Nakajima--Grojnowski operators and derived autoequivalences of Hilbert schemes of points on surfaces

• Applicant: Dr. Andreas Krug
• Subject Area: Mathematics
• Term: from 2014 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 257237495

For every smooth algebraic surface X there is a series of higher-dimensional smooth varieties, namely the Hilbert schemes of points on X. A classical invariant of varieties is given by their cohomology. A central result in the theory of Hilbert schemes of points on surfaces is the description of their cohomology by means of the Nakajima--Grojnowski operators. The derived category is a more modern invariant of a variety. These categories and in particular their groups of autoequivalences have been object to intensive research in the area of algebraic geometry for about the last 15 years. One reason for the interest lies in conjectured connections to physical string theory by means of the homological mirror symmetry. The goal of the first part of the project is the construction of analogues of the Nakajima--Grojnowski operators as P-functors on the level of the derived categories. A P-functor, as defined by Addington, is a Fourier--Mukai transform of a special shape. Every P-functor automatically yields an autoequivalence of its target category. There is reason to hope that these new autoequivalences allow to give a description of the full group of autoequivalences of the Hilbert schemes in easier cases. It is the main goal of the second part of the project to achieve such descriptions.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Participating Institution: University of Warwick ; Mathematics Institute
• Host: Professor Dr. Miles Reid