TITLE: Holomorphic vector fields acting on Dolbeault cohomology
QUESTION [20 upvotes]: The question.
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) decomposition and also that it commutes with $\bar{\partial}$. From this it follows that $u$ acts infinitesimally on the Dolbeault cohomology groups $H^{p,q}(X)$ of $X$. My question is, does anyone know of an example in which this action is non-trivial?
Some context.
To give some context, first note that the analgous action for de Rham cohomology is always trivial: If $M$ is any smooth manifold and $v$ any vector field, then the formula $L_v = d \circ i_v + i_v \circ d$ shows that the infinitesimal action of $v$ on de Rham cohmology is trivial. (This is an instance of the more general fact that homotopic maps induce the same homomorphisms on singular cohomology. The field $v$ generates diffeomorphisms which are by construction isotopic to the identity map.)
Returning to Dolbeault cohomology, suppose we know that each Dolbeault class is represented by a $d$-closed form. (For example, this is true if $X$ is a compact Kähler manifold, by Hodge theory.) Then the action is necessarily trivial. The proof is as follows. Let $\alpha$ be a $\bar{\partial}$-closed (p,q)-form which is also $d$-closed. Then we know that $L_u \alpha = d(i_u \alpha)$ is also of type (p,q). So, 
$$
L_u\alpha = \bar{\partial}\left((i_u\alpha)^{p, q-1}\right) + \partial\left((i_u \alpha)^{p-1, q}\right)
$$
and the other contributions $\bar{\partial}((i_u\alpha)^{p-1,q}$) and $\partial((i_u\alpha)^{p,q-1})$ vanish. Now the fact that $\bar\partial((i_u\alpha)^{p-1,q}) = 0$ and our hypothesis imply that there is a (p-1, q-1)-form $\beta$ such that 
$$
(i_u\alpha)^{p-1,q}+ \bar\partial \beta
$$
is closed. Hence
$$
\partial \left((i_u\alpha)^{p-1,q}\right) = \bar\partial \partial \beta
$$
and so
$$
L_u\alpha = \bar \partial \left( (i_u \alpha)^{p,q-1} + \partial \beta\right)
$$
which proves the action of $u$ on $H^{p,q}(X)$ is trivial.

REPLY [2 votes]: Klemyatin proved that this action is trivial if the corresponding
${\Bbb C}$-flow is compatible with some metric (hence can be extended
to a compact torus action),
https://arxiv.org/abs/1909.04075,
(N. Klemyatin, Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group, J. Geom. Phys. 157 (2020), 103823.)
Also, this action is trivial whenever the Hodge to de Rham spectral sequence degenerates in $E_1$. This happens on all complex surfaces, on Vaisman manifolds, Oeljeklaus-Toma manifolds, and on some other interesting classes of non-Kahler complex manifolds.
Examples when the action is non-trivial are given by Akhiezer in this paper:
Akhiezer, Dmitri
Group actions on the Dolbeault cohomology of homogeneous manifolds.
Math. Z. 226 (1997), no. 4, 607–621.