It is known that K3 surfaces are never geometrically formal [1]. That is, the wedge product of two harmonic forms on an arbitrary K3 surface is in general not harmonic, or equivalently, the space $\mathscr H^{1,1}(X)$ of harmonic $(1,1)$-forms depends on the Ricci-flat Kähler metric. I am interested in gaining some control over this failure of geometric formality in the following sense.
Suppose $X$ is a compact K3 surface and $\omega$ a Ricci-flat Kähler form on it. We may choose a basis $\{\alpha_0:=\omega,\alpha_1,\ldots,\alpha_{19}\}$ of $\mathscr H^{1,1}(X)$ that is orthonormal in the sense
$$\int_X \alpha_i\wedge\star_\omega\alpha_j =\delta_{ij}\int_X \omega^2.$$
Then we can uniquely define functions $t_{ij}=t_{ji}$ on $X$ via the equations
$$\alpha_i\wedge \star_\omega\alpha_j = (\delta_{ij} + \Delta_\omega t_{ij})\,\omega^2,\qquad \int_X t_{ij}\, \omega^2 = 0.$$
The geometric interpretation is that if the Kähler class $[\omega]$ is deformed along $[\alpha_i]$, then the harmonic form in the class $[\alpha_j]$ with respect to the new Ricci-flat Kähler metric differs from the old one by $\mathrm{dd}^{\mathrm{c}}t_{ij}$. So, in particular, the functions $t_{0j}$ identically vanish, and if $X$ is geometrically formal, then all the functions $t_{ij}$ identically vanish. My first question then is the following:
Is anything known about the possible constraints the functions $t_{ij}$ need to satisfy? For example, is it known whether the constants $$C_{ijkl}:=\int_X (t_{ij}\Delta_\omega t_{kl}+t_{jk}\Delta_\omega t_{il}+t_{ki}\Delta_\omega t_{jl})\,\omega^2$$ have an enumerative interpretation in the case when $[\omega]$ is integral?
Here is why I think an affirmative answer is plausible. Huybrecht's proof of the fact that K3 surfaces are never geometrically formal relies on the existence of a smooth rational curve in a deformation of an arbitrary K3 surface. So, it would make sense if the above constants counted smooth rational curves satisfying certain conditions.
My second question has to do with generalising the definition of the constants $C_{ijkl}$ to an arbitrary Kähler surface $(X,\omega)$, where we no longer have a preferred Kähler metric in a given Kähler class.
Given an arbitrary Kähler surface $(X,\omega)$, do the analogously defined constants $C_{ijkl}$ depend only on the Kähler class $[\omega]$ and not on the representative $\omega$?
The reason why I think this is plausible is that the constants $C_{ijkl}$ seem very similar in spirit to the classical Futaki invariant which indeed depends only on the Kähler class and provides an obstruction to the existence of Kähler–Einstein metrics in a given Kähler class. In fact, if the constants $C_{ijkl}$ indeed depend only on the Kähler class $[\omega]$, they would provide an obstruction to the existence of a formal metric in a given Kähler class.
In any case, the above questions seem rather natural to ask and I suspect people have thought about them before. However, I haven't been successful in finding anything in the literature on geometric formality addressing this. I suppose this is probably due to me not looking for the right keywords or failing to translate what is there into language I am familiar with (i.e. differential geometry rather than rational homotopy theory). I would be grateful if someone could point me to any relevant references.
[1] Daniel Huybrechts, "Products of harmonic forms and rational curves", Documenta Mathematica 6 (2001) 227–239, arXiv: math/0003202
