Research

I work in Algebraic Geometry, one of the oldest and at the same time one of the most active research areas in pure mathematics. A short description of this research area can be found here. Some of my lectures are available on the Internet.

Recently, I became interested in computer-aided formalization of mathematics. I am currently working to formalize the main theorems of Nevanlinna theory using the interactive theorem prover Lean.

Publications

Preprints

2024	Entire curves in \(\mathcal C\)-pairs with large irregularity (with Erwan Rousseau). Preprint arXiv:2410.01245.
	The Albanese of a \(\mathcal C\)-pair (with Erwan Rousseau). Preprint arXiv:2410.00405.
	\(\mathcal C\)-pairs and their morphisms (with Erwan Rousseau). Preprint arXiv:2407.10668.

Refereed Papers

Published

2024	Miyaoka-Yau inequalities and the topological characterization of certain klt varieties (with Daniel Greb and Thomas Peternell). Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 141-157. DOI:10.5802/crmath.580. Preprint arXiv:2309.14121.
2022	Failure of the Brauer-Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell (with Jorge Vitório Pereira and Arne Smeets). Duke Math. J. 171 (2022), no. 17, 3515–3591. DOI:10.1215/00127094-2022-0045. Preprint arXiv:1905.02795.
	Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor (with Daniel Greb and Thomas Peternell). Journal of Algebraic Geometry, Vol. 31, pp. 467-496, 2022. DOI:10.1090/jag/785. Preprint arXiv:2006.08769.
2021	Survey paper: Generic positivity and applications to hyperbolicity of moduli spaces (with Benoît Claudon and Behrouz Taji). In Hyperbolicity Properties of Algebraic Varieties, volume 56 of Panoramas & Synthèses, pages 169–208. Société Mathématique de France, 2021. Preprint arXiv:1610.09832.
	Projectively flat KLT varieties (with Daniel Greb and Thomas Peternell). Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1005-1036. DOI:10.5802/jep.164. Preprint arXiv:2010.06878.
	Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities (with Christian Schnell) Journal of the American Mathematical Society. Volume 34, Number 2, April 2021, Pages 315–368. DOI:10.1090/jams/962. Preprint arXiv:1811.03644.
2020	Survey paper: Boundedness results for singular Fano varieties and applications to Cremona groups Séminaire Nicolas Bourbaki. Vol. 2018/2019. Exp. No. 1157, 253--290 (2020) DOI:10.24033/ast.1129. Preprint arXiv:1812.04506.
	Harmonic metrics on Higgs sheaves and uniformization of varieties of general type (with Daniel Greb, Thomas Peternell and Behrouz Taji) Mathematische Annalen volume 378 (2020), 1061–1094. DOI:10.1007/s00208-019-01906-4. Preprint arXiv:1804.01266.
2019	The Miyaoka-Yau inequality and uniformisation of canonical models (with Daniel Greb, Thomas Peternell and Behrouz Taji) Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 6, 1487–1535. DOI:10.24033/asens.2414. Preprint arXiv:1511.08822.
	Klt varieties with trivial canonical class - Holonomy, differential forms, and fundamental groups (with Daniel Greb and Henri Guenancia) Geometry & Topology 23-4 (2019), 2051–2124. DOI:10.2140/gt.2019.23.2051. Preprint arXiv:1704.01408.
	Nonabelian Hodge Theory for klt spaces and descent theorems for vector bundles (with Daniel Greb, Thomas Peternell and Behrouz Taji) Compositio Math. 155(2019), 289–323. DOI:10.1112/S0010437X18007923. Preprint arXiv:1711.08159.
2018	Survey paper: Uniformisation of higher-dimensional varieties (with Daniel Greb and Behrouz Taji) In: Proceedings of 2015 AMS Summer Institute on Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, Volume 97.1, 2018. DOI:10.1090/pspum/097.1/01676. Preprint arXiv:1608.06644.
2017	Differential forms in positive characteristic avoiding resolution of singularities (with Annette Huber and Shane Kelly) Bull. Soc. Math. France 145 (2017), no. 2, 305–343. DOI:10.24033/bsmf.2739. Preprint arXiv:1407.5786.
	Comparing \(\mathbb A^1\)-\(h\)-cobordism and \(\mathbb A^1\)-weak equivalence (with Aravind Asok and Matthias Wendt) Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVII (2017), 531-572. DOI:10.2422/2036-2145.201503_007. Preprint arXiv:1410.3038.
2016	Singular spaces with trivial canonical class (with Daniel Greb und Thomas Peternell). in "Minimal Models and Extremal Rays (Kyoto, 2011)". Advanced Studies in Pure Mathematics 70, Pages 67-113, Mathematical Society of Japan, Tokyo, 2016. Preprint arXiv:1110.5250.
	Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties (with Daniel Greb und Thomas Peternell). Duke Math. J., Volume 165, Number 10 (2016), 1965-2004. DOI:10.1215/00127094-3450859. Preprint arXiv:1307.5718.
	Survey paper: Aspects of the geometry of varieties with canonical singularities (with Thomas Peternell) Simons Symposia: Foliation Theory in Algebraic Geometry, Editors: Paolo Cascini, James McKernan and Jorge Vitório Pereira. Springer, New York, 2016. DOI:10.1007/978-3-319-24460-0. Available from this URL.
	Movable curves and semistable sheaves (with Daniel Greb and Thomas Peternell). Int Math Res Notices, 2016 (2): 536-570 DOI:10.1093/imrn/rnv126. Preprint arXiv:1408.4308 contains an extended version with additional figures.
2014	Reflexive differential forms on singular spaces -- Geometry and Cohomology (with Daniel Greb and Thomas Peternell). Journal für die reine und angewandte Mathematik (Crelles Journal). Vol 697, pp 57–89 (2014). DOI:10.1515/crelle-2012-0097. Preprint arXiv:1202.3243.
2013	Pull-back morphisms for reflexive differential forms Advances in Mathematics, Volume 245, pp. 78-112, 2013. DOI:10.1016/j.aim.2013.06.013. Preprint arXiv:1210.3255.
	Survey paper: Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks In: Handbook of Moduli, Volume II, Editors: Gavril Farkas and Ian Morrison. Number 26 in Advanced Lectures in Mathematics, International Press, Beijing Preprint arXiv:1107.4239.
	Survey paper: Uniruledness criteria and applications, In: Simons Symposia: Birational Geometry, Rational Curves, and Arithmetic, Editors: Fedor Bogomolov, Brendan Hassett and Yuri Tschinkel, pp. 147-162, Springer, New York, 2013. DOI:10.1007/978-1-4614-6482-2. Available from this URL.
2012	Deformations along subsheaves, II (with Clemens Jörder) EMS Series of Congress Reports, Contributions of Algebraic Geometry, Impanga Lecture Notes (Trends in Mathematics), EMS Publishing House, 2012. DOI:10.4171/114. Preprint arXiv:1103.5630.
2011	Positive sheaves of differentials coming from coarse moduli spaces (with Kelly Jabbusch) Annales de l'institut Fourier, 61 no. 6 (2011), p. 2277-2290, DOI:10.5802/aif.2673. Preprint arXiv:0904.2445.
	Differential Forms on Log Canonical Spaces (with Daniel Greb, Sándor Kovács and Thomas Peternell) Publications Mathématiques de l'IHÉS, Volume 114, Number 1 (2011), 87-169. DOI:10.1007/s10240-011-0036-0. Preprint arXiv:1003.2913 contains an extended version with additional graphics.
	Families over special base manifolds and a conjecture of Campana (with Kelly Jabbusch) Mathematische Zeitschrift: Volume 269, Issue 3 (2011), Page 847-878. DOI:10.1007/s00209-010-0758-6. Preprint arXiv:0905.1746.
	Deformations along subsheaves (with Stavros Kousidis and Daniel Lohmann) L’Enseignement Mathématique, Vol. 56, pp. 287-313, 2010. DOI:10.4171/LEM/56-3-3. Preprint arXiv:0905.2749.
2010	The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized manifolds (with Sándor Kovács) Duke Mathematical Journal, Vol. 155, No 1, pp. 1-33, 2010. DOI:10.1215/00127094-2010-049. Preprint arXiv:0812.2305.
	Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties (with Daniel Greb and Sándor Kovács) Compositio Mathematica, Vol. 146, pp. 193–219, 2010. DOI:10.1112/S0010437X09004321. Preprint arXiv:0808.3647.
2008	A refinement of Stein factorization and deformations of surjective morphisms (with Thomas Peternell) Asian Journal of Mathematics, Vol. 12, No. 3, pp. 365-389, 2008. DOI:10.4310/AJM.2008.v12.n3.a6. Preprint arXiv:math/0508285.
	Families of varieties of general type over compact bases (with Sándor Kovács) Advances in Mathematics, Vol. 218, No. 3, pp. 649-652, 2008. DOI:10.1016/j.aim.2008.01.005. Preprint arXiv:math/0704.2556.
	Families of canonically polarized varieties over surfaces (with Sándor Kovács) Inventiones Mathematicae, Vol. 172, No. 3, pp. 657-682, 2008. DOI:10.1007/s00222-008-0128-8. Preprint arXiv:math/0511378.
	Galois coverings and endomorphisms of projective varieties (with Marian Aprodu and Thomas Peternell) Mathematische Zeitschrift,  Vol. 260, No. 2, pp. 431-449, 2008. DOI:10.1007/s00209-007-0282-5. Preprint arXiv:arXiv:0705.4602.
2007	Rationally connected foliations after Bogomolov and McQuillan (with Luis Solá and Matei Toma) Journal Algebraic Geometry, Vol. 16, No. 1, pp. 65-81, 2007. DOI:10.1090/S1056-3911-06-00435-8. Preprint arXiv:math/0505222.
2006	Survey paper: Existence of rational curves on algebraic varieties, minimal rational tangents, and applications (with Luis Solá) In Global Aspects of Complex Geometry, pp. 359-416. Springer, 2006 DOI:10.1007/3-540-35480-8_10. Preprint arXiv:math/0512487.
	Holomorphic maps onto varieties of non-negative Kodaira dimension (with Jun-Muk Hwang and Thomas Peternell) Journal of Algebraic Geometry, Vol. 15, pp. 551-561, 2006. DOI:10.1090/S1056-3911-05-00411-X. Preprint arXiv:math/0307220.
2005	Geometry of chains of minimal rational curves (with Jun-Muk Hwang) Journal für die reine und angewandte Mathematik, Vol. 584, pp. 173-194, 2005. DOI:10.1515/crll.2005.2005.584.173. Preprint arXiv:math/0403352.
	Lines on complex contact manifolds, II Compositio Mathematica, Vol. 141, No. 1, pp. 227-252, 2005. DOI:10.1112/S0010437X04000880. Preprint arXiv:math/0306260.
2004	Are minimal degree rational curves determined by their tangent vectors? (with Sándor Kovács) Annales de l'Institut Fourier, Vol. 54, No. 1, pp. 53-80, 2004. Available from this URL. Preprint arXiv:math/0206193.
2002	Projective bundles of singular plane cubics Mathematische Nachrichten, Vol. 242, No. 1, pp. 119-131, 2002. DOI:10.1002/1522-2616(200207)242:13.0.CO;2-R. Preprint arXiv:math/0009083.
	Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron In Complex Geometry, Collection of Papers dedicated to Hans Grauert, pp. 147-156. Springer, 2002. Preprint arXiv:math/0107069.
	A reduction map for NEF line bundles (with Thomas Bauer, Frédéric Campana, Thomas Eckl, Thomas Peternell, Sławomir Rams, Tomasz Szemberg and Lorenz Wotzlaw) In Complex Geometry, Collection of Papers dedicated to Hans Grauert, pp. 27-36. Springer, 2002. Preprint arXiv:math/0106147.
	Manifolds with nef rank 1 subsheaves in \(\Omega^1_X\) (with Thomas Peternell and Andrew J. Sommese) In Complex Geometry, Collection of Papers dedicated to Hans Grauert, pp. 157-164. Springer, 2002.
	Families of singular rational curves Journal of Algebraic Geometry, Vol. 11, pp. 245-256, 2002. DOI:10.1090/S1056-3911-01-00308-3. Preprint arXiv:math/0004023.
2001	Lines on contact manifolds Journal für die reine und angewandte Mathematik, Vol. 539, pp. 167-177, 2001. DOI:10.1515/crll.2001.072. Preprint arXiv:math/0004103.
2000	Projective Contact Manifolds  (with Thomas Peternell, Andrew J. Sommese and Jarosław A. Wiśniewski) Inventiones Mathematicae, Vol. 142, No. 1, pp. 1-15, 2000. DOI:10.1007/PL00005791. Preprint arXiv:math/9810102.
	Group Actions on \(S^6\) and complex structures on \(\mathbb P_3(\mathbb C)\) (with Alan T. Huckleberry and Thomas Peternell) Duke Mathematical Journal, Vol. 102, No. 1, pp. 101-124, 2000. DOI:10.1215/S0012-7094-00-10214-1. Preprint arXiv:math/9812076.
	On the Classification of 3-dimensional \(\operatorname{SL}_2(\mathbb C)\)-varieties Nagoya Mathematical Journal, Vol. 157, pp. 129-147, 2000. Available from this URL. Preprint arXiv:math/9805043.
	Relatively Minimal Quasihomogeneous Projective 3-Folds Nagoya Mathematical Journal, Vol. 157, pp. 149-176, 2000. Available from this URL. Preprint arXiv:math/9805042.
1998	Simple Models of Quasihomogeneous Projective 3-Folds Documenta Mathematica, Vol. 3, pp. 15-26, 1998. Preprint arXiv:math/9805041.

Non-refereed papers

2004	Survey paper: Holomorphe Abbildungen auf Mannigfaltigkeiten mit nicht-negativer Kodaira-Dimension (German) Mathematisches Institut, Seminars, Ed.: Y. Tschinkel, Universität Göttingen, pp. 157-166, 2004. Available from this URL.
2000	Survey paper: Bounds for Families of Singular Rational Curves Conference proceedings of the Saitama Conference in Algebraic Geometrye, Saitama University, Urawa City (Japan), 2000. Available from this URL.

Other publications

2023	Komplexe Analysis - Differential and Algebraic methods in Kähler spaces Abstracts from the workshop held April 9--April 14, 2023. Organized by Philippe Eyssidieux, Jun-Muk Hwang, Stefan Kebekus and Mihai Păun. Oberwolfach Rep. 20 (2023), no. 2, pp. 965–1030. DOI:10.4171/OWR/2023/18.
2020	Komplexe Analysis - Algebraicity and Transcendence Abstracts from the workshop held August 16–August 22, 2020. Organized by Philippe Eyssidieux, Jun-Muk Hwang, Stefan Kebekus and Mihai Păun. Oberwolfach Rep. 24 (2020), no. 3, 1263–1309. DOI:10.4171/OWR/2020/24.
2017	Komplexe Analysis Abstracts from the workshop held August 27–September 2, 2017. Organized by Philippe Eyssidieux, Jun-Muk Hwang, Stefan Kebekus and Mihai Păun. Oberwolfach Rep. 14 (2017), no. 3, 2427–2473. DOI:10.4171/OWR/2017/39.
2014	Komplexe Analysis Abstracts from the workshop held August 24–30, 2014. Organized by Philippe Eyssidieux, Jun-Muk Hwang, Stefan Kebekus and Mihai Păun. Oberwolfach Rep. 11 (2014), no. 3, 2177–2234. DOI:10.4171/OWR/2014/39.
2013	Moduli spaces in algebraic geometry Abstracts from the workshop held February 3–9, 2013. Organized by Dan Abramovich, Lucia Caporaso, Gavril Farkas and Stefan Kebekus. Oberwolfach Rep. 10 (2013), no. 1, 343–392. DOI:10.4171/OWR/2013/06.
2010	Moduli spaces in algebraic geometry Abstracts from the workshop held January 10–16, 2010. Organized by Dan Abramovich, Gavril Farkas and Stefan Kebekus. Oberwolfach Reports. Vol. 7, no. 1. Oberwolfach Rep. 7 (2010), no. 1, 55–107. DOI:10.4171/OWR/2010/02.
2009	Multiplier ideal sheaves in algebraic and complex geometry Abstracts from the workshop held April 12–18, 2009. Organized by Stefan Kebekus, Mihai Păun, Georg Schumacher and Yum-Tong Siu. Oberwolfach Reports. Vol. 6, no. 2. Oberwolfach Rep. 6 (2009), no. 2, 1101–1155. DOI:10.4171/OWR/2009/21.

Ph.D. thesis and Habilitation

2001	Rationale Kurven auf projektiven Mannigfaltigkeiten (German) Habilitationsschrift, Universität Bayreuth, 2001. Available from this URL.
1996	Almost Homogeneous Projective 3-Folds Ph.D. thesis, Ruhr-Universität Bochum, 1996. Available from this URL.

Links

• My papers in the zbMath Open publication database
• My papers in Mathematical Reviews
• Papers on the arXiv-Preprint Server