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Mirror symmetry for blow-ups (Abouzaid-Auroux-Katzarkov, in progress)
 

Summary: Mirror symmetry for blow-ups
(Abouzaid-Auroux-Katzarkov, in progress)
Goal: construct mirror of ^XY = blow-up of X along a codimension 2
subvariety Y X (need Y D | - KX |)
Motivation: a mirror of ^XY is almost as good as a mirror of Y .
DbCoh(^XY ) DbCoh(Y ), DbCoh(X) (semiorthogonal decomp.)
(Bondal-Orlov)
also expect F(^XY ) related to F(Y ) (esp. if X = D C and Y fiber
of a pencil in D)
Simplification: assume (X, D) toric (but not Y ).
Motivating example: what's the mirror of a genus 2 curve ?
Answer: blow up (CP1
)3 along P1 P1 {0}, take mirror, restrict.
Denis Auroux (MIT) Special Lagrangians and mirror symmetry January 2009 - U. of Miami 1 / 7
Blowing up a point
Local model in dim. 2:
X = C C, D = C {0}, = d log x d log y, = 0
^X = blowup at (1, 0), ^D = proper transform, ^ = , ^ = ^ ( E ^ = )
S1 action (y eiy) lifts, fixed point set ^D {pt}. := moment map.
S1-invariant S.Lag. fibration on ^X \ ^D: Lt1,t2 = {log |x| = t1, = t2}.

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics