Following the notation of Moroianu's *Lectures on Kähler Geometry*, we let $(M,g,\Omega,J)$ denote the metric $g$, symplectic form $\Omega$, and complex structure $J$ of a Kähler manifold $M$, satisfying the compatibility condition $g(X,Y) = g(JX,JY) = \Omega(X,JY)$ for vector fields $X,Y$.

Supposing further that $V\lrcorner\ \Omega = dH$ for $H$ a real-valued (explicitly biholomorphic) Hamiltonian potential $H:M\to\mathbb R$, such that V is real-holomorphic (is this correct?) then we immediately have the following Lie derivative relations:

$$ \mathcal{L}_V \Omega = \mathcal{L}_V J = 0\quad\Rightarrow\quad \mathcal{L}_V g = 0 $$

or equivalently this proposition:

PropositionEvery real-holomorphic Hamiltonian vector field on a Kähler manifold is Killing.

This proposition is (in essence) a Hamiltonian converse to the following proposition of Moroianu's

Proposition 9.5(Moroianu)Every Killing vector field on a compact Kähler manifold is real holomorphic.

Three specific questions are asked:

**Q1** Is the proposition $\mathcal{L}_V g = 0$ correct (for the assumptions given)?

**Q2** Where can it be found in the literature?

**Q3** Does it "trivially" imply $\mathcal{L}_V \mathcal{R} = 0$, where $\mathcal{R}$ is the scalar curvature?

**Note:** My numerical calculations suggest $\mathcal{L}_V \mathcal{R} \ne 0$. The *practical* question is simply which is buggy: the formal reasoning associated to the proposition, or the code, or the "trivial" expectation that **Q3**'s answer is "yes"?

Deficiencies in my understanding of terms like "real-holomorphic" are plausible (and even likely). Answers/references/general advice are *very* welcome!

not(in general) suffice to ensure that the flow is real-holomorphic. I will grind out some (index-heavy) calculations to check this, but if anyone wants to post an answer, please do so! $\endgroup$