New developments in the moduli theory of decorated sheaves

• Applicant: Professor Dr. Alexander Schmitt
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 471183228

Moduli spaces are fundamental objects of algebraic geometry. They solve numerous classification problems. An interesting class of examples is formed by "decorated'' principal bundles, i.e., principal bundles endowed with an extra structure, such as a section in an associated vector bundle. Examples of decorated principal bundles comprise Higgs bundles, holomorphic triples, and coherent systems. A successful tool for constructing moduli spaces has been geometric invariant theory which was developed by Mumford, based on fundamental work by Hilbert. Several modern moduli problems, e.g., those concerning Bridgeland semistable objects in a triangulated category, are not suited for the approach via geometric invariant theory.In this project, we will investigate two related classification problems for decorated principal bundles which exhibit interesting new phenomena or require new techniques. The first one deals with coherent systems for decorated principal bundles on curves. This emerging concept has potential applications to the geometry of moduli spaces of Higgs bundles and Brill--Noether theory of vector bundles. Although there is a classical geometric invariant theory set-up for the construction of the moduli spaces, there is a surprising restriction which seems worthwhile investigating from several perspectives. The second problem deals with moduli spaces of Bridgeland semistable objects in derived categories of quiver sheaves. We will look at holomorphic chains on curves and surfaces. New sophisticated techniques have to be applied to obtain moduli spaces. This work will also lead to a better understanding of the "classical'' moduli spaces. We will also consider important techniques for investigating the geometry of the moduli spaces, such as Hitchin maps and wall crossing.

• DFG Programme: Research Grants
• International Connection: Italy
• Cooperation Partner: Dr. Alejandra Rincón Hidalgo