Lagrangian submanifolds in elliptic and log symplectic manifolds

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1 Lagrangian submanifolds in elliptic and log symplectic manifolds joint work with arco Gualtieri (University of Toronto) Charlotte Kirchhoff-Lukat University of Cambridge Geometry and ynamics in Interaction C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

2 Generalized complex geometry 2n smooth manifold H Ω 3 cl () Generalized Geometry (Hitchin, Gualtieri, 2000 s): T T := T T Generalized complex structures A GC structure on is J End(T T ) s.t. J 2 = 1. J orthogonal w.r.t X + ξ, Y + η = η(x) + ξ(y ). +i-eigenbundle L T C is integrable w.r.t. X + ξ, Y + η = [X, Y ] + L X η i Y dξ + i Y i X H C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

3 Generalized complex geometry (2) Examples I complex( structure ) ( ) ( ) I 0 T T J I = 0 I : T T is GC. ( ) 0 ω 1 ω symplectic form J ω = is GC. ω 0 In general: ( ) Q J =, Q Poisson. C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

4 Outline 1 Introduction: Generalized complex geometry 2 Stable GC structures 3 Logarithmic and elliptic symplectic geometry 4 Real-oriented blow-up: Relation between log and elliptic symplectic geometry 5 Local neighbourhoods of Lagrangian submanifolds 6 Outlook C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

5 Stable GC structures Cavalcanti, Gualtieri 2006/2015 Examples: GC, neither complex nor symplectic (Up to gauge equivalence) determined by Poisson structure Q. Q non-denerate except on = codim-2 submanifold. (On : rank dim 4) Write: ω = Q 1 r θ C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

6 Logarithmic and elliptic symplectic geometry Logarithmic Elliptic x r θ Z Z = codim. 1 = codim. 2 (x, y 2,..., y 2n ), Z = {x = 0} (r, θ, y 3,... ), = {r = 0} T( log Z) T( log ) = x x y 2,..., y 2n = r,, r θ y 3,..., y 2n T (log Z) = dx, dy x 2,..., dy 2n ω Γ( 2 T (log Z)) non-degenerate and dω = 0 T (log ) = dr, dθ, dy r 3,..., dy 2n ω Γ( 2 T (log )) non-degenerate and dω = 0 C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

7 Stable GC structures Cavalcanti, Gualtieri 2006/2015 Examples: GC, neither complex nor symplectic (Up to gauge equivalence) determined by Poisson structure Q. Write: ω = Q 1 co-oriented, ω elliptic symplectic with res ell ω = 0. ω = dr r Ω I + dθ Ω R + σ, Ω I, Ω R, σ Ω () dr dθ r r θ C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

8 Lagrangian branes L n 2n Lagrangian L smooth, T (L ) = TL T L and ι Lω = 0. Case 1: L, dim(l ) = n 2. elliptic structure, Generalized complex brane Case 2: L = L, dim(l ) = n 1. logarithmic structure, Lagrangian brane with boundary C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

9 Examples: Lefschetz thimbles in stable GC Lefschetz fibrations f Σ Boundary Lefschetz fibration (Cavalcanti, Klaasse 17): fibres over boundary Σ. Well-defined notion for stable GC Lefschetz fibration: ω compatible with f. Can extend Lefschetz thimbles into. C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

10 Real oriented blow-up ~ θ ~ r r θ β : is U(1)-principal bundle {log vector fields & forms} ω = β (ω) log symplectic with: i res ω = 0 θ ω = 0 di θ N oriented N complex. {elliptic vector fields & forms} ω stable GC C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

11 Branes under real oriented blow-up β ~ L Lagrangian, L, L = L. Case 1: T ( L ) L := β( L), smooth brane θ without boundary. Case 2: not tangent to L, θ β L injective. L := β( L) brane with boundary. C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

12 Local neighbourhoods and small deformations Lagrangian neighbourhood theorems: log symplectic geometry: ( L, L) (,, ω) compact, L. (Ũ, Ũ ) = (T L(log L), ω0 ). elliptic symplectic geometry: L compact, L. (U, U ) = (T L(log L ), ω 0 ). wedge neighbourhood for Lag. branes with boundary: (L, L) (,, ω) compact. β(ũ), (Ũ, Ũ ) = T L(log L) wedge neighbourhood. ~ β L C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

13 Small deformations of Lagrangians Small = Inside Lagrangian neighbourhood, graph of one-form Small deformations (up to Ham. isotopy) = First cohomology L L log Lagrangians: H 1 (Ω (L, log L)) elliptic Lagrangian, L : H 1 (Ω (L, log L )) Lagrangian brane with boundary: H 1 (Ω (L, log L)) Natural differential complex associated to brane. C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

14 Fukaya category for stable GC manifolds? Stable Hamiltonian system in neighbourdhood of Neighbourhood of puncture in \ : ω = dr r α + β, α, β Ω ( ), α β n 1 0 S 1 (N) L L R Well-defined wrapped Fukaya category? How to take and full GC structure into account? C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

15 Bibliography. Gualtieri, C.Kirchhoff-Lukat. Lagrangian branes with boundary and symplectic methods for stable generalized complex manifolds, in preparation. G. Cavalcanti,. Gualtieri. Stable generalized complex structures, 2015 G. Cavalcanti, R. Klaasse. Fibrations and stable generalized complex structures, Gualtieri. Generalized complex geometry, 2003 V. Guillemin, E. iranda, A. Pires. Symplectic and Poisson geometry on b-manifolds, 2012 C. Kirchhoff-Lukat (Cambridge) Lagrangians in elliptic and log sympl. mfds / 15

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