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CATEGORIES:Algebraic Geometry Seminar
SUMMARY:Moduli theory\, stability of fibrations and optima
l symplectic connections - Ruadhaí Dervan
DTSTART;TZID=Europe/London:20191023T141500
DTEND;TZID=Europe/London:20191023T151500
UID:TALK132700AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/132700
DESCRIPTION:The two basic motivations for moduli theory are fi
rstly to construct a space whose points are in bij
ection with the varieties under consideration\, an
d secondly to precisely understand how these varie
ties vary in families. The notion of a coarse modu
li space gives a complete solution of the first pa
rt but only a very weak solution of the second par
t. I will describe a new\, mostly conjectural\, ap
proach to moduli theory where one focuses only on
the second part and drops the first completely. Th
is is most interesting for varieties with large au
tomorphism group. As usual in moduli theory there
is a notion of stability required\, and the main n
ovelty is a notion of a "stable fibration" over a
fixed base variety\, where each fibre of the fibra
tion is assumed to be (K-)polystable. The definiti
on extends the usual notion of slope stability for
vector bundles\, viewed as fibrations via the pro
jectivisation construction (with the point being t
hat each fibre of the projectivisation is projecti
ve space\, which is K-polystable). The main result
\, rather than a construction of a moduli space of
stable fibrations\, is a result showing how stabi
lity of fibrations is related to the existence of
certain canonical metrics called optimal symplecti
c connections\, generalising the Hitchin-Kobayashi
correspondence. This is work (in progress!) with
Lars Sektnan.
LOCATION:CMS MR3
CONTACT:Dhruv Ranganathan
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