  We propose new constraints for 6d (1, 0) supergravity theories based on
consistency conditions on the Kahler moduli spaces of their 5d reductions. The
requirement that both the metric and the BPS string tensions in the Kahler
moduli space are positive imposes specific restrictions on the Chern-Simons
coefficients in the 5d effective Lagrangians that are derived from the
Kaluza-Klein reductions of 6d theories. Moreover, the emergence of local
interacting 5d CFTs when the moduli space metric degenerates introduces
additional constraints coming from the analysis of 5d SCFTs. Focusing on the
moduli spaces of 6d supergravity theories without a tensor multiplet and their
Higgsings, we show that these constraints require the presence of certain
primary states in the 2d worldvolume CFTs on 1/2 BPS strings. We specifically
analyze a class of SU(2) models and infinite families of U(1) models using
these constraints, and demonstrate that the theories featuring a 1-form
symmetry in their massless spectra, unless the 1-form symmetry is gauged, fail
to satisfy the constraints and therefore belong to the Swampland.
