Analysis auf singulären komplexen Räumen

• Applicant: Professor Dr. Jean Ruppenthal
• Subject Area: Mathematics
• Term: from 2011 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 193570782

• Final Report Year: 2017

Final Report Abstract

The main purpose of the research project was to gain a better understanding for the natural interplay of analysis and geometry on singular complex spaces by • the study of differential operators (connections between properties of the ∂ and the ∂-Neumann operator and the geometry/topology of the underlying space), • the development of analytic tools (L2-theory, integral formulas). Analytic methods have led to fundamental advances in geometry on complex manifolds, but are still not very well developed for singular complex spaces. The main achievements are • the development of an L2-theory for the ∂-operator, including • the study of L2-canonical sheaves (resolution, adjunction formula) and extension of L2-cohomology classes, • analytic realizations of Grothendieck-Serre-duality, • the investigation of complex Laplace operators (spectrum, compactness) and of the ∂-Neumann problem (subelliptic estimates), and • the development of integral formulas on singular complex spaces, as well as the investigation of auxiliary singular analytic structures appearing in this context: • (proper) modifications of coherent analytic sheaves, the Grauert–Riemenschneider canonical sheaf with values in coherent analytic sheaves, and • Chern forms for singular Hermitian metrics. An important, crucial insight was the discovery that the L2 -theory for the ∂-operator and integral formulas are particularly fruitful on spaces with canonical singularities which play a prominent role in the Minimal Model Program. The further investigation of this connection is a very interesting objective for future research.

Publications

• Koppelman formulas on affine cones over smooth projective complete intersections, Indiana University Mathematical Journal
  R. Lärkäng, J. Ruppenthal
  
• L2 -Serre duality on singular complex spaces and rational singularities, International Mathematics Research Notices
  J. Ruppenthal
  
• L2-properties of the ∂ and the ∂-Neumann operator on spaces with isolated singularities, Mathematische Annalen 359 (2014), no. 3–4, 803–838
  N. Øvrelid, J. Ruppenthal
  
• L2-theory for the ∂-operator on compact complex spaces, Duke Mathematical Journal 163 (2014), no. 15, 2887–2934
  J. Ruppenthal
  
• Subelliptic estimates for the ∂-problem on a singular space, Journal of Geometric Analysis 24 (2014), no. 4, 1844–1859
  D. Ehsani, J. Ruppenthal
  
• Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L2-cohomology classes, Indiana University Mathematical Journal 64 (2015), no. 2, 533–558
  J. Ruppenthal, H. Samuelsson Kalm, E. Wulcan
  
• Koppelman formulas on the A1 -singularity, Journal of Mathematical Analysis and Applications 437 (2016), 214–240
  R. Lärkäng, J. Ruppenthal
  
• Parabolicity of the regular locus of complex varieties, Proceedings of the AMS 144 (2016), 225–233
  J. Ruppenthal
  
• Explicit Serre duality on complex spaces, Advances in Mathematics 305 (2017), 1320–1355
  J. Ruppenthal, H. Samuelsson Kalm, E. Wulcan
  
• Modifications of torsion-free coherent analytic sheaves, Annales de l’Institut Fourier 67 (2017), 237–265
  J. Ruppenthal, M. Sera