Profile
Profile

Prof. Dr. Daniel Huybrechts

E-mail: huybrech(at)math.uni-bonn.de
Phone: +49 228 73 3135
Fax: +49 228 73 3257
Homepage: http://www.math.uni-bonn.de/people/huybrech/
Room: 3.005
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Research Area DE (Leader)
Former Research Area E (Leader)
Research Area C
Date of birth: 09.Nov 1966
Mathscinet-Number: 344746

Academic Career

1985 - 1992

Studies of Mathematics, HU Berlin and Max Planck Institute, Bonn

1992

PhD, HU Berlin

1993 - 1994

Postdoc, Max Planck Institute, Bonn

1994 - 1995

Postdoc, Institute for Advanced Study, Princeton, NJ, USA

1995 - 1996

Postdoc, Max Planck Institute, Bonn

1996 - 1997

Assistant Professor (C1), University-GH Essen

1997 - 1998

Marie-Curie Fellow, ENS Paris, France

1998

Habilitation, University-GH Essen

1998 - 2002

Professor (C3), University of Cologne

2002 - 2005

Professor, Paris Diderot University (Paris 7), France

Since 2005

Professor (C4/W3), University of Bonn

Research Profile

My main focus is on K3 surfaces and higher dimensional analogues which can be studied in terms of algebraic invariants like Hodge structures and derived categories. K3 surfaces and related moduli spaces are particularly interesting test cases for some of the central conjectures in algebraic geometry (e.g. Tate, Hodge, Bloch-Beilinson). I have studied Chow groups of K3 surfaces from a geometric and a categorical perspective. In particular, I have introduced the notion of constant cycle curves and studied the action of symplectic automorphisms on Chow groups, providing further evidence for one of Bloch's elusive conjectures. Finite group of symplectic derived auto-equivalences have been classified completely in terms of the Conway group, one of the exotic sporadic simple groups. For Kuznetsov's K3 category associated with any cubic fourfold I have extended work of Addington and Thomas to the twisted case and described the group of auto-equivalences in the generic case. This has subsequently led to a new proof of the global Torelli theorem for cubic fourfolds (with Rennemo).

It has been conjecture that rationality of cubic fourfolds is determined by the structure of the associated K3 category. Further investigations of the structure of Kuznetsov's category should shed more light on the role of derived techniques on rationality questions in broader generality. The bearing of derived techniques on our understanding of cycles on K3 surfaces and cubics hypersurfaces needs to be clarified. Cohomological methods relating classical invariants like the Jacobian ring of a hypersurface with categorical invariants similar to Hochschild cohomology may lead to global Torelli theorems for cubics of higher dimensions. The role of mirror symmetry needs to be explored. Further foundational questions concerning the motivic nature of K3 surfaces shall be addressed.

Research Projects and Activities

DFG Collaborative Research Center SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”
Local coordinator, since 2006

Oberwolfach Workshops on “Algebraic Geometry”
Organizer, 2015, 2017

DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Principal investigator
Vice-Coordinator (until 2017)

Contribution to Research Areas

Research Area C
Homological mirror symmetry relates symplectic and algebraic geometry as an equivalence of categories (Fukaya category of Lagrangians resp. derived category of coherent sheaves). Fundamental aspects of both sides can thus be seen also from the mirror perspective which has led to new insight. In [1], we have proved the mirror analogue of a theorem of Donaldson on the action of the diffeomorphism group of a K3 surface. The conjectured braid group like description of the group of autoequivalences of the derived category of Calabi-Yau varieties of dimension two is an example and one of the main open problems in the area.
Former Research Area E
Spaces of stability conditions on abelian and triangulated categories form a new kind of moduli spaces with an intriguing wall and chamber structure reflecting the change of moduli spaces of stable objects. The main open questions in the are concern the global geometry of the space of stability conditions and the change of numerical and motivic invariants of the associated moduli spaces of stable objects. The case of the derived category of coherent sheaves on a K3 surface is of particular interest as moduli spaces of sheaves and complexes yield higher dimensional varieties with special geometries. A surprising relation to conjectures on the structure of Chow groups has been discovered in [2].

Selected Publications

[1] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Derived equivalences of K3 surfaces and orientation
Duke Math. J. , 149: (3): 461--507
2009
DOI: 10.1215/00127094-2009-043
[2] Daniel Huybrechts
Chow groups of K3 surfaces and spherical objects
J. Eur. Math. Soc. (JEMS) , 12: (6): 1533--1551
2010
DOI: 10.4171/JEMS/240
[3] D. Huybrechts
Curves and cycles on K3 surfaces
With an appendix by C. Voisin
Algebr. Geom. , 1: (1): 69--106
2014
DOI: 10.14231/AG-2014-005
[4] Daniel Huybrechts, Richard P. Thomas
Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes
Math. Ann. , 346: (3): 545--569
2010
DOI: 10.1007/s00208-009-0397-6
[5] Daniel Huybrechts, Manfred Lehn
The geometry of moduli spaces of sheaves
Cambridge Mathematical Library : xviii+325
Publisher: Cambridge University Press, Cambridge
2010
ISBN: 978-0-521-13420-0
DOI: 10.1017/CBO9780511711985
[6] D. Huybrechts
Fourier-Mukai transforms in algebraic geometry
Oxford Mathematical Monographs : viii+307
Publisher: The Clarendon Press, Oxford University Press, Oxford
2006
ISBN: 978-0-19-929686-6; 0-19-929686-3
DOI: 10.1093/acprof:oso/9780199296866.001.0001
[7] Daniel Huybrechts, Richard Thomas
\Bbb P-objects and autoequivalences of derived categories
Math. Res. Lett. , 13: (1): 87--98
2006
DOI: 10.4310/MRL.2006.v13.n1.a7
[8] Daniel Huybrechts
Compact hyper-Kähler manifolds: basic results
Invent. Math. , 135: (1): 63--113
1999
DOI: 10.1007/s002220050280
[9] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Stability conditions for generic K3 categories
Compos. Math. , 144: (1): 134--162
2008
DOI: 10.1112/S0010437X07003065
[10] Daniel Huybrechts
Lectures on K3 surfaces
of Cambridge Studies in Advanced Mathematics : xi+485
Publisher: Cambridge University Press, Cambridge
2016
ISBN: 978-1-107-15304-2
DOI: 10.1017/CBO9781316594193

Publication List

Editorships

• Bulletin et Mémoires de la SMF (2005 - 2013)
• Kyoto Journal of Mathematics (since 2010)
• Crelle Journal (since 2012)
• Inventiones mathematicae (since 2014)

Selected Invited Lectures

2008

Algebro-Geometric Derived Categories and Applications, Institute for Advanced Study, Princeton, NJ, USA

2009

Classical Algebraic geometry today, Mathematical Sciences Research Institute (MSRI), Berkeley, CA, USA

2010

International Congress of Mathematicians, Hyderabad, India

2011

Moduli spaces and moduli stacks, Columbia University, New York, USA

2011

Spring lectures in algebraic geometry, Ann Arbor, MI, USA

2015

Perspectives on Complex Algebraic Geometry, Columbia University, New York, USA

2015

Schrödinger Lecture, ESI, Vienna, Austria

2016

Homological Mirror Symmetry, Methods and Structures, IAS, Princeton, NJ, USA

2016

Generalised Geometry and Noncommutative Algebra, Clay Mathematics Institute, Oxford, England, UK

Offers

2008

University of Heidelberg

Habilitations

Vladimir Lazic (2015), now Professor, Saarland University

Selected PhD students

Marc Nieper-Wißkirchen (2002): “Characteristic Classes and Rozansky-Witten Invariants of Compact Hyperkähler Manifolds”,
now Professor, University of Augsburg

Sven Meinhardt (2008): “Stability conditions on derived categories”,
now Research Assistant, University of Sheffield, England, UK

Michael Kemeny (2015): “Stable maps and singular curves on K3 surfaces”,
now Szegö Assistant Professor, Stanford University, CA, USA

Stefan Schreieder (2015): “Construction problems in algebraic geometry and the Schottky problem”,
now Postdoc, University of Bonn

Ulrike Riess (2016): “On irreducible symplectic varieties: Chow rings and base loci of certain line bundles”

Supervised Theses

  • Master theses: 17, currently 2
  • Diplom theses: 12
  • PhD theses: 12, currently 1
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