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• W1937840078 abstract "In arXiv:1503.00125, the authors proved that every complete intersection smooth projective variety $Y$ is a Fano visitor, i.e. its derived category $D^b(Y)$ is equivalent to a full triangulated subcategory of the derived category $D^b(X)$ of a smooth Fano variety $X$, called a Fano host of $Y$. They also introduced the notion of Fano dimension of $Y$ as the smallest dimension of a Fano host $X$ and obtained an upper bound for the Fano dimension of each complete intersection variety. In this paper, we provide a Hodge-theoretic criterion for the existence of a Fano host which enables us to determine the Fano dimensions precisely for many interesting examples, such as low genus curves, quintic Calabi-Yau 3-folds and general complete intersection Calabi-Yau varieties. Next we initiate a systematic search for more Fano visitors. We generalize the methods of arXiv:1503.00125 to prove that smooth curves of genus at most 4 are all Fano visitors and general curves of genus at most 9 are Fano visitors. For surfaces and higher dimensional varieties, we find more examples of Fano visitors and raise natural questions. An interesting recent discovery is the existence of quasi-phantom subcategories in derived categories of some surfaces of general type with $p_g=q=0$. But no examples of Fano with quasi-phantom have been found. Applying a result of Orlov and Cayley's trick used in arXiv:1503.00125, we construct a Fano orbifold whose derived category contains a quasi-phantom category." @default.
• W1937840078 created "2016-06-24" @default.
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• W1937840078 date "2015-04-29" @default.
• W1937840078 modified "2023-09-27" @default.
• W1937840078 title "Fano visitors, Fano dimension and quasi-phantom categories" @default.
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• W1937840078 hasPublicationYear "2015" @default.
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