[Paper Review] Fukaya categories and deformations

Paul Seidel|arXiv (Cornell University)|Jun 15, 2002

Homotopy and Cohomology in Algebraic TopologyMathematics15 references89 citations

TL;DR

This paper proposes a deformation-theoretic framework to relate the Fukaya category of a Calabi-Yau projective variety $X$ to that of its affine open subset $M = X \setminus D$, where $D$ is a smooth hyperplane section. By deforming the Fukaya category of $M$ via a formal parameter $t$ encoding intersection numbers with $D$, the construction yields a new $A_\infty$-category $\mathcal{F}(M \subset X)$ that interpolates between $\mathcal{F}(M)$ and $\mathcal{F}(X)$, with a conjectural equivalence linking the derived category of the generic fiber to $D^\pi(\mathcal{F}(X))$. The key contribution is a strategy to compute $D^\pi(\mathcal{F}(X))$ via deformation of $\mathcal{F}(M)$, under finiteness and split-generation assumptions.

ABSTRACT

This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.

Motivation & Objective

• To develop a deformation-theoretic method for computing the derived Fukaya category $D^\pi(\mathcal{F}(X))$ of a Calabi-Yau projective variety $X$.
• To relate the Fukaya category of the affine open subset $M = X \setminus D$ to that of the full $X$ via an $A_\infty$-deformation over $\mathbb{Q}[[t]]$.
• To conjecture that the generic fiber of this deformation recovers $D^\pi(\mathcal{F}(X))$ up to parameter reparametrization.
• To establish conditions under which the deformation space of $\mathcal{F}(M)$ is finite-dimensional, enabling computability.

Proposed method

• Use symplectic cohomology $SH^*(M)$ as a finite-dimensional invariant of the affine manifold $M = X \setminus D$, leveraging the Bott-Morse spectral sequence to compute its cohomology.
• Construct a deformation of the Fukaya category $\mathcal{F}(M)$ by introducing a formal parameter $t$ that tracks holomorphic polygons intersecting the divisor $D$ with multiplicity $k$ in the $t^k$ term.
• Define the $A_\infty$-category $\mathcal{F}(M \subset X)$ as a deformation of $\mathcal{F}(M)$ over $\mathbb{Q}[[t]]$, where composition maps encode holomorphic polygons in $X$ intersecting $D$.
• Use the Hochschild cohomology $HH^*(\mathcal{F}(M), \mathcal{F}(M))$ to classify $A_\infty$-deformations, with finite dimensionality implying uniqueness up to parameter reparametrization.
• Apply the spectral sequence (1) to show that $\dim SH^2(M) \leq b_2(X)$, establishing finiteness of deformation spaces.
• Conjecture that the derived category of the generic fiber $\mathcal{F}(M \subset X)_{\text{gen}}$ tensored with the Novikov ring $\Lambda_t$ is equivalent to $D^\pi(\mathcal{F}(X))$.

Experimental results

Research questions

• RQ1 Can the derived Fukaya category $D^\pi(\mathcal{F}(X))$ of a Calabi-Yau projective variety $X$ be computed via deformation of the Fukaya category of its affine open subset $M = X \setminus D$?
• RQ2 Under what conditions is the Hochschild cohomology $HH^2(\mathcal{F}(M), \mathcal{F}(M))$ finite-dimensional, ensuring uniqueness of $A_\infty$-deformations?
• RQ3 Is there a canonical equivalence between the derived category of the generic fiber of the deformation $\mathcal{F}(M \subset X)$ and $D^\pi(\mathcal{F}(X))$?
• RQ4 How does the symplectic cohomology $SH^*(M)$ relate to the geometry of $X$ and its divisor $D$, particularly in terms of spectral sequences and Betti numbers?
• RQ5 Can the deformation $\mathcal{F}(M \subset X)$ be constructed explicitly using holomorphic polygons intersecting $D$, and does it interpolate between $\mathcal{F}(M)$ and $\mathcal{F}(X)$?

Key findings

• The symplectic cohomology $SH^*(M)$ of the affine manifold $M = X \setminus D$ is finite-dimensional, with $\dim SH^2(M) \leq b_2(X)$, under the assumption $\dim_{\mathbb{C}}(X) > 2$.
• A Bott-Morse spectral sequence computes $SH^*(M)$ with $E_1^{pq}$ terms given by $H^q(M)$ for $p=0$ and $H^{q+3p}(\partial M)$ for $p<0$, providing a computational tool.
• The $A_\infty$-deformation $\mathcal{F}(M \subset X)$ of $\mathcal{F}(M)$ over $\mathbb{Q}[[t]]$ is constructed by encoding holomorphic polygons in $X$ intersecting $D$ with multiplicity $k$ in the $t^k$ coefficient of composition maps.
• If $HH^2(\mathcal{F}(M), \mathcal{F}(M)) \cong \mathbb{Q}$, then any nontrivial $A_\infty$-deformation is unique up to reparametrization of $t$, implying a one-dimensional versal deformation space.
• Conjecture 5 posits a canonical equivalence $D^\pi(\mathcal{F}(M \subset X)_{\text{gen}} \otimes_{\mathbb{Q}[t^{-1}][[t]]} \Lambda_t) \cong D^\pi(\mathcal{F}(X))$, suggesting that $D^\pi(\mathcal{F}(X))$ can be recovered from the generic fiber of the deformation.
• For $X \subset \mathbb{CP}^{n+1}$ a degree $n+2$ hypersurface with $n \geq 3$, $D^\pi(\mathcal{F}(M))$ is split-generated by finitely many objects, making $Tw^\pi(\mathcal{F}(M))$ computationally accessible under finiteness assumptions.