Stability conditions in geometric invariant theory

ABSTRACT We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. In the appendix by Ibáñez Núñez, it is explained how central charges can be viewed through the graded points of a stack. We use these ideas to introduce an axiomatic notion of a stability condition for polarized schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf–Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang–Mills connections, Z-critical connections and Z-critical Kähler metrics.