Prof. Dr. Daniel Greb - Publications
Databases
My preprints on arXiv. My published papers on MathSciNet (subscription required). My Google Scholar entry.
Publications
PREPRINTS
- Miyaoka-Yau inequalities and the topology of klt varieties, arXiv:2309.14121
(with Stefan Kebekus and Thomas Peternell)
Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka-Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are also characterized topologically: if a complex, projective manifold $X$ is homeomorphic to a variety of this type, then $X$ is itself of this type. In this paper, similar results are established for projective varieties with klt singularities that are homeomorphic to singular ball quotients, quotients of Abelian varieties, or projective spaces. - Milnor-Wood inequality for klt varieties of general type and uniformization, arXiv:2308.05586
(with Matteo Costantini)
We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations. -
Reductive quotients of klt singularities, arXiv:2111.02812
(with Lukas Braun, Kevin Langlois, and Joaquín Moraga)
We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ edowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$, we can find a boundary $B$ on $X/\!/G$ so that the pair $(X/\!/G,B)$ is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$-dimensional K-polystable Fano manifolds of volume $v$ has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type. - 1-rational singularities and quotients by reductive groups, arXiv:0901.3539
This preprint 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.
PEER-REVIEWED PUBLICATIONS
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Momentum maps and the Kähler property for base spaces of reductive principal bundles
(with Christian Miebach)
Proceedings of the Edinburgh Mathematical Society 66 (2023), no. 1, 218-230. DOI:10.1017/S0013091523000123
Preprint arXiv:2209.00521
- Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor
(with Stefan Kebekus and Thomas Peternell)
Journal of Algebraic Geometry 31 (2022), 467-496, DOI: 10.1090/jag/785
Preprint arXiv:2006.08769
The results were also discussed in my contribution to Oberwolfach Report 24/2020.
- Projectively flat klt varieties
(with Stefan Kebekus and Thomas Peternell)
Journal de l'École Polytechnique - Mathématiques 8 (2021), 1005-1036, DOI: 10.5802/jep.164
Open Access Publication
Preprint arXiv:2010.06878
- Complex algebraic compactifications of the moduli space of Hermitian-Yang-Mills connections on a projective manifold
(with Ben Sibley, Matei Toma, and Richard Wentworth)
Geometry & Topology 25 (2021), issue 4, 1719-1818, DOI: 10.2140/gt.2021.25.1719
Preprint arXiv:1810.00025
- Canonical complex extensions of Kähler manifolds
(with Michael Lennox Wong)
Journal of the London Mathematical Society 101 (2020), no. 2, 786-827, DOI: 10.1112/jlms.12287
Open Access Publication
Preprint arXiv:1807.01223 - Harmonic metrics on Higgs sheaves and the uniformization of varieties of general type
(with Stefan Kebekus, Thomas Peternell, and Behrouz Taji)
Mathematische Annalen 378 (2020), 1061-1094, DOI: 10.1007/s00208-019-01906-4
Open Access Publication
Preprint arXiv:1804.01266 - Moduli of sheaves that are semistable with respect to a Kähler polarisation
(with Matei Toma)
Journal de l'École polytechnique - Mathématiques 7 (2020), 233-261, DOI: 10.5802/jep.116
Open Access Publication
Preprint arXiv:1807.05928
- Klt varieties with trivial canonical class - Holonomy, differential forms, and fundamental groups
(with Henri Guenancia and Stefan Kebekus)
Geometry & Topology 23 (2019), no. 4, 2051-2124, DOI: 10.2140/gt.2019.23.2051
Preprint arXiv:1704.01408
The results are also discussed in my contribution to Oberwolfach Report 39/2017. - Nonabelian Hodge Theory for klt spaces and descent theorems for vector bundles
(with Stefan Kebekus, Thomas Peternell, and Behrouz Taji)
Compositio Mathematica 155 (2019), vol. 2, 289-323, DOI: 10.1112/S0010437X18007923
Free read-only version available here.
Preprint arXiv:1711.08159
- The Miyaoka-Yau inequality and uniformisation of canonical models
(with Stefan Kebekus, Thomas Peternell, and Behrouz Taji)
Annales scientifiques de l' École normale supérieure (4) 52 (2019), no.6, 1487-1535, DOI: 10.24033/asens.2414
Preprint arXiv:1511.08822
- Hamiltonian actions of unipotent groups on compact Kähler manifolds
(with Christian Miebach)
Épijournal de Géométrie Algébrique 2 (2018), article 10
Open Access Publication
Preprint arXiv:1805.02551
- A master space for moduli spaces of Gieseker-stable sheaves
(with Julius Ross and Matei Toma)
Transformation Groups 24 (2019), no. 2, 379-401, DOI: 10.1007/s00031-018-9477-6
Free read-only version available here.
Preprint arXiv:1605.06642 - Uniformisation of higher-dimensional minimal varieties
(with Stefan Kebekus and Behrouz Taji)
"Algebraic Geometry: Salt Lake City 2015", Proceedings of Symposia in Pure Mathematics 97.1, American Mathematical Society, 2018, pp. 277-308, DOI: 10.1090/pspum/097.1/01676
Preprint arXiv:1608.06644 - Compact moduli spaces for slope-semistable sheaves
(with Matei Toma)
Algebraic Geometry 4 (2017), Issue 1, 40-78, DOI: 10.14231/AG-2017-003
Open Access Publication
Preprint arXiv:1303.2480 - Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces
(with Julius Ross and Matei Toma)
Journal für die Reine und Angewandte Mathematik 749 (2019), 227-265, DOI: 10.1515/crelle-2016-0022
Preprint arXiv:1501.04440
- Variation of Gieseker moduli spaces via quiver GIT
(with Julius Ross and Matei Toma)
Geometry & Topology 20 (2016), no. 3, 1539-1610, DOI: 10.2140/gt.2016.20.1539
Preprint arXiv:1409.7564
- Moduli of vector bundles on higher-dimensional base manifolds - Construction and Variation
(with Julius Ross and Matei Toma)
International Journal of Mathematics 27 (2016), vol. 07, Special Issue: Proceedings of VBAC 2014 "Algebraic Varieties: Bundles, Topology, Physics", 1650054 (27 pages), DOI: 10.1142/S0129167X16500543
Preprint arXiv:1503.00319 - Étale fundamental groups of klt spaces, flat sheaves, and quotients of Abelian varieties
(with Stefan Kebekus and Thomas Peternell)
Duke Mathematical Journal 165 (2016), no. 10, 1965-2004, DOI:10.1215/00127094-3450859
Preprint arXiv:1307.5718 - Movable curves and semistable sheaves
(with Stefan Kebekus and Thomas Peternell)
International Mathematics Research Notices 2016 (2016), no. 2, 536-570, DOI:10.1093/imrn/rnv126
Preprint arXiv:1408.4308 - Complex-analytic quotients of algebraic G-varieties
Mathematische Annalen 363 (2015), no. 1, 77-100, DOI: 10.1007/s00208-014-1163-y
Preprint arXiv:1403.2097
- Partial positivity: geometry and cohomology of q-ample line bundles
(with Alex Küronya)
"Recent Advances in Algebraic Geometry" (A Volume in Honor of Rob Lazarsfeld's 60th Birthday), London Mathematical Society Lecture Note Series 417, Cambridge University Press, 2015, pp. 207-239.
Preprint arXiv:1307.5568 - Base manifolds for Lagrangian fibrations on hyperkähler manifolds
(with Christian Lehn)
International Mathematics Research Notices 2014 (2014), no. 19, 5483-5487, DOI:10.1093/imrn/rnt133
Preprint arXiv:1303.3919 -
Singular spaces with trivial canonical class
(with Stefan Kebekus and Thomas Peternell)
"Minimal Models and Extremal Rays" (Kyoto 2011), proceedings of a conference in honor of Shigefumi Mori's 60th birthday, Advanced Studies in Pure Mathematics 70, Mathematical Society of Japan, Tokyo, 2016, pp. 67-114. DOI: 10.2969/aspm/07010067
Preprint arXiv:1110.5250 -
Lagrangian fibrations on hyperkähler fourfolds
(with Christian Lehn and Sönke Rollenske)
Izvestiya: Mathematics 78 (2014), no. 1, 22-33, DOI: 10.1070/IM2014v078n01ABEH002678
Preprint arXiv:1110.2680 -
Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville
(with Christian Lehn and Sönke Rollenske)
Annales scientifiques de l'École normale supérieure (4) 46 (2013), no. 3, 375-403, DOI: 10.24033/asens.2191
Preprint arXiv:1105:3410
Some of the results of the paper were announced in Oberwolfach Report 47/2011 -
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) 697 (2014), 57-89, 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, 1983-2011, DOI:10.5802/aif.2740
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, DOI:10.4310/MRL.2011.v18.n6.a14
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 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
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
CONFERENCE PROCEEDINGS (invited, without peer review)
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The cycle space of the K3 period domain and deformations of complex hyperkähler metrics
Oberwolfach Report 40/2022, Complex Geometry and Dynamical Systems, DOI: 10.4171/OWR/2022/40, 2022. -
Projective flatness over klt spaces and characterisation of finite quotients of projective spaces
Oberwolfach Report 24/2020, Komplexe Analysis - Algebraicity and Transcendence, DOI: 10.4171/OWR/2020/24, 2020. -
Holonomy of singular Kähler-Einstein metrics on klt varieties with trivial canonical divisor
Oberwolfach Report 39/2017, Komplexe Analysis, DOI: 10.4171/OWR/2017/39, 2017. -
Étale fundamental groups of klt spaces, flat sheaves, and quotients of Abelian varieties
Oberwolfach Report 27/2013, Complex Algebraic Geometry, DOI: 10.4171/OWR/2013/27, 2013. -
Compact moduli spaces for slope-semistable sheaves on higher-dimensional projective manifolds
Oberwolfach Report 06/2013, Moduli Spaces in Algebraic Geometry, DOI:10.4171/OWR/2013/6, 2013. -
Oberwolfach Report 6/2013 "Moduli Spaces in Algebraic Geometry"
I was the reporter for this workshop. Moreover, I contributed an abstract entitled "Compact moduli spaces for slope-semistable sheaves on higher-dimensional projective manifolds" (see previous item), DOI:10.4171/OWR/2013/6, 2013. -
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'' (see subsequent item), DOI:10.4171/OWR/2010/2, 2010. -
Differential forms on singular varieties
Oberwolfach Report 2/2010, Moduli Spaces in Algebraic Geometry, DOI:10.4171/OWR/2010/2, 2010.
Habilitationsschrift
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Holomorphic Invariant Theory and Differential Forms in Higher-dimensional Algebraic Geometry
(Zusammenfassende Darstellung der wissenschaftlichen Veröffentlichungen)
Mathematisches Institut der Albert-Ludwigs-Universität Freiburg, February 2012.
Dissertation
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Projectivity of analytic Hilbert quotients
Ruhr-Universität Bochum, January 2008.
MathSciNet-Reviews
Reviews written by myself for Mathematical Reviews can be found here (subscription required).