end{aligned}$$

And it was pointed out by Finski in [1] that there is a gap in the proof of Theorem 3.3. In the proof, we tried to use the Ohsawa–Takegoshi–Manivel extension theorem in the following way. We first endow the line bundle $L^k-K_M$ with the metric $\text {e}^{-k\varphi }\otimes \text {d}^V_M=\text {e}^{-k\varphi +\kappa }$, where $\text {d}V_M=\frac{\omega ^m}{m!}$ is the volume form. Let $r(p)={{\text {dist}}}(x,V)$. Then we choose a nonnegative smooth function $\chi$ on $[0,\infty )$, which is concave and satisfies the following conditions:

• (1) $\chi (x)=x$ for $x\le \frac{(\log k)^2}{k}$;

• (2) $\chi (x)$ is constant for $x\ge \frac{(10\log k)^2}{k}$.

So $\chi (r^2)$ can be seen as a smooth function on M, which is constant away from V. Then we twist the metric on $L^k-K_M$ by $\text {e}^{\beta _k \chi (r^2)}$ for $\beta _k$ to be determined. We then claimed that we can choose $\beta _k=k-O\Big (\frac{\log k}{\sqrt{k}}\Big )$, which is a mistake, since in order to apply the Ohsawa–Takegoshi–Manivel theorem, we need to make sure that $k\varphi -\kappa -\beta _k \chi (r^2)$ is plurisubharmonic, which cannot be guaranteed if $\beta _k=k-O\Big (\frac{\log k}{\sqrt{k}}\Big )$. We can only have $\beta _k=k-O(\sqrt{k}\log k)$, which is not enough to prove Theorem 3.3. And this mistake makes the proof of Theorem 3.3 invalid.

In Theorem 4.4 of [1], Finski proved that there exist $C>0$ (independent of k) such that for each $g\in H^0(V,L^k)$, there is $f\in H^0(M,L^k)$ such that for $f|_V=g$

$$\begin{aligned} \parallel f\parallel ^2_{L^2(M)}\le Ck^{n-m} \parallel g\parallel ^2_{L^2(V)} \end{aligned}$$

for k large enough. Since $R^{-1}g$ is the extension of g with minimal $L^2$-norm, this theorem clearly implies the corrected statement of our Theorem 3.3. So Theorem 3.3 is still valid and the rest of the article is not affected.