The symplectic topology of Ramanujam's surface

• Autores: Paul Seidel, Ivan Smith
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 80, Nº 4, 2005, págs. 859-881
• Idioma: inglés
• DOI: 10.4171/cmh/37
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 $L$.