  We investigate the large curvature perturbations which can lead to the
formation of primordial black holes (PBHs) in the context of no-scale
supergravity. Our study does not depend on any exotic scenario, such as scalar
potentials with inflection points or bulks, and aims to avoid the fine-tuning
of model parameters to achieve the formation of PBHs. This formation relies on
the quantum fluctuations of a light spectator stochastic field after the
inflationary period. Our analysis is based on the SU(2,1)/SU(2)$\times$U(1)
symmetry, considering both the inflaton and the spectator field. Specifically,
we examine existing no-scale models with Starobinsky-like scalar potentials
that are consistent with observable constraints on inflation from measurements
of the cosmic microwave background (CMB). These models involve two chiral
fields: the inflaton and the modulus field. We propose a novel role for the
modulus field as a spectator field, responsible for generating PBHs. Our
hypothesis suggests that while the inflaton field satisfies the CMB constraints
of inflation, it is the modulus field acting as the spectator that leads to
large curvature perturbations, capable to explain the production of PBHs.
Additionally, we prioritize retaining the inflationary constraints from the CMB
through the consideration of spectator fluctuations. Therefore, by exploring
the relationship between these fields within the framework of the
SU(2,1)/SU(2)$\times$U(1) symmetry, our aim is to unveil their implications for
the formation of PBHs.
