Strong-subsolution conjecture for solutions of the Z-critical equation

From papers

Let EE be an object carrying a ZZ-critical equation, and let hh be a metric on EE. A strong subsolution is a metric satisfying the positivity conditions on all formal matrix derivatives of the central charge, namely

Im(eiφ(E)Z~(p)(h))>0\operatorname{Im}\bigl(e^{-i\varphi(E)}\widetilde Z^{(p)}(h)\bigr)>0

as an EndE\operatorname{End}E-valued (np,np)(n-p,n-p)-form for every p=1,,np=1,\dots,n. Strong-subsolution conjecture. A solution to the ZZ-critical equation is a strong subsolution.

This is singled out as the principal analytical conjecture concerning subsolutions, within a proposed diagram relating solutions, strong subsolutions, ordinary subsolutions, and corresponding stability notions. Its resolution is not specified in the source.

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Sources & referencesView supporting material

Primary source

John Benjamin McCarthy, “Stability conditions and canonical metrics”, arXiv:2302.04966 (2023).

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