Document Zbl 1255.14032

Floer cohomology and pencils of quadrics. (English) Zbl 1255.14032

Invent. Math. 189, No. 1, 149-250 (2012).

This paper is an important contribution to the the broader program of Kontsevich’s Homological Mirror Symmetry conjecture. It relates the Fukaya category \( F(\Sigma_g)\) of a closed surface \(\Sigma_g\) of genus \(g\geq2\) and the Fukaya category \( F(Q_0^{2g}\cap Q_1^{2g})\) of the complete intersection \(Q_0 \cap Q_1\) of two smooth quadric hypersurfaces in \(\mathbb P^{2g+1}\). The main result of the paper, stated in Theorem 1.1, is the following \(\mathbb C\)-linear equivalence of \(\mathbb Z_2\)-graded split-closed triangulated categories: \[ D^{\pi} F(\Sigma_g) \simeq D^{\pi} F(Q_0^{2g} \cap Q_1^{2g};0), \] where \(D^{\pi} C\) denotes the cohomological category \(H(Tw^{\pi} C)\) underlying the split-closure of the category of twisted complexes of an \(A_{\infty}\)-category \( C\) and \( F(\bullet;0)\) denotes the summand corresponding to the \(0\)-eigenvalue of quantum cup-product by the first Chern class. The proof constructs equivalences with quasi-isomorphic images \[ D^{\pi} F(\Sigma_g) \;\hookrightarrow \;D^{\pi} F(Z) \;\hookleftarrow \;D^{\pi} F(Q_0\cap Q_1;0), \] where the relative quadric \(Z\) is given by blowing up \(\mathbb P^{2g+1}\) along \(Q_0\cap Q_1\).
The author also gives nice applications of the main result to representation varieties and instanton Floer homology, making the paper of wide interest to experts on geometric topology and mathematical physics. The paper is well organized: Section 1 is a very readable summary of the motivation and background of the paper, as well as an outline of the proof. Section 2 contains the applications to topology, assuming the main theorem. Section 3 is an exposition of Fukaya category. Finally the main theorem is proved in Sections 4 and 5.
The paper could be read together with [P. Seidel, J. Algebr. Geom. 20, No. 4, 727–769 (2011; Zbl 1226.14028)].

Reviewer: Hao Xu (Cambridge)

Cited in 33 Documents

MSC:

14J33	Mirror symmetry (algebro-geometric aspects)
53D40	Symplectic aspects of Floer homology and cohomology
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30	Derived categories, triangulated categories (MSC2010)

Citations:

Zbl 1237.53080; Zbl 1226.14028

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References:

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