Abstract

The two basic motivations for moduli theory are firstly to construct a space whose points are in bijection with the varieties under consideration, and secondly to precisely understand how these varieties vary in families. The notion of a coarse moduli space gives a complete solution of the first part but only a very weak solution of the second part. I will describe a new, mostly conjectural, approach to moduli theory where one focuses only on the second part and drops the first completely. This is most interesting for varieties with large automorphism group. As usual in moduli theory there is a notion of stability required, and the main novelty is a notion of a “stable fibration” over a fixed base variety, where each fibre of the fibration is assumed to be (K-)polystable. The definition extends the usual notion of slope stability for vector bundles, viewed as fibrations via the projectivisation construction (with the point being that each fibre of the projectivisation is projective space, which is K-polystable). The main result, rather than a construction of a moduli space of stable fibrations, is a result showing how stability of fibrations is related to the existence of certain canonical metrics called optimal symplectic connections, generalising the Hitchin-Kobayashi correspondence. This is work (in progress!) with Lars Sektnan.