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Exceptional collections in categories of singularities of 3-dimensional Landau-Ginzburg models. (English. Russian original) Zbl 1375.13020

Russ. Math. Surv. 68, No. 6, 1136-1138 (2013); translation from Usp. Mat. Nauk 68, No. 6, 173-174 (2013).
This is an announcement of a proof for a conjecture of Orlov in dimension less than or equal to 3. Given an invertible polynomial, it constructs a quiver with relations whose derived category of modules is equivalent to the derived category of singularities of the given polynomial.
The key step is constructing a full strongly exceptional collection \((E_1,\ldots,E_m)\) of the derived category of singularities. Then by Bondal, the morphisms among \(E_i\) produce the quiver with the required property. There are two cases in dimension three: chain type and loop type. An exceptional collection is explicitly given in each case.
The result is important to understand singularity theory and mirror symmetry for invertible polynomials.

MSC:

13D09 Derived categories and commutative rings
16E35 Derived categories and associative algebras
14J33 Mirror symmetry (algebro-geometric aspects)
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