L2-Dolbeault cohomology and canonical sheaves on singular complex spaces

• Applicant: Dr. Martin Sera
• Subject Area: Mathematics
• Term: from 2016 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 317654309

The classification and the study of complex manifolds are central in complex analysis. Both, the Dolbeault differential operator and the canonical line bundle, are essential therein since they represent many different geometrical properties. Whereas both tools are well-understood on complex manifolds, there are many open questions in the singular setting.Originating from different characterizations of the canonical bundle, which is as well represented by the canonical sheaf, in the smooth case, there exist different possibilities to define the canonical sheaf on complex spaces, which are not equivalent in general. Yet, each of them represents different geometrical properties. So far, there is not much known about the relation between them. Therefore, it is a goal of the presented project to figure out the relation between the different definitions of canonical sheaves on singular complex spaces. To do so, we would like to study Hermitian metrics on multiplier ideal sheaves.Since Takegoshi's relative vanishing theorem is very helpful to comprehend canonical sheaves, we want to generalize Takegoshi's result to line bundles with semi-positive singular metrics. Furthermore, we are going to prove local L²-vanishing results on singular complex spaces to infer the equivalence of L²- and L²-loc-Dolbeault cohomologies using Grauert's bumping method.

• DFG Programme: Research Fellowships
• International Connection: Sweden
• Host: Dr. Elizabeth Wulcan