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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Cubic fourfolds\, K3 surfaces\, and mirror symmetr
y - Nick Sheridan\, Cambridge
DTSTART;TZID=Europe/London:20171025T160000
DTEND;TZID=Europe/London:20171025T170000
UID:TALK74561AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/74561
DESCRIPTION:While many cubic fourfolds are known to be rationa
l\, it is expected that the very general cubic fou
rfold is irrational (although none have been prove
n to be so). There is a conjecture for precisely w
hich cubics are rational\, which can be expressed
in Hodge-theoretic terms (by work of Hassett) or i
n terms of derived categories (by work of Kuznetso
v). The conjecture can be phrased as saying that o
ne can associate a `noncommutative K3 surface' to
any cubic fourfold\, and the rational ones are pre
cisely those for which this noncommutative K3 is `
geometric'\, i.e.\, equivalent to an honest K3 sur
face. It turns out that the noncommutative K3 asso
ciated to a cubic fourfold has a conjectural sympl
ectic mirror (due to Batyrev-Borisov). In contras
t to the algebraic side of the story\, the mirror
is always `geometric': i.e.\, it is always just an
honest K3 surface equipped with an appropriate Kä
hler form. After explaining this background\, I wi
ll state a theorem: homological mirror symmetry ho
lds in this context (joint work with Ivan Smith).
LOCATION:MR13
CONTACT:Ivan Smith
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