Note 8
Discreteness of Language
Flux Conjecture
(Lalonde-McDuff-Polterovich 1998)
Image of Flux homomorphism is discrete at H1(M; R).
Lemma1
Next two are equivalent.
(i) Flux conjecture is correct.
(ii) All the complete symplectic homeomorphism is C1 topological closed at symplectic
transformation group.
Lemma 2
Next two are equivalent.
(1) Flux conjecture is correct.
(ii) Diagonal set M
M×M is stable by the next definition.
M×M is stable by the next definition.
Definition
L is stable at the next condition.
(i) There exist differential 1 form u1, u2 over L that is sufficiently small.
(ii) When sup|u1|, sup|u2| is Lu1
Lu2 for u1, u2 ,there existsf that satisfies u1 - u2 = df .
Lu2 for u1, u2 ,there existsf that satisfies u1 - u2 = df .
Explanation
0 <For conjecture and lemmas>
is de Rham cohomology class.
Symplectic manifold (M, w)
Group's connected component of complete homeomorphism Ham (M, w)
Flux isomorphism Flux: π1(Ham(M, w) )→ R
Road of Ham (M, w) γ(t)
δγ / δt = Xu(t) that is defined bu closed differential form Utover M
Explanation
1
Symplectic manifold M
n-dimensional submanifold L 
M
M
L that satisfies next condition is called special Lagrangian submanifold.
Ω's restriction to L is L's volume.
2
M's special Lagrangian submanifold L
Flat complex line bundle L
LAGsp(M) (L, L)
3
Complex manifold M†
p 
M†
M†
Sheaf over M† fp
fp (U) = C ( p
U)
U)
fp (U) = 0 ( p 
U)
U)
4
Special Lagrangian fiber bundle π : M → N
Complementary dimension 2's submanifold S(N) 
N
N
π-1 (p) = LP
Pair (Lp, Lp)
p
N-S(N)
N-S(N)
Lp Complex flat line bundle
All the pair (Lp, Lp) s is M0† .
5
(Geometric mirror symmetry conjecture Strominger-Yau-Zaslow 1996)
Mirror of M is diffeomorphic with compactification of M0† .
6
Pairs of Lagrangian submanifold of M and flat U(1) over the submanifold (L1, L1), (L2, L2)
(L1, L1) 
(L2, L2) means the next.
(L2, L2) means the next.
There exists complete symplectic homeomorphism that is ψ(L2 ) = L2
and
ψ*L2 is isomorphic with L1.
Impression
Discreteness of language is possible by Flux conjecture 1998.
[References]
To be continued
Tokyo July 19, 2009