9. Differential Embedding Problems over Complex Function Fields

  • Annette Bachmayr Mathematisches Institut der Universit\"at Bonn, 53115 Bonn, Germany
  • David Harbater Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
  • Julia Hartmann Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
  • Michael Wibmer Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
Keywords: Differential algebra, linear algebraic groups and torsors, patching, Picard-Vessiot theory, embedding problems, inverse differential Galois problem, Riemann surfaces


We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new insight on the structure of the absolute differential Galois group, i.e., the fundamental group of the underlying Tannakian category.


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