The biran decomposition: A symplectic lefshetz hyperplane theorem
Summary:
The symplectic structure of a Kahler manifold decomposes into two pieces: A symplectic disc bundle over a hyperplane section, and an isotropic skeleton. This Biran decomposition is a symplectic refinement of the Lefshetz hyperplane theorem. The proof uses the morse theory of the norm squared of a holomorphic section of the prequantum line bundle. We will use this to prove some lagrangian barrier phemonena. For example, every symplectic ball of radius $1/\sqrt{2}$ embedded in $\mathbb{C} \mathbb{P}^n$ interesects $\mathbb{R} \mathbb{P}^n$
Presented at:
- Student symplectic seminar
🔗 Link to file
Sources:
Paul Biran 2001, LAGRANGIAN BARRIERS AND SYMPLECTIC EMBEDDINGS
- Original source for the Biran decomposition, and my main source this talk.
Olguta Buse, Richard Hind, 2011, Ellipsoid embeddings and symplectic packing stability
- Application of the Biran decomposition to symplectic embedding problems. They prove that any closed, integral kahler manifold has packing stability.
Emmanuel Giroux, 2017, Remarks on Donaldson’s symplectic submanifolds
- Proves the Biran decomposition for non-Kahler manifolds, using asymptotic holomorphic technique. Biran essentially did this in dimension 4 a while ago, but this works in any dimension.