- August 1, 2014 - May 31, 2018
- total support EUR 407.190,--

**Project leader**: Andreas Cap, Faculty of Mathematics, University of
Vienna.

**People supported by the project**:

- Travis Willse (Post-Doc) from October 2015 to August 2017 and from January 2018 to May 2018
- Christoph Harrach PhD student from August 2014 to June 2017 and Post doc from July 2017 to May 2018
- Callum Sleigh (Post-Doc) January 2015 to August 2015
- Chiara de Zanet (PhD student) September 2014 to June 2015

- A. Rod Gover (University of Auckland, New Zealand)
- Tomas Salac (Charles University, Prague, Czech Republic)
- Vladimir Soucek (Charles University, Prague, Czech Republic)

**Scientific Aims**: The BGG-sequences (short for
Bernstein-Gelfand-Gelfand-sequences studied in this project are
sequences of differential operators intrinsic to certain geometric
structures. Under certain types of flatness assumptions they are known
to be complexes respectively to contain certain subcomplexes. The name
draws from a duality relating these sequences to homomorphisms of
generalized Verma modules, which form Lepowski's generalizations of
the Bernstein-Gelfand-Gelfand resolutions of finite dimensional
representations of semi-simple Lie algebras.

After some initial constructions, in particular for the case of conformal structures, a general theory of BGG sequences in the setting of so-called parabolic geometries was developed around the year 2000. Since then, a large number of geometric applications of BGG sequences have been found and they are among the central tools for the theory of parabolic geometries. In particular, there is a close connection between BGG sequences and overdetermined systems of PDEs which are intrinsic to parabolic geometries. Among the solutions to these systems, there is a special subclass of normal solutions, which give rise to holonomy reductions of parabolic geometries, for which a general theory has been developed recently.

The basic aim of the project is to apply the techniques that have led to the construction of BGG-sequences for the developement of new tools which can be applied to geometric problems beyond the realm of parabolic geometries. The four main directions of study planned for the project are:

**Publications related to the project**:

- A. Cap, A.R. Gover, M. Hammerl: "Holonomy reductions of Cartan geometries and curved orbit decompositions", Duke Math. J.
**163**, no. 5 (2014) 1035-1070, available at arXiv:1103.4497 - A. Cap, T. Salac: "Pushing down the Rumin complex to locally conformally symplectic quotients ", Differential Geom. Appl.
**35**Supplement (2014) 255-265, available at arXiv:1312.2712. - A. Cap, A.R. Gover: "Scalar Curvature and projective compactness", J. Geom. Phys.
**98**(2015) 475-481, available online at arXiv:1409.1698. - C. de Zanet: "Generic one-step bracket-generating distributions of rank four", Arch. Math. (Brno)
**51**(2015) 257-264, available online via the EMIS electronic library - C. de Zanet: "Generic one-step bracket generating distributions of rank four", doctoral thesis, University of Vienna, April 2016i, available online vie http://othes.univie.ac.at/41905/.
- A. Cap, A.R. Gover: "Projective Compactifications and Einstein Metrics", J. Reine Angew. Math.
**717**(2016) 47-75, available online at arXiv:1304.1869. - A. Cap, V. Soucek: "Relative BGG sequences; I. Algebra", J. Algebra
**463**(2016) 188-210, available online at arXiv:1510.03331. - C. Harrach: "Poisson transforms for differential forms adapted to the flat parabolic geometries on spheres", doctoral thesis, University of Vienna, March 2017.
- A. Cap, A.R. Gover: "Projective Compactness and Conformal Boundaries", Math. Ann.
**366**no. 3-4 (2016), 1587-1620, published version (via SharedIt), also available online at arXiv:1406.4225. - A. Cap, A.R. Gover, C.R. Graham, M. Hammerl: "Conformal Holonomy Equals Ambient Holonomy", Pacific J. Math.
**285**no. 2 (2016), 303-318, available online at arXiv:1504.00914. - C. Harrach: "Poisson transforms for differential forms", Arch. Math. (Brno)
**52**(2016) 303-311, available online via the EMIS electronic library. - K. Sagerschnig, T. Willse: "The geometry of almost Einstein (2,3,5) distributions", SIGMA Symmetry Integrability Geom. Methods Appl.
**13**(2017) paper 004, 56 pp., published version available online here - A. Cap, V. Soucek: "Relative BGG sequences; II. BGG machinery and invariant operators", Adv. Math.
**320**(2017) 1009-1062, available online as preprint arXiv:1510.03986. - K. Sagerschnig, T. Willse: "The almost Einstein operator for (2,3,5) distributions ", Arch. Math. (Brno)
**53**(2017), 347-370, published version available online here. - A. Cap, T. Salac: "Parabolic conformally symplectic structures I; definition and distinguished connections", Forum Math.
**30**, 3 (2018) 733-751, available online as arXiv:1605.01161. - T. Willse: "Cartan's incomplete classification and an explicit ambient metric of holonomy G
_{2}^{*}", Eur. J. Math.**4**2 (2018) 622-638, published version available online here. - A. Cap, T. Salac: "Parabolic conformally symplectic structures II; parabolic contactification", Ann. Mat. Pura Appl.
**197**no. 4 (2018) 1175-1199, available online at http://link.springer.com/article/10.1007/s10231-017-0719-3 (open access). - C. Harrach: "Poisson transforms adapted to BGG-complexes", Differential Geom. Appl.
**64**(2019) 92-113, availabble online as preprint arXiv:1806.08599 - A. Cap, A.R. Gover: "C-Projective Compactification; (quasi-)Kähler Metrics and CR boundaries", Amer. J. Math.
**141**3 (2019) 813-856, available online as preprint arXiv:1603.07039. - A. Cap, T. Salac: "Parabolic conformally symplectic structures III; Invariant differential operators and complexes",Doc. Math.
**24**(2019) 2203-2240, published version available online via elibm. - A. Cap: "On canonical Cartan connections associated to filtered G-structures", preprint arXiv:1707.05627.
- A. Cap, B. Doubrov, D. The: "On C-class equations", preprint arXiv:1709.01130.
- A. Cap, A.R. Gover, M. Hammerl: "Parabolic Compactification of Homogeneous Spaces", to appear in J. Inst. Math. Jussieu, available online as preprint arXiv:1807.04556.
- A. Cap, C. Harrach, P. Julg: "A Poisson transform adapted to the Rumin complex", preprint arXiv:1904.00635.

**Talks related to the project**:

- A. Cap: "Geometry at infinity", (plenary lecture), ECC-Seminar, Trest, Czech Republic, October 2014.
- A. Cap: "Conformal and projective compactifications", Central European Seminar on differential geometry, Brno, Czech Republic, November 2014.
- A. Cap: "Projective compactness", Workshop "Equivalence, invariants, and symmetries of vector distributions and related structures : from Cartan to Tanaka and beyond ", Institut Henri Poincaré, Paris, France, December 2014
- A. Cap: "A relative version of Kostant's theorem", 35th Winter School Geometry and Physics, Srni, Czech Republic, January 2015
- C. de Zanet: "Dual Darboux distributions", 35th Winter School Geometry and Physics, Srni, Czech Republic, January 2015
- C. Harrach: "A Poisson transform for the Rumin complex", 35th Winter School Geometry and Physics, Srni, Czech Republic, January 2015
- A. Cap: "Parabolic almost conformally symplectic structures", University of Auckland, New Zealand, Febrauary 2015
- C. Sleigh: "Cohomology of BGG complexes", Central European Seminar on differential geometry, Brno, Czech Republic, March 2015
- A. Cap: "Projective compactifications", Princeton-Tokyo workshop on Geometric Analysis, University of Tokyo, Japan, March 2015
- C. Sleigh: "Introduction to tractor calculus and BGG complexes", Geometric Analysis and Physics Seminar, University of Vienna, April 2015
- A. Cap: "Relative BGG sequences", ECI-Seminar, Trest, Czech Republic, October 2015
- T. Willse: "Generic distributions and Einstein geometry", Ernst Moritz Arndt University, Greifswald, November 2015
- A. Cap: "PACS-structures and special symplectic connections", Central European Seminar on differential geometry, Brno, Czech Republic, December 2015
- A. Cap: "The (relative) BGG machinery", series of 3 plaenary lectures, 36th Winter School Geometry and Physics, Srni, Czech Republic, January 2016
- C. Harrach: "Poisson transforms of differential forms", 36th Winter School Geometry and Physics, Srni, Czech Republic, January 2016
- A. Cap: "PACS structures and special symplectic connections", Univeristy of Auckland, New Zealand, February 2016
- A. Cap: "c-projective compactness", Central European Seminar on differential geometry, Brno, Czech Republic, April 2016
- A. Cap: "Geometry of higher order ODEs and C-class equations", Central European Seminar on differential geometry, Brno, Czech Republic, May 2016
- A. Cap: "C-projective compactness", Conference Differential Geometry and its Applications, Brno, Czech Republic, July 2016
- T. Willse: "Almost Einstein (2,3,5) conformal structures" (poster), Conference Differential Geometry and its Applications, Brno, Czech Republic, July 2016
- A. Cap: "Desending invariant operators and BGG sequences to PCS structures", Seminar of the Eduard Czech Institute, Telc, Czech Republic, October 2016
- T. Willse: "A missing distribution and a metric of holonomy G
_{2}^{*}", Ernst Moritz Arndt University, Greifswald, October 2016 - A. Cap: "From holonomy reductions of Cartan geometries to geometric compactifications", Workshop on Conformal geometry and Spectral Theory, Humboldt University, Berline, November 2016.
- A. Cap: "BGG complexes associated to PACS-structures and their cohomology", Central European Seminar on differential geometry, Brno, Czech Republic, December 2016
- A. Cap: "PCS-structures and differential complexes", 37th Winter School Geometry and Physics, Srni, Czech Republic, January 2017
- T. Willse: "Almost Einstein (2,3,5) conformal structures", 37th Winter School Geometry and Physics, Srni, Czech Republic, January 2017
- A. Cap: "Introduction to BGG sequences", University of Auckland, New Zealand, March 2017.
- A. Cap: "Parabolic compactifications", Central European Seminar on differential geometry, Brno, Czech Republic, March 2017
- A. Cap: "On ODEs of C-class", 19th ÖMG congress and Annual DMV meeting, Salzburg, Austria, September 2017
- C. Harrach: "Poisson transforms for differential forms adapted to homogeneous parabolic geometries", IMPAN, Warsaw, Poland, September 2017
- A. Cap: "Canonical Cartan connections associated to filtered G-structures", ECI Workshop, Telc, Czech Republic, October 2017.
- T. Willse: "Curved orbit decompositions and the ambient metric construction", Simmons Semester "Symmetry and Geometric Structures" , Banach Center Warsaw, Poland, October 2017.
- C. Harrach: "Poisson transforms for differential forms on homogeneous parabolic geometries", Fall School "Lie Theory, Geometry and Differential Equations", Rauischholzhausen, Germany, October 2017.
- A. Cap: "On (systems of) ODEs of C-class", International Conference on Symmetry and Geometric Structures, Banach Center Warsaw, Poland, November 2017
- A. Cap: "Canonical Cartan connections associated to filtered G
0 -structures", 38th Winter School Geometry and Physics, Srni, Czech Republic, January 2018. - T. Willse: "Sasaki-Einstein metrics and their compactifications via projective geometry", 38th Winter School Geometry and Physics, Srni, Czech Republic, January 2018.
- A. Cap: "Parabolic contact structures with a view towards symplectic geometry", Conference Conformal and symplectic geometry, University of Auckland, New Zealand, February 2018.
- A. Cap: "Path geometries and chains for parabolic contact structures", Whitiroa Workshop, New Zealand, February 2018.
- A. Cap: "A slice theorem for a compactification of a symmetric space", Central European Seminar on differential geometry, Brno, Czech Republic, April 2018.
- T. Willse: "Special geometries via projective holonomy", Sophus Lie Seminar, Univeristy of Tromso, Norway, May 2018.