Prof. Dr. Daniel Greb  Publications
Databases
My preprints on arXiv. My published papers on MathSciNet (subscription required). My Google Scholar entry.
Publications
PREPRINTS
 MiyaokaYau 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 MiyaokaYau 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.  MilnorWood 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 MilnorWood 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 quasiprojective 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 GITquotients 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 Kpolystable 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 GITquotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.  1rational 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.
PEERREVIEWED PUBLICATIONS

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, 218230. DOI:10.1017/S0013091523000123
Preprint arXiv:2209.00521
 Projective flatness over klt spaces and uniformisation of varieties with nef anticanonical divisor
(with Stefan Kebekus and Thomas Peternell)
Journal of Algebraic Geometry 31 (2022), 467496, 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), 10051036, DOI: 10.5802/jep.164
Open Access Publication
Preprint arXiv:2010.06878
 Complex algebraic compactifications of the moduli space of HermitianYangMills connections on a projective manifold
(with Ben Sibley, Matei Toma, and Richard Wentworth)
Geometry & Topology 25 (2021), issue 4, 17191818, 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, 786827, 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), 10611094, DOI: 10.1007/s00208019019064
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), 233261, 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, 20512124, 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, 289323, DOI: 10.1112/S0010437X18007923
Free readonly version available here.
Preprint arXiv:1711.08159
 The MiyaokaYau 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, 14871535, 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 Giesekerstable sheaves
(with Julius Ross and Matei Toma)
Transformation Groups 24 (2019), no. 2, 379401, DOI: 10.1007/s0003101894776
Free readonly version available here.
Preprint arXiv:1605.06642  Uniformisation of higherdimensional 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. 277308, DOI: 10.1090/pspum/097.1/01676
Preprint arXiv:1608.06644  Compact moduli spaces for slopesemistable sheaves
(with Matei Toma)
Algebraic Geometry 4 (2017), Issue 1, 4078, DOI: 10.14231/AG2017003
Open Access Publication
Preprint arXiv:1303.2480  Semicontinuity 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), 227265, DOI: 10.1515/crelle20160022
Preprint arXiv:1501.04440
 Variation of Gieseker moduli spaces via quiver GIT
(with Julius Ross and Matei Toma)
Geometry & Topology 20 (2016), no. 3, 15391610, DOI: 10.2140/gt.2016.20.1539
Preprint arXiv:1409.7564
 Moduli of vector bundles on higherdimensional 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, 19652004, DOI:10.1215/001270943450859
Preprint arXiv:1307.5718  Movable curves and semistable sheaves
(with Stefan Kebekus and Thomas Peternell)
International Mathematics Research Notices 2016 (2016), no. 2, 536570, DOI:10.1093/imrn/rnv126
Preprint arXiv:1408.4308  Complexanalytic quotients of algebraic Gvarieties
Mathematische Annalen 363 (2015), no. 1, 77100, DOI: 10.1007/s002080141163y
Preprint arXiv:1403.2097
 Partial positivity: geometry and cohomology of qample 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. 207239.
Preprint arXiv:1307.5568  Base manifolds for Lagrangian fibrations on hyperkähler manifolds
(with Christian Lehn)
International Mathematics Research Notices 2014 (2014), no. 19, 54835487, 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. 67114. 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, 2233, 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, 375403, 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), 5789, DOI:10.1515/crelle20120097
Preprint arXiv:1202.3243 
Invariant meromorphic functions on Stein spaces
(with Christian Miebach)
Annales de l'Institut Fourier (Grenoble) 62 (2012), no. 5, 19832011, 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/s1024001100360
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 BogomolovSommese 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/S0002994710050002
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:10.1007/9780817648756
Preprint arXiv:0709.0205
CONFERENCE PROCEEDINGS (invited, without peer review)

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 KählerEinstein 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 slopesemistable sheaves on higherdimensional 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 slopesemistable sheaves on higherdimensional 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

Holomorphic Invariant Theory and Differential Forms in Higherdimensional Algebraic Geometry
(Zusammenfassende Darstellung der wissenschaftlichen Veröffentlichungen)
Mathematisches Institut der AlbertLudwigsUniversität Freiburg, February 2012.
Dissertation

Projectivity of analytic Hilbert quotients
RuhrUniversität Bochum, January 2008.
MathSciNetReviews
Reviews written by myself for Mathematical Reviews can be found here (subscription required).