Volume form under holomorphic automorphisms

Question

$(M,\omega)$ is a compact Kaehler manifold and $f_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V_s$. My question is whether the function $\frac{f_{t,s}^* \omega^n}{\omega^n}$ is a bounded independent of $t$ and $s$. If not, can we control the $L^p$($p>1$) norm of $\frac{f_{t,s}^* \omega^n}{\omega^n}$ uniformly with respect to $t$ and $s$?

Answer 1

There is no such bound. As a counterexample take $(M,\omega)=(\mathbb{CP}^1, \omega_{FS})$. Identify $\mathbb{CP}^1$ with $\mathbb{C}\cup \infty$; in such co-ordinates $$\omega_{FS}=\frac{i dz\wedge d\overline z}{(1+|z|^2)^2}.$$ The radial vector field $V=z\partial/\partial z$ on $\mathbb{C}$ extends smoothly to a holo vector field on $\mathbb{CP}^1$, and the pullbacks of $\omega_{FS}$ by the family of diffeomorphisms generated by $V$ are $$\omega_{\lambda}=\frac{i dz\wedge d\overline z}{(\lambda^{-1}+\lambda|z|^2)^2},$$ where $\lambda$ varies in $\mathbb{R}^+$. The ratio $$ \frac{\omega_\lambda}{\omega_1}=\frac{(1+|z|^2)^2}{(\lambda^{-1}+\lambda|z|^2)^2} $$ is clearly unbounded. The $L^p$ norm of this quantity is also unbounded ($p>1$), since for $\lambda$ large, \begin{align*} \int_\mathbb{C}\left[\frac{\omega_\lambda}{\omega_1}\right]^{2p}\omega_1 &=\int_\mathbb{C}\frac{(1+|z|^2)^{2p-2}}{(\lambda^{-1}+\lambda|z|^2)^{2p}}i dz\wedge d\overline z\\ &\geq\int_{|z|<\lambda^{-1}}\frac{(1+|z|^2)^{2p-2}}{(\lambda^{-1}+\lambda|z|^2)^{2p}}i dz\wedge d\overline z\\ &\geq\int_{|z|<\lambda^{-1}}\frac{i dz\wedge d\overline z}{(2\lambda^{-1})^{2p}}\\ &=C\lambda^{2p-2}. \end{align*} One remark: I don't understand why there are two parameters $t$ and $s$ in your notation -- usually the notation for a 1-parameter subgroup features only one parameter! Did I misunderstand your question?