What does it mean for a Ricci form to be "basic" with respect to a Killing vector?

Question

In a paper on superconformal anomalies, Cassani and Martelli say (at the bottom of page 16) that given the expression for the Ricci form of a metric (let us call it $\mathcal{R}$), it is "straightforward to verify that it is "basic" with respect to a complex Killing vector $\partial_{w}$, namely

$$\partial_{w}\_ _{|}\mathcal{R} = 0$$

What does this mean? (Oh by the way, I'm not drawing this correctly in $\LaTeX$. The symbol is a laterally flipped L).

Answer 1

I think this is related to the definition given above equation (2.21) of the paper arXiv:hep-th/0603021("Sasaki-Einstein Manifolds and Volume Minimization", by Dario Martelli, James Sparks and Shing-Tung Yau).

Recall that a $p$-form $\alpha$ on $L$ is said to be basic with respect to the foliation induced by $\xi$ if and only if $$\mathcal{L}_{\xi}\alpha = 0,\qquad \qquad \xi\lrcorner\alpha=0$$

(The relevant $\LaTeX$ symbol is \lrcorner, in case you were wondering.)

I'd still welcome a more comprehensive explanation of course.