Lemma 11.10.3.047ADFD98AAF In a holomorphic bundle chart, write $v=(v',v_ r)$ and use the affine projective coordinate $w=v'/v_ r$ on the cone $v_ r\ne 0$. Suppose that a positive continuous complex homogeneous function $Q$ of degree two is strictly plurisubharmonic and locally strongly convex in the fibers on this cone, and put $q(z,w)=Q(z,(w,1))$. For a nonnegative smooth approximate identity $\eta _\delta (s,t)$ of integral one, define
\[ q_\delta (z,w)=\int q(z-s,w-t)\eta _\delta (s,t)\, \mathrm ds\, \mathrm dt, \qquad Q_\delta (z,v)=|v_ r|^2q_\delta (z,v'/v_ r). \]
On every relatively compact projective subchart, for sufficiently small $\delta $, $Q_\delta $ is a smooth positive complex homogeneous function of degree two that is strictly plurisubharmonic and locally strongly convex in the fibers. As $\delta \to 0$, it converges to $Q$ uniformly on compact subsets of the cone consisting of vectors of norm one, and in every finite derivative norm wherever $Q$ is already smooth.
Proof.
Complex homogeneity gives the identity
\[ Q_\delta (z,v) =\int Q(z-s,T_ tv)\eta _\delta (s,t)\, \mathrm ds\, \mathrm dt, \qquad T_ t(v',v_ r)=(v'-v_ rt,v_ r). \]
For fixed $(s,t)$, the map $(z,v)\mapsto (z-s,T_ tv)$ is holomorphic and invertible, and its restriction to each fiber is linear. On a compact subset of the cone consisting of vectors of norm one, all these maps with $(s,t)$ sufficiently small stay in a fixed compact subset of the original cone, and their differentials and inverses are uniformly bounded there. A positive local lower bound for the Levi form of $Q$, interpreted through quadratic barriers, therefore gives a positive lower bound for every pullback and for their average. The same argument with real quadratic barriers in the fibers preserves local strong convexity. Only fiber segments contained in the local cone neighborhood are used here.
Ordinary convolution makes $q_\delta $ smooth, and the formula for $Q_\delta $ makes it smooth on the cone and complex homogeneous. Uniform convergence, derivative convergence on compact regions where $Q$ is smooth, and positivity for small $\delta $ follow from the corresponding properties of convolution and from the positive minimum of $Q$ on compact sets of vectors of norm one.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.