Non-liftable Calabi-Yau spaces

We construct many new non-liftable three-dimensional Calabi-Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi-Yau space over some number field as introduced by the first author and D. van Straten.Comment: 16 pages, 5 tables; v2: minor corrections and addition

Fourier Mukai Transforms and Applications to String Theory

We give an introductory review of Fourier-Mukai transforms and their application to various aspects of moduli problems, string theory and mirror symmetry. We develop the necessary mathematical background for Fourier-Mukai transforms such as aspects of derived categories and integral functors as well as their relative version which becomes important for making precise the notion of fiberwise T-duality on elliptic Calabi-Yau threefolds. We discuss various applications of the Fourier-Mukai transform to D-branes on Calabi-Yau manifolds as well as homological mirror symmetry and the construction of vector bundles for heterotic string theory.Comment: 52 pages. To appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. Minor changes, reference of conjecture in section 7.5 changed, references update

Stability conditions on Kuznetsov components

We introduce a general method to induce Bridgeland stability conditions on semiorthogonal decompositions. In particular, we prove the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of many Fano threefolds (including all but one deformation type of Picard rank one), and of cubic fourfolds. As an application, in the appendix, written jointly with Xiaolei Zhao, we give a variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.Comment: 52 pages. Appendix about the Torelli theorem for cubic fourfolds by A. Bayer, M. Lahoz, E. Macri', P. Stellari, and X. Zhao. v2: main results also for characteristic p; updated discussion about related wor