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SUMMARY:Cubic fourfolds\, K3 surfaces\, and mirror symmetry - Nick Sherida
 n\, Cambridge
DTSTART:20171025T150000Z
DTEND:20171025T160000Z
UID:TALK74561@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:While many cubic fourfolds are known to be rational\, it is ex
 pected that the very general cubic fourfold is irrational (although none h
 ave been proven to be so). There is a conjecture for precisely which cubic
 s are rational\, which can be expressed in Hodge-theoretic terms (by work 
 of Hassett) or in terms of derived categories (by work of Kuznetsov). The 
 conjecture can be phrased as saying that one can associate a `noncommutati
 ve K3 surface' to any cubic fourfold\, and the rational ones are precisely
  those for which this noncommutative K3 is `geometric'\, i.e.\, equivalent
  to an honest K3 surface. It turns out that the noncommutative K3 associat
 ed to a cubic fourfold has a conjectural symplectic mirror (due to  Batyre
 v-Borisov). In contrast to the algebraic side of the story\, the mirror is
  always `geometric': i.e.\, it is always just an honest K3 surface equippe
 d with an appropriate Kähler form. After explaining this background\, I w
 ill state a theorem: homological mirror symmetry holds in this context (jo
 int work with Ivan Smith). 
LOCATION:MR13
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