Commentarii Mathematici Helvetici
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The symplectic topology of Ramanujam's surface
Paul Seidel (1) and Ivan Smith (2)
(1) Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, MA 02139-4307, CAMBRIDGE, UNITED STATES(2) Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WB, CAMBRIDGE, UNITED KINGDOM
Ramanujam's surface $M$ is a contractible affine algebraic surface which is not homeomorphic to the affine plane. For any $m>1$ the product $M^m$ is diffeomorphic to Euclidean space ${mathbb R}^{4m}$. We show that, for every $m>0$, $M^m$ cannot be symplectically embedded into a subcritical Stein manifold. This gives the first examples of exotic symplectic structures on Euclidean space which are convex at infinity. It follows that any exhausting plurisubharmonic Morse function on $M^m$ has at least three critical points, answering a question of Eliashberg. The heart of the argument involves showing a particular Lagrangian torus $L$ inside $M$ cannot be displaced from itself by any Hamiltonian isotopy, via a careful study of pseudoholomorphic discs with boundary on $L$.
Keywords: Stein manifold, contractible affine surface, exotic symplectic structure, subcritical Stein manifold