We introduce an analogue of Bridgeland's stability conditions for polarised
varieties. Much as Bridgeland stability is modelled on slope stability of
coherent sheaves, our notion of Z-stability is modelled on the notion of
K-stability of polarised varieties. We then introduce an analytic counterpart
to stability, through the notion of a Z-critical K\"ahler metric, modelled on
the constant scalar curvature K\"ahler condition. Our main result shows that a
polarised variety which is analytically K-semistable and asymptotically
Z-stable admits Z-critical K\"ahler metrics in the large volume regime. We also
prove a local converse, and explain how these results can be viewed in terms of
local wall crossing. A special case of our framework gives a manifold analogue
of the deformed Hermitian Yang-Mills equation.Comment: v2: linear analysis section corrected, minor changes otherwise. 65
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