Tags: #geomtop/Morse-theory #projects/my-talks #projects/notes/reading

Actual Talk

Goal: reduce the complex geometry of Stein manifolds (hard) to the symplectic geometry of Weinstein manifolds (less hard). Study the space of structures up to homotopy.

Weinstein to Stein

Theorem: there is a “weak homotopy inverse“to \(\mathfrak{M}_\phi\), i.e. given a Weinstein structure \((\omega, X, \phi)\) there is a Stein structure \((J, \phi)\) such that \(\mathfrak{M}(J, \phi)\) is Weinstein-homotopic (homotopy-equivalent?) to \((\omega, X, \phi)\) rel \(\phi\), i.e. \(\phi\) is fixed through the homotopy.

Questions

Do these structures satisfy an h principle? Origins (Gromov, 1970): given a PDE, a “partially specified” solution can be deformed into an actual solution. The former have topological properties (and are amenable to algebraic topology), the latter analytic. Recast, the inclusion of the space of “partially specified” solutions into the space of solutions is a weak homotopy equivalence.