Differential forms on singular spaces in arbitrary characteristic

• Applicant: Dr. Patrick Graf
• Subject Area: Mathematics
• Term: from 2017 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 364017874

An algebraic variety is given locally as the zero set of finitely many polynomials with coefficients in a field. It makes a substantial difference whether that field has characteristic zero (e.g. the complex numbers) or positive characteristic (i.e. 1+1+...+1=0).A variety is called regular if locally it looks like affine space. In classification theory, also singular varieties are interesting. We investigate reflexive differential forms on varieties with mild singularities, i.e. differential forms on the smooth locus. More precisely, we consider the question of their extendability to a resolution of singularities.In positive characteristic, F-pure and strongly F-regular singularities are relevant here. In general, extension does not hold, but maybe extension with logarithmic poles does hold. This would contribute to a better understanding of the failure of the Lipman-Zariski conjecture in positive characteristic: there are singular varieties with locally free tangent sheaf.In characteristic zero we investigate extendability of reflexive pluri-differential forms, i.e. sections of tensor powers of the cotangent sheaf. Here extendability with logarithmic poles should also hold. This would yield a geometric criterion for a rationally connected variety not to carry any reflexive pluri-differentials.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Karl Schwede, Ph.D.