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If    (  X  ,  ω  )    is a closed symplectic manifold, and   Σ   is a smooth symplectic submanifold Poincaré dual to a positive multiple of   ω  , then    X  ∖  Σ    can be completed to a Liouville manifold    (  W  ,  d  λ  )   .  Under monotonicity assumptions on   X   and on   Σ  , we construct a chain complex whose homology computes the symplectic homology of   W  . We show that the differential is given in terms of Morse contributions, Gromov–Witten invariants of   X   relative to   Σ   and Gromov–Witten invariants of   Σ  .  We use a Morse–Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to   Σ  .

Symplectic homology of complements of smooth divisors