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Homological mirror symmetry for punctured spheres. (English) Zbl 1276.53089
Kontsevich’s homological mirror symmetry conjecture for pairs of Calabi-Yau varieties “predicts an equivalence between the derived category of coherent sheaves of one variety and the derived Fukaya category of the other”. This conjecture has been proved so far for elliptic curves, the quartic \(K_3\) surface and their products. Later this conjecture was extended for Fano varieties, where in many cases the mirrors are Landau-Ginzburg models: pairs \((X, W)\) consisting of a variety \(X\) and a holomorphic function \(W:X\to\mathbb{C}\) called superpotential.
More recently L. Katzarkov suggested that mirror symmetry also extends to surfaces of general type, many of which also admit mirror Landau-Ginzburg models. The first positive result in this setting was established by P. Seidel for the genus-\(2\) curve [J. Algebr. Geom. 20, No. 4, 727–769 (2011; Zbl 1226.14028)]. The argument used by Seidel was then extended to higher-genus curves, to pairs of pants and their higher-dimensional analogs and to Calabi-Yau hypersurfaces in the projective space.
In the work under review the authors study homological mirror symmetry for an open genus-\(0\) curve \(C\), which is the complement of a finite set of \(n\geq 3\) points in \(\mathbb{P}^1\). They describe a Landau-Ginzburg model mirror \((X(n),W)\) to \(C\), where \(X(n)\) is a non-compact toric 3-fold and \(W:X(n)\to \mathbb{C}\) is a superpotential. Then they consider the wrapped Fukaya category of \(C\) and the associated triangulated category \(D\mathcal{W}(C)\). The main result states that “this triangulated category is equivalent to the triangulated category of singularities of the singular fiber \(W^{-1}(0)\) of \((X(n), W)\).” A similar result in the case of unramified cyclic covers of punctured spheres is also proved.
The idea of the proof is inspired by that used by Seidel for the genus-2 curve. Only the symplectic side of the homological mirror symmetry is considered in this work. The other side is generally considered to be out of reach of current technologies for the studied examples, due to the singular nature of the critical locus of \(W\).

MSC:
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33 Mirror symmetry (algebro-geometric aspects)
53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
18E30 Derived categories, triangulated categories (MSC2010)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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