Symplectic Geometry of Vector Bundle Maps of Tangent Bundles

Essentially Finite Vector Bundles on Varieties with Trivial Tangent Bundle

Let X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle TX is trivial. Let FX : X −→ X be the absolute Frobenius morphism of X. We prove that for any n ≥ 1, the n–fold composition Fn X is a torsor over X for a finite group–scheme that depends on n. For any vector bundle E −→ X, we show that the direct image (Fn X)...

Lecture 2: Symplectic Vector Bundles

Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over R) and ω is a bilinear form which satisfies • Skew-symmetry: for any u, v ∈ V , ω(u, v) = −ω(v, u). • Nondegeneracy: for any u ∈ V , ω(u, v) = 0 ∀v ∈ V implies u = 0. A linear symplectomorphism of a symplectic vector space (V, ω) is a vector space isomorphism ψ : V → V such that ω(ψu, ψ...

Finsler Geometry of Projectivized Vector Bundles

Differential Geometry of Complex Vector Bundles

The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow [28, 27] and the authors [14], before discussing a new approach using family Floer cohomology [10] and the “wrapped Fukaya category”. The latter, inspired by Viterbo’s symplec...