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...ll abelian varieties. (Recent work of Orlov [41] describes Auteq(Db (X)) completely in this case.) Our own interest in self-equivalence groups comes from Kontsevich’s homological mirror conjecture in =-=[30]-=-. One consequence of this conjecture is that, for Calabi-Yau varieties to which mirror symmetry applies, the group Auteq(Db (X)) should be related to the symplectic automorphisms of the mirror manifol...

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...ification exchanges the two foliations, so that the Kähler moduli space of X becomes identified with the moduli of complex structures on ˇ X and vice versa. Kontsevich’s homological mirror conjecture =-=[13]-=- predicts that the derived category D(X) is equivalent to the derived Fukaya category D Fuk( ˇ X). Roughly speaking, this equivalence is expected to take the subcategory P(φ) ⊂ D(X) at a particular po...

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...arities is proper then all Hom’s between objects are finite-dimensional vector spaces (Corollary 1.24). The investigation of such categories is inspired by the Homological Mirror Symmetry Conjecture (=-=[21]-=-). Works on topological string theory are mainly concerned with the case of N=2 superconformal sigma-models with a Calabi-Yau target space. In this case the field theory has two topologically twisted ...

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...hown here is that there is a homotopy action of chains on all-genus moduli spaces of Riemann surfaces on the Hochschild chains. Results of this form were conjectured by Kontsevich as far back as 1994 =-=[8]-=-, and have also been conjectured by Getzler, Segal, Kapustin and Rozansky [4].... A different approach to the proof of this result has been given by Kontsevich [9] using the ribbon graph decomposition...

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...To find such structure is essential for the extension of the predictions of Mirror Symmetry in the dimensions n > 3. 0.1 Background philosophy. The Mirror Symmetry conjecture, as it was formulated in =-=[K1]-=-, states that the derived category of coherent sheaves on a Calabi-Yau manifold M is equivalent to the derived category constructed from (conjectured) Fukaya category associated with the dual Calabi-Y...

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...see, e.g. [19]). The investigation of triangulated categories of singularities is not only connected with a study of singularities but is mainly inspired by the Homological Mirror Symmetry Conjecture =-=[20]-=-. More precisely, the objects of these categories are directly related to D-branes of type B (B-branes) in Landau-Ginzburg models. Such models arise as a mirrors to Fano varieties [15]. For Fano varie...