 
Summary: Mirror symmetry for blowups
(AbouzaidAurouxKatzarkov, in progress)
Goal: construct mirror of ^XY = blowup 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.)
(BondalOrlov)
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}. µ := moment map.
S1invariant S.Lag. fibration on ^X \ ^D: Lt1,t2 = {log x = t1, µ = t2}.
