Spherical Dehn twist Spherical vs Projective and more Institute for Advanced Study September 23, 2016
Spherical Dehn twist Spherical vs Projective and more Symplectic manifolds A symplectic manifold (M 2n, ω) is a smooth manifold M equipped with ω Ω 2 (M) such that dω = 0, ω n nowhere vanishing eg. (R 2n, n i=1 dx i dy i ), (T Q, ω can ), etc
Spherical Dehn twist Spherical vs Projective and more Symplectomorphism A symplectomorphism φ : (M, ω M ) (N, ω N ) is a diffeomorphism φ : M N, such that φ ω N = ω M eg. when (M, ω M ) = (N, ω N ) compact, time 1 flow along a vector field X on M such that L X ω M = 0
Spherical Dehn twist Spherical vs Projective and more Dehn twist Given a Lagrangian sphere S (M, ω), one can perform Dehn twist τ S : (M, ω) (M, ω) which is a symplectomorphism Figure : S is purple, L and τ S (L) are green
Spherical Dehn twist Spherical vs Projective and more Fukaya category Fukaya category Fuk(M, ω) is an A category objects: Lagrangians submanifolds L (with additional structures/restrictions) morphism: hom(l 0, L 1 ) = p L0 L 1 K < p > A operations µ k : hom(l k 1, L k ) hom(l 0, L 1 ) hom(l 0, L k )
Spherical Dehn twist Spherical vs Projective and more Quasi-equivalence Given an A category A, one can take its cohomological category H(A). An A functor F : A B is a quasi-equivalence if the induced functor on the cohomological category is an equivalence.
Spherical Dehn twist Spherical vs Projective and more Seidel s exact triangle Theorem (Seidel) For any graded compact exact Lagrangian L, there is an exact triangle in D π Fuk(M, ω) HF (S, L) S L τ S (L) HF (S, L) S[1] The induced autoequivalence T S : D π Fuk(M, ω) D π Fuk(M, ω) by τ S can be formulated purely algebraically.
Spherical Dehn twist Spherical vs Projective and more Spherical twist Let X be a smooth projective variety. An object E in D b (X ) is spherical if E ω X E, and Ext (E, E) = H (S dim(x ) ) A spherical object determines an autoequivalence T E on D b (X ).
Spherical Dehn twist Spherical vs Projective and more Summary Table : From symplectomorphism to autoequivalence (M, ω) D π Fuk(M, ω) D b (X ) S S E τ S T S T E
Spherical Dehn twist Spherical vs Projective and more Questions Are there other symplectomorphisms supported near Lagrangian submanifolds? What can one say about the induced auto-equivalences?
Spherical Dehn twist Spherical vs Projective and more A parallel story for projective space: Table : From symplectomorphism to autoequivalence (M, ω) D π Fuk(M, ω) D b (X ) P P P τ P T P T P Here, P is a Lagrangian (real/complex) projective space and P is a P-object in D b (X ).
Spherical Dehn twist Spherical vs Projective and more The definition of P-object and P-twist is due to Huybrechts-Thomas and is motivated by the symplectomorphism τ P. However, the relation between τ P and T P is still conjectural. Conjecture (Huybrechts-Thomas) The induced autoequivalence on D π Fuk(M, ω) by τ P is T P.
Spherical Dehn twist Spherical vs Projective and more Partial result In monotone setting, using Mau-Wehrheim-Woodward functor and Biran-Cornea Lagrangian cobordism theory, we have Theorem (M-Wu) In Tw Fuk(M, ω), there is a natural quasi-isomorphism of objects τ P (L) = Cone(Cone(hom(P, L) P[ 2] hom(p, L) P) L) for every L It looks similar to T P (L) but the morphisms in the theorem are not explicitly determined.
Spherical Dehn twist Spherical vs Projective and more Question Are spherical twists and P-twists related?
Spherical Dehn twist Spherical vs Projective and more A hybrid A Lagrangian S 2 = CP 1 in D π Fuk(M, ω) is both spherical and projective (similarly S 1 = RP 1 ) Table : From symplectomorphism to autoequivalence (M, ω) D π Fuk(M, ω) D b (X ) S = P S = P S = P τs 2 = τ P TS 2 = T P TS 2 = T P
Spherical Dehn twist Spherical vs Projective and more Another hybrid A Lagrangian P = RP 2n+1 is Question a P-object when char(k) = 2 a spherical object when char(k) 2 What is the induced autoequivalence of τ P on D π Fuk(M, ω) when char(k) 2?
Spherical Dehn twist Spherical vs Projective and more Hints When char(k) 2, the induced autoequivalence by τ P on D π Fuk(M, ω) is well-defined when P is (relatively) spin not a projective twist not a spherical twist not a square of a spherical twist
Spherical Dehn twist Spherical vs Projective and more Work in progress Theorem (M-Wu) Let P = RP 4n+3 be a monotone Lagrangian in a close monotone (M, ω). The induced autoequivalence by τ P is a simultaneous spherical twist by two spherical objects when char(k) 2. Goal: Explain to you autoequivalences obtained by S n /Γ twists in various characteristic Observe new phenomena of autoequivalences of D b (X )
Spherical Dehn twist Spherical vs Projective and more THANK YOU Thank you very much for your attention!