In this paper, we introduce the interested reader to homological mirror symmetry. After recalling a little background knowledge, we tackle the simplest cases of homological mirror symmetry: curves of genus zero and one. We close by outlining the current state of the field and mentioning what homo-logical mirror symmetry has to say about other aspects of… (More)
We investigate sheaves supported on the zero section of the total space of a locally-free sheaf E on a smooth, projective variety X when E satisfies rank E E ∼ = ω X. We rephrase this construction using the language of A∞-algebra and provide a simple characterisation of the case E = ω X .
We give a new upper bound for the generation time of a tilting object and use it to verify, in some new cases, a conjecture of Orlov on the dimension of the derived category of coherent sheaves on a smooth variety.
We extend Orlov's result on representability of equivalences to schemes projective over a field. We also investigate the quasi-projective case.
We extend the result of Bondal and Orlov on reconstruction of a variety from its derived category to the case when one of the varieties involved is Gorenstein and Fano or of general type and the other possesses enough locally-free coherent sheaves.