PomeTheorem 24.4.28.5C15639B8E53 Suppose that $(M,J_0)$ carries a pluriclosed Hermitian form $\omega _0$ and that $h_ A^{1,1}(M,J_ t)$ is constant near $0$. Then for small $t$ there is a smooth family of pluriclosed Hermitian forms $\omega _ t$ on $(M,J_ t)$ with the prescribed value $\omega _0$ at $t=0$.
Theorem 24.4.285C15639B8E53
Proof. Let $\pi _ t^{1,1}$ denote the projection onto forms of type $(1,1)$ for $J_ t$, and use the operators of lemma 24.4.24. Define
\[ \omega _ t=\mathcal P_ t\bigl (\pi _ t^{1,1}\omega _0\bigr ). \]
These are smooth real $(1,1)$-forms satisfying $\partial _ t\bar\partial _ t\omega _ t=0$. At $t=0$ the projection fixes $\omega _0$, since $\omega _0$ already satisfies this equation. The tensors $\omega _ t(\cdot ,J_ t\cdot )$ converge uniformly to the positive definite tensor $\omega _0(\cdot ,J_0\cdot )$, which has a uniform positive lower bound by compactness. Hence they remain positive definite after shrinking $U$. $\square$
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