Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients

Abstract

Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $CO_λ: QH^*(X) \to HH^*(\mathcal{F}(X)_λ)$ to a variant of the length-zero closed-open map on $L$ incorporating $\mathbf{k}[H_1(L; \mathbb{Z})]$ coefficients, denoted $CO^0_\mathbf{L}$. The former is categorically important but very difficult to compute, whilst the latter is geometrically natural and amenable to calculation. We further show that, after a suitable completion, injectivity of $CO^0_\mathbf{L}$ implies injectivity of $CO_λ$. Via Sheridan's version of Abouzaid's generation criterion, this gives a powerful tool for proving split-generation of the Fukaya category. We illustrate this by showing that the real part of a monotone toric manifold (of minimal Chern number at least 2) split-generates the Fukaya category in characteristic 2. We also give a short new proof (modulo foundational assumptions in the non-monotone case) that the Fukaya category of an arbitrary compact toric manifold is split-generated by toric fibres.

Figures (10)

• Figure 1: The $A_\infty$-operation $\mu_\mathcal{F}^5$ (left), the squifferential on $\operatorname{\mathcal{LM}}_{}(K) = \operatorname{CF}_S^*(\mathbf{L}, K)$ (centre), and the component $\operatorname{\mathcal{LM}}_{}^3$ (right).
• Figure 2: Codimension-$1$ degenerations of a $\operatorname{\mathcal{LM}}_{}^3$ disc.
• Figure 3: The operations $\mu_\mathbf{L}$ on $\operatorname{CF}_S^*(\mathbf{L}, \mathbf{L})^\mathrm{op}$.
• Figure 4: The cocycles $\sigma$ (left) and $\sigma_\mathbf{L}$ (right), representing $\operatorname{\mathcal{CO}}_\lambda(\alpha)$ and $\operatorname{\mathcal{CO}}^0_\mathbf{L}(\alpha)$ respectively.
• Figure 5: The discs defining $\Theta(c_l, \dots, c_1)(a_k, \dots, a_1)(x)$.

Theorems & Definitions (183)

• Theorem 1.1: Generation criterion, Abouzaid AbouzaidGeometricCriterion, Sheridan SheridanFano
• Remark 1.2
• Remark 1.3
• Theorem A: \ref{['propThetaHom', 'lemThetaModuleAction', 'corThetaUnital']}
• Theorem B: \ref{['corThmBCommutes']}
• Remark 1.4
• Theorem C: \ref{['thmEMcomparison', 'propThmC']}
• Theorem D: \ref{['propTonkonogInterpretation']}
• Theorem E: \ref{['propKerFrob']}
• Remark 1.5