Research Interests

• Geometry of higher-dimensional algebraic varieties and Kähler manifolds, in particular
  • Differential forms on singular varieties
  • Vanishing theorems
  • Singularities of the Minimal Model Program
  • Hyperkähler manifolds and Lagrangian fibrations
  • Minimal varieties with trivial canonical bundle


• Symmetries in Complex Algebraic and Kählerian Geometry, in particular
  • Algebraic transformation groups
  • Geometric Invariant Theory and geometry of good quotients
  • Hamiltonian actions of compact and complex-reductive Lie groups
  • Symplectic and Kählerian reduction
  
  
• Applications to moduli problems
• Stability of vector bundles with respect to movable curves
• Moduli spaces for slope-semistable sheaves on higher-dimensional varieties
• Momentum maps and stability of vector bundles

In preparation

• Compact moduli spaces for slope-semistable sheaves on higher-dimensional projective manifolds
  (with Matei Toma)
  the results were announced in Oberwolfach Report 43/2012


• Complex-analytic quotients of algebraic G-varieties


• q-Kodaira vanishing for q-ample line bundles
  (with Alex Küronya)


• Singular projective varieties with numerically trivial first and second Chern classes
  (with Stefan Kebekus and Thomas Peternell)

Preprints

• 1-rational singularities and quotients by reductive groups
  arXiv:0901.3539, 8 pages, 2009.
  (will not be submitted; the results follow easily from those in my later paper ''Rational singularities and quotients by holomorphic group actions'' that appeared in Annali della Scuola Normale Superiore di Pisa; however, the preprint gives an independent and technically simpler proof in the algebraic case and avoids the technical difficulties encountered in the analytic setup)

Publications

• Lagrangian fibrations on hyperkähler fourfolds
  (with Christian Lehn and Sönke Rollenske)
  will appear in Izvestiya: Mathematics.
  Preprint: arXiv:1110.2680


• Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville
  (with Christian Lehn and Sönke Rollenske)
  will appear in Annales scientifiques de l'École normale supérieure.
  Preprint: arXiv:1105.3410
  some of the results of the paper were announced in Oberwolfach Report 47/2011


• Singular spaces with trivial canonical class
  (with Stefan Kebekus and Thomas Peternell)
  will appear in "Minimal models and extremal rays" -
  Proceedings of the conference in honor of Shigefumi Mori's 60th birthday,
  Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo.
  Preprint: arXiv:1110.5250


• Reflexive differential forms on singular spaces - Geometry and Cohomology
  (with Stefan Kebekus and Thomas Peternell)
  Journal für die Reine und Angewandte Mathematik (Crelle's Journal).
  Electronic version - ahead of print: here.
  DOI:10.1515/crelle-2012-0097
  Preprint: arXiv:1202.3243


• Invariant meromorphic functions on Stein spaces
  (with Christian Miebach)
  Annales de l'Institut Fourier (Grenoble) 62 (2012), no. 5, p. 1983-2011.
  Electronic version: here.
  Preprint arXiv:1010.2935
  some of the results of the paper were announced in Oberwolfach Report 39/2010


• Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities
  (with Sönke Rollenske)
  Mathematical Research Letters 18 (2011), no. 6, 1259 - 1269.
  Electronic version: here.
  Preprint arXiv:1012.5940


• Differential Forms on Log Canonical Spaces
  (with Stefan Kebekus, Sándor Kovács, and Thomas Peternell)
  Publications Mathématiques de l'IHÉS 114 (2011), no. 1, 87 - 169.
  DOI:10.1007/s10240-011-0036-0
  Preprint arXiv:1003.2913
  some of the results of the paper were announced in Oberwolfach Report 2/2010


• Rational singularities and quotients by holomorphic group actions
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), Vol. X (2011), issue 2, 413 - 426.
  DOI:10.2422/2036-2145.2011.2.07.
  Preprint arXiv:0906.4623.
  
• Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory
  Advances in Mathematics 224 (2010), 401 - 431.
  DOI:10.1016/j.aim.2009.11.013.
  Preprint arXiv:0812.2643.
• Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties
  (with Stefan Kebekus and Sándor Kovács)
  Compositio Mathematica 146 (2010), 193 - 219.
  DOI:10.1112/S0010437X09004321.
  Preprint arXiv:0808.3647.
• Projectivity of analytic Hilbert and Kähler quotients
  Transactions of the American Mathematical Society 362 (2010), 3243 - 3271.
  DOI:10.1090/S0002-9947-10-05000-2.
  Preprint arXiv:0805.3973.
• Kählerian Reduction in Steps
  (with Peter Heinzner)
  ''Symmetry and Spaces'' (In Honor of Gerry Schwarz), Progress in Mathematics 278, Birkhäuser, Boston, 2010, pp. 63 - 82.
  DOI:10.1007/978-0-8176-4875-6.
  Preprint arXiv:0709.0205.
  

• Invariant meromorphic functions on Stein manifolds
  Oberwolfach Report 39/2010, Komplexe Analysis, DOI:10.4171/OWR/2010/39, 2010


• Oberwolfach Report 2/2010 "Moduli Spaces in Algebraic Geometry"
  I was the reporter for this workshop. Furthermore, I contributed an abstract entitled ''Differential forms on singular varieties''.
  DOI:10.4171/OWR/2010/2, pdf


• Differential forms on singular varieties
  Oberwolfach Report 2/2010, Moduli Spaces in Algebraic Geometry, DOI:10.4171/OWR/2010/2, 2010

Habilitationsschrift

Dissertation