Smooth categories and motivic measures

• Applicant: Dr. Olaf Schnürer
• Subject Area: Mathematics
• Term: from 2014 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 259130191

Non-commutative geometry studies differential graded (dg) categories instead of geometric objects. This is justified and motivated by the fact that any complex algebraic variety gives rise to several dg categories of sheaves describing interesting derived categories. The dictionary between the geometric and the dg language is, unfortunately, not in a satisfactory condition. Even though notions like smoothness and compactness exist in the dg world, their precise relation to geometry is to be clarified. In the first part of our project we plan to work out these relations and to provide precise translations. To achieve this goal we will use generalized Cech enhancements. These questions are motivated by our previous collaboration with Professor Valery Lunts in which we constructed motivic measures using categories of matrix factorizations. A motivic measure is a ring morphism from a Grothendieck ring of varieties to a suitable target ring. The target ring of our measure is a Grothendieck ring of smooth dg categories. In the second part of our project we want to compare our measure with two other motivic measures. On the one hand with the known motivic measure that sends a smooth projective complex variety to the dg version of its bounded derived category, and on the other hand with a motivic measure that we plan to construct using motivic vanishing fibers.

• DFG Programme: Research Fellowships
• International Connection: USA
• Participating Institution: University of Indiana ; Department of Mathematics
• Host: Professor Dr. Valery Lunts