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SUMMARY:Moduli theory\, stability of fibrations and optimal symplectic con
 nections - Ruadhaí Dervan
DTSTART:20191023T131500Z
DTEND:20191023T141500Z
UID:TALK132700@talks.cam.ac.uk
CONTACT:Dhruv Ranganathan
DESCRIPTION:The two basic motivations for moduli theory are firstly to con
 struct a space whose points are in bijection with the varieties under cons
 ideration\, 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 wil
 l describe a new\, mostly conjectural\, approach to moduli theory where on
 e 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 novel
 ty is a notion of a "stable fibration" over a fixed base variety\, where e
 ach fibre of the fibration is assumed to be (K-)polystable. The definition
  extends the usual notion of slope stability for vector bundles\, viewed a
 s fibrations via the projectivisation construction (with the point being t
 hat each fibre of the projectivisation is projective space\, which is K-po
 lystable). 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 sympl
 ectic connections\, generalising the Hitchin-Kobayashi correspondence. Thi
 s is work (in progress!) with Lars Sektnan. 
LOCATION:CMS MR3
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