Prof. Dr. Daniel Greb  Publications
Databases
My preprints on arXiv. My published papers on MathSciNet (subscription required). My Google Scholar entry.
Publications
PREPRINTS
 Complex algebraic compactifications of the moduli space of HermitianYangMills connections on a projective manifold, arXiv:1810.00025
(with Ben Sibley, Matei Toma, and Richard Wentworth)
In this paper we study the relationship between three compactifications of the moduli space of HermitianYangMills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the DonaldsonUhlenbeckYau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by GiesekerMaruyama semistable torsionfree sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space.  Moduli of sheaves that are semistable with respect to a Kähler polarisation, arXiv:1807.05928
(with Matei Toma)
Using an existence criterion for good moduli spaces of Artin stacks by AlperFedorchukSmyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are GiesekerMaruyamasemistable with respect to a fixed Kähler class.  Canonical complex extensions of Kähler manifolds, arXiv:1807.01223
(with Michael Lennox Wong)
Given a complex manifold X, any Kähler class defines an affine bundle over X, and any Kähler form in the given class defines a totally real embedding of X into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of X. For compact Kähler manifolds of nonnegative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of LempertSzőke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having nonnegative holomorphic bisectional curvature and big tangent bundle.  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
 Harmonic metrics on Higgs sheaves and the uniformization of varieties of general type
(with Stefan Kebekus, Thomas Peternell, and Behrouz Taji)
Mathematische Annalen, Online First, 2019, will appear in print, DOI: 10.1007/s00208019019064
Open Access Publication
Preprint arXiv:1804.01266  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
 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)
accepted for publication in Annales scientifiques de l' École normale supérieure
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
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)

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).