Let $\omega $ be an LCK metric with Lee form $\lambda $.
The Kähler covering provides a potential calculus compatible with the Lee equation. Conjugating the ordinary differential operators by the local conformal factor produces a twisted complex Hessian on $X$.
For the fixed closed one-form $\lambda $, define
\[ \mathrm{d}_\lambda =\mathrm{d}-\lambda \wedge , \qquad \mathrm d^{\mathrm c}_\lambda =\mathrm d^{\mathrm c}-\lambda ^{\mathrm c}\wedge , \qquad D_\lambda =\mathrm{d}_\lambda \mathrm d^{\mathrm c}_\lambda . \]
On a chart where $\lambda =\mathrm{d}f$, these operators satisfy
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23.2.19
\begin{equation} \mathrm{d}_\lambda =\mathrm{e}^ f\mathrm{d}\mathrm{e}^{-f}, \qquad \mathrm d^{\mathrm c}_\lambda =\mathrm{e}^ f\mathrm d^{\mathrm c}\mathrm{e}^{-f}, \qquad D_\lambda =\mathrm{e}^ f\mathrm{dd}^{\mathrm{c}}\mathrm{e}^{-f}. \end{equation}
These identities show that $D_\lambda u$ is a real $(1,1)$-form for every smooth real function $u$, and that $\mathrm{d}_\lambda D_\lambda u=0$.
Proposition 23.2.20.B365A2A75E69 Let $u$ be a smooth real function. Then
\[ \mathrm{d}(\omega +\mathrm{dd}^{\mathrm{c}}u)-\lambda \wedge (\omega +\mathrm{dd}^{\mathrm{c}}u) =-\lambda \wedge \mathrm{dd}^{\mathrm{c}}u, \]
whereas
\[ \mathrm{d}(\omega +D_\lambda u) =\lambda \wedge (\omega +D_\lambda u). \]
Consequently, an ordinary potential preserves the LCK equation with the same Lee form precisely when $\lambda \wedge \mathrm{dd}^{\mathrm{c}}u=0$. A twisted potential always preserves that equation, and it gives an LCK metric whenever $\omega +D_\lambda u>0$.
Proof.
The first identity follows from $\mathrm{d}\mathrm{dd}^{\mathrm{c}}u=0$ and $\mathrm{d}\omega =\lambda \wedge \omega $. The second follows from $\mathrm{d}_\lambda \omega =0$ and (23.2.19).
$\square$
On the Kähler covering, a twisted potential becomes the automorphic function $\widetilde u=\mathrm{e}^{-F}\pi ^*u$, and
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\begin{equation} \mathrm{e}^{-F}\pi ^*(\omega +D_\lambda u) =\widetilde\omega +\mathrm{dd}^{\mathrm{c}}\widetilde u. \end{equation}
The top power $(\widetilde\omega +\mathrm{dd}^{\mathrm{c}}\widetilde u)^ n$ transforms by $\chi ^ n$ under deck transformations, so its ratio to $\widetilde\omega ^ n$ is invariant and descends to $X$. A measure on $X$ is obtained from this ratio only after a metric in the conformal class has been fixed; the automorphic top-degree form cannot simply be integrated over the noncompact cover.
The two kinds of potentials also differ in their normalization. Expanding the twisted Hessian gives
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\begin{equation} D_\lambda u =\mathrm{dd}^{\mathrm{c}}u-\mathrm{d}u\wedge \lambda ^{\mathrm c} -\lambda \wedge \mathrm d^{\mathrm c}u +u\bigl (\kappa _\lambda +\lambda \wedge \lambda ^{\mathrm c}\bigr ). \end{equation}
In particular, $D_\lambda 1=\kappa _\lambda +\lambda \wedge \lambda ^{\mathrm c}$ need not vanish, so adding a constant to $u$ can change the metric $\omega +D_\lambda u$. The usual normalization $\sup _ Xu=0$ is therefore not harmless for twisted equations.
For a chosen Lee form $\lambda $ in its de Rham class, the real twisted Bott–Chern group in bidegree $(1,1)$ is
\[ H^{1,1}_{\mathrm{BC},\lambda }(X,\mathbb R) =\frac{\{ \eta \in C^\infty (X,\Lambda ^{1,1}_{\mathbb R}): \mathrm{d}_\lambda \eta =0\} }{D_\lambda C^\infty (X,\mathbb R)}. \]
This is Bott–Chern cohomology with values in the weight bundle, OV24LCK]. It differs both from ordinary Bott–Chern cohomology and from the quotient of all backgrounds by ordinary $\mathrm{dd}^{\mathrm{c}}$-terms considered in section 10.30.
Pome
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