\(Z\)-critical connections and Bridgeland stability conditions-critical connections and Bridgeland stability conditions (Q6639653)

This paper is devoted to \(Z\)-critical connections and Bridgeland stability conditions, a recent notion of stability arising from [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. There are many aspects to Bridgeland's notion of a stability condition, but here the most important point will be that for a given stability condition, one can still ask for a vector bundle to be stable. Bridgeland stability conditions have themselves had many spectacular applications in algebraic geometry, and these applications have emphasised that there is typically no canonical choice of stability condition [\textit{A. Bayer}, Proc. Symp. Pure Math. 97, 3--27 (2018; Zbl 1451.14048)]. Many of the applications arise from the very fact that one can vary the stability condition. \N\NIn this paper, the authors associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. They call solutions to these equations \(Z\)-critical connections, with \(Z\) a central charge. Deformed Hermitian Yang-Mills connections are a special case. They explain how their equations arise naturally through infinite-dimensional moment maps. The main result obtained shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a \(Z\)-critical connection if and only if it is asymptotically \(Z\)-stable. Even for the deformed Hermitian Yang-Mills equation, this provides the first examples of solutions in higher rank. \N\NThe paper is organized as follows: Section 1 is an introduction to the subject and statement of results. Section 2 deals with some preliminaries about stability conditions, \(Z\)-critical connections and moment maps, subsolutions and existence on surfaces. The authors turn to complex differential geometry, by introducing a geometric partial differential equation, involving various curvature quantities related to connections on holomorphic vector bundles, to each polynomial central charge. They discuss subsolutions, which will allow them to discuss not only moment maps, but also ellipticity conditions away from the large volume limit, as well as a complete understanding of the existence of \(Z\)-critical connections on line bundles over complex surfaces. Section 3 deals with asymptotic \(Z\)-stability of \(Z\)-critical vector bundles. This section proves one direction of the following result: the existence of \(Z\)-critical connections on a sufficiently smooth holomorphic vector bundle \(E\) over a compact Kähler manifold \((X,k\omega)\) for all \(k\gg0\) implies asymptotic \(Z\)-stability. The authors follow a general strategy for gauge-theoretic equations, and use the basic principle that curvature decreases in subbundles and increases in quotients. Section 4 is devoted to the existence of \(Z\)-critical connections on \(Z\)-stable bundles. The authors consider a holomorphic vector bundle \(E\) over a compact Kähler manifold \((X,\omega)\), together with a polynomial central charge \(Z\). They prove their main result: a simple, sufficiently smooth holomorphic vector bundle \(E\) admits uniformly bounded \(Z_k\)-critical connections for all \(k\gg0\) if and only if it is asymptotically \(Z\)-stable. \N\NIt should be emphasized that these results are new even in the case of deformed Hermitian Yang-Mills connections. In particular, this gives the first construction of deformed Hermitian Yang-Mills connections on a vector bundle of rank at least two. It is worth remarking that there has been no derivation of the deformed Hermitian Yang-Mills equation on a holomorphic vector bundle of rank at least two, neither in the physics literature nor the mathematics literature, and the results obtained in this paper thus give strong mathematical justification that the higher rank deformed Hermitian Yang-Mills equation suggested by \textit{T. C. Collins} and \textit{S.-T. Yau} [Ann. PDE 7, No. 1, Paper No. 11, 73 p. (2021; Zbl 1469.58007)] is indeed the appropriate equation.