THE SYMPLECTIC TOPOLOGY OF RAMANUJAM'S SURFACE

By P. Seidel, I. Smith. 2004

53 citations recorded by the source corpus.

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 $L$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9e1482d9c9bfb19e07eba632013b965fea217791

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In Mathematical Physics

Citations in the corpus

Altering symplectic manifolds by homologous recombination (2010, background): “Some other references covering similar basic material are [20, 19, 53, 6] (there is also a forthcoming book [13], which looks likely”
Lagrangian spheres, symplectic surfaces and the symplectic mapping class group (2010, unknown): “1 in [9] and Lemma 6 in [52], (U0, J0, φh) is a complete Stein manifold of finite type with Kähler form ωh = −d(dφh ◦ J0).”
Legendrian fronts for affine varieties (2016, background): “[26, 59, 73] but the underlying Stein structures are not quite understood; see [54, 55, 66] for progress in this direction.”
The diffeomorphism type of symplectic fillings (2016, methodology): “Stein manifolds of this kind exist by the work of Seidel–Smith [41] and McLean [31]: in all even dimensions 2n ≥ 8 there are infinitely many distinct finite type Stein manifolds diffeomorphic to R2n; they all have Stein handle decompositions involving critical handles.”
A biased view of symplectic cohomology (2007, background): “2, applied to the crossing torus introduced in [49], shows that this surface has nonzero symplectic cohomology.”