Generalised Einstein equations and prescribed relations for the first Chern weil form

Generalised Einstein equations (Einstein equations with sources in the physicist's grammar) can, in the Kähler setup, be seen as cohomological equations within the first Chern class. Introducing a two parameter secondary class (or source term) to prescribe such a cohomological relation, we characterise regions and paths of those parameters to ensure that the associated equation admits at least one solution. Those regions could be seen, in the context of geometric analysis, as a measure of the metric flexibility allowed within the Kählerian rigidity. When the first Chern class is positive, the Aubin Tian constant and the bounds for the pluriharmonic concavity and convexity of the source term characterise the bounds of that region. Taking into account the minimal regularity of the secondary class to ensure the existence of classical solutions, we observe, in particular, an improvement of some results quoted in the literature in the context of Calabi's conjecture.