PomeTheorem 11.2.4.7D069B347672 For $n\geq 2$, the one-form $\alpha $ has a primitive on $\mathcal H_\omega $ if and only if $\mathrm{d}\omega =0$. In fact, a primitive on a $C^2$-open neighborhood of $0$ already forces $\mathrm{d}\omega =0$.
Theorem 11.2.47D069B347672
Proof. Suppose that a primitive $E$ exists on such a neighborhood $U$. For $u\in U$ and smooth real directions $v,w$, the mixed second derivatives of $(s,t)\mapsto E(u+sv+tw)$ at the origin agree. After division by $n$, this equality reads
B4B9D0AD937E
11.2.5
\begin{equation} \int _ Xw\mathrm{dd}^{\mathrm{c}}v\wedge \omega _ u^{n-1} =\int _ Xv\mathrm{dd}^{\mathrm{c}}w\wedge \omega _ u^{n-1}. \end{equation}
By lemma 11.2.2,
\[ 0=\mathrm{d}(\omega _ u^{n-1}) =(n-1)\mathrm{d}\omega \wedge (\omega +\mathrm{dd}^{\mathrm{c}}u)^{n-2}. \]
Fix a point $x$. A real quadratic polynomial in local coordinates, multiplied by a cutoff function equal to one near $x$, realizes any prescribed real $(1,1)$-form $H$ as $\mathrm{dd}^{\mathrm{c}}u(x)$. These functions can be chosen to depend linearly on $H$, with global $C^2$ norms bounded by a fixed constant times the coefficient norm of $H$. Thus every sufficiently small $H$ is realized by a potential in $U$, and the preceding identity holds at $x$ with $\omega (x)+H$ ranging over an open neighborhood of the positive form $\omega (x)$. Applying lemma 11.2.3 with $T=\mathrm{d}\omega (x)$, we conclude that $\mathrm{d}\omega (x)=0$.
Since $x$ was arbitrary, $\mathrm{d}\omega =0$. Conversely, proposition 11.2.8 below constructs a primitive for a closed background. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.