Cluster tilting is a well studied notion appearing in various subjects in representation theory, categorification of cluster algebras, higher Auslander-Reiten theory, and non-commutative crepant resolutions.

The aim of this talk is to explain that cluster tilting subcategories naturally appear also in projective geometry. For example, a projective space of dimension d has a cluster tilting subcategory consisting of (direct sums of) all line bundles, by Horrocks splitting criterion. We show that if a projective variety X has a tilting bundle T such that dim X=gl.dim End(T) and the anti-canonical bundle is ample, then X has a cluster tilting subcategory. We give some examples from surfaces, flag varieties and Geigle-Lenzing projective spaces. We explain applications including a construction of Calabi-Yau algebras.

This talk is based on joint works with Buchweitz and Hille, and with Herschend, Minamoto and Oppermann.