Minifolds and Phantoms

Sergey Galkin; Ludmil Katzarkov; Anton Mellit; Evgeny Shinder

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arXiv · 1305.4549

Minifolds and Phantoms

Sergey Galkin·Ludmil Katzarkov·Anton Mellit·Evgeny Shinder

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Abstract

A minifold is a smooth projective $n$-dimensional variety such that its bounded derived category of coherent sheaves $\D^b(X)$ admits a semi-orthogonal decomposition into an exceptional collection of $n+1$ exceptional objects. In this paper we classify minifolds of dimension $n \leq 4$. We conjecture that the derived category of fake projective spaces have a similar semi-orthogonal decomposition into a collection of $n+1$ exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group. We construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.

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Sergey Galkin, Ludmil Katzarkov, Anton Mellit, Evgeny Shinder. 2013-10-17. Minifolds and Phantoms. https://arxiv.org/abs/1305.4549

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