Symplectic Geometry Seminar

Dylan Cant (Université de Montréal)

Title: On the spectral diameter of the Grassmannians

Abstract: For a compact symplectic manifold 𝑀 there is a spectral pseudometric on the
universal cover of 𝐻𝑎𝑚(𝑀), built from Floer theory. A folklore conjecture
says its diameter should be infinite whenever the symplectic form vanishes
on 𝑝𝑖2(𝑀), on the other hand the presence of symplectic spheres can force
the diameter to be finite, as happens for 𝐶𝑃𝑛. I will discuss what happens
for the complex Grassmannians 𝐺𝑟(𝑘, 𝑛). We show that the spectral diam-
eter of 𝐺𝑟(2, 𝑛) is finite when 𝑛 is prime, and that the spectral diameter of
𝐺𝑟(2𝑘, 2𝑛) is infinite when 𝑘 < 𝑛 (both over a field of characteristic zero; in
nonzero characteristic there is a more refined statement for 𝐺𝑟(2, 𝑛)). The
finiteness comes from the quantum cohomology of the Grassmannian, and
the infiniteness comes from Lagrangian submanifolds. This is joint work
with Habib Alizadeh, Marcelo Atallah and Jianqiao Shang.