Fourier transforms and a filtration on the Lagrangian cobordism group of tori

TL;DR

This paper introduces a finite filtration on the fibered Lagrangian cobordism group of symplectic tori and constructs a Fourier transform between Fukaya categories, linking it to mirror symmetry and algebraic geometry.

Contribution

It establishes a natural finite filtration on the Lagrangian cobordism group and constructs a Fourier transform between Fukaya categories of dual tori, connecting symplectic and algebraic geometry.

Findings

The fibered Lagrangian cobordism group admits a finite geometric filtration.

A Fourier transform between Fukaya categories of dual tori is constructed.

The filtration is shown to be mirror to the Bloch filtration on Chow groups.

Abstract

Given a polarized tropical affine torus, we show that the fibered Lagrangian cobordism group of the corresponding symplectic manifold admits a natural geometric filtration of finite length. This contrasts with results of Sheridan-Smith in dimension four and the present author in higher dimensions, who showed that such group is infinite-dimensional. In the second half of this paper, we construct a Fourier transform between Fukaya categories of dual symplectic tori. We show that, under homological mirror symmetry, it corresponds to the Fourier transform between derived categories of coherent sheaves of dual abelian varieties due to Mukai. We use this to show how our filtration is mirror to the Bloch filtration on Chow groups of abelian varieties, but the results may be of broader interest.