The anticanonical complex - a combinatorial tool for Fano varieties

• Applicant: Dr. Milena Wrobel
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 520425292

Fano varieties are projective varieties with an ample anticanonical divisor which play an important role in many areas of algebraic geometry as for example in the minimal model program or in mirror symmetry. In the toric case, there is a one-to-one correspondence between Fano varieties and the so-called Fano polytopes, which opens the possibility to describe algebraic geometric properties of the variety in purely combinatorial terms. The anticanonical complex is a polyhedral complex that has been introduced to extend the features of the Fano polytope to wider classes of varieties and so far has been successfully used as a combinatorial tool for detecting singularity types as they appear for instance in the context of the minimal model program. The intention of this project is to develop further combinatorial methods based on the anticanonical complex by mimicking the toric case where possible. For this, we intend to give an intrinsic convex-geometric description of anticanonical complexes, aiming on a deeper understanding of their structure. Moreover, we aim on linking convex-geometric properties of these polyhedral complexes to geometric ones of the corresponding Fano varieties. Concrete applications will be classification results for Fano varieties with certain constraints on their geometry, as for instance their Gorenstein index, which will be obtained by classifying polyhedral complexes.

• DFG Programme: Research Grants