Symplectic cohomology relative to a smooth anticanonical divisor

For a monotone symplectic manifold and a smooth anticanonical divisor, there is a formal deformation of the symplectic cohomology of the divisor complement, defined by allowing Floer cylinders to intersect the divisor. We compute this deformed symplectic cohomology, in terms of the ordinary cohomology of the manifold and divisor; and also describe some additional structures that it carries.