A non exceptionally general cyclic point configuration and Floer theory of the non-exact torus

Abstract

The main results of this thesis are twofold: one concerns the existence of a non-exceptional three-dimensional cyclic point configuration, while the other concerns the Floer-theoretic structure of the non-exact torus. The Fukaya category has emerged as one of the central objects in modern symplectic geometry and homological mirror symmetry. In particular, the wrapped Fukaya category of a cotangent bundle encodes deep geometric and algebraic information about the underlying manifold and its symplectic structure. The purpose of this work is to investigate how this picture changes when the cotangent bundle is equipped with a non-exact symplectic form. Specifically, we study the Lagrangian Floer theory of a cotangent fiber inside the cotangent bundle of the two–torus endowed with a perturbed symplectic form. Our first goal is to analyze the resulting moduli spaces of pseudo-holomorphic curves and to establish structural properties of the associated Floer complex. In particular, we show that the Floer differential vanishes for the cotangent fiber, so that the Floer cohomology coincides with the underlying complex. The main result of this work is an explicit computation of the Floer product for the cotangent fiber in the presence of a perturbed symplectic form. While in the exact case this algebra is known to be the commutative ring of Laurent polynomials in two variables, we show that the perturbation produces a non commutative deformation whose commutation relations are determined directly by the cohomology class of the base symplectic form. The result included here is folklore and experts have speculated on it. However, a specific proof using the methods described here has not been written. We also study secondary polytopes and regular subdivisions associated with cyclic point configurations. A central question is whether cyclic configurations satisfy exceptional generality, a property asserting that the dimensions of spaces of piecewise affine functions on subdivisions behave in the expected generic manner. While cyclic configurations in dimension two exhibit this property, the situation changes in higher dimensions. We construct an explicit three-dimensional cyclic point configuration that is not exceptionally general. More precisely, we exhibit a regular subdivision of the configuration for which the space of piecewise affine functions has dimension larger than the dimension predicted by genericity. The construction is analyzed through the combinatorics of the subdivision and the affine compatibility conditions imposed on its maximal cells. This provides a concrete obstruction to extending the two-dimensional exceptional generality phenomenon to three-dimensional cyclic configurations. The example shows that the combinatorial structure of a cyclic configuration alone does not force exceptional generality and highlights additional geometric constraints that arise in the study of secondary polytopes in dimension three.