Langer, Adrian

On boundedness of semistable sheaves

Doc. Math. 27, 1-16 (2022)
DOI: 10.25537/dm.2022v27.1-16
Communicated by Gavril Farkas


We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for semistable sheaves on a smooth projective variety of any dimension \(\ge 2\) without using any restriction theorems.

Mathematics Subject Classification

14F06, 14D20, 14J60


semistable sheaves, Bogomolov's inequality, bounded families


  • 1. Bogomolov, Fedor Alekseevich. Holomorphic tensors and vector bundles on projective manifolds. (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 1227-1287, 1439, MR0522939.
  • 2. Gieseker, David. On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), 77-85, DOI 10.2307/2373939, zbl 0431.14005, MR0527826.
  • 3. Huybrechts, Daniel; Lehn, Manfred. The geometry of moduli spaces of sheaves. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010. xviii+325 pp., zbl 1206.14027, MR2665168.
  • 4. Langer, Adrian. Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251-276; Addendum to: ``Semistable sheaves in positive characteristic'', Ann. of Math. (2) 160 (2004), 1211-1213, zbl 1080.14014, zbl 1080.14015 (Addendum) MR2051393, MR2144978 (Addendum).
  • 5. Langer, Adrian. Moduli spaces of sheaves in mixed characteristic, Duke Math. J. 124 (2004), 571-586, DOI 10.1215/S0012-7094-04-12434-0, zbl 1086.14036, MR2085175.
  • 6. Langer, Adrian. Moduli spaces and Castelnuovo-Mumford regularity of sheaves on surfaces, Amer. J. Math. 128 (2006), 373-417, DOI 10.1353/ajm.2006.0014, zbl 1102.14030, MR2214897.
  • 7. Langer, Adrian. The Bogomolov-Miyaoka-Yau inequality for logarithmic surfaces in positive characteristic, Duke Math. J. 165 (2016), 2737-2769, DOI 10.1215/00127094-3627203, zbl 1386.14160, MR3551772.
  • 8. Maruyama, Masaki. Moduli spaces of stable sheaves on schemes. Restriction theorems, boundedness and the GIT construction. With the collaboration of T. Abe and M. Inaba. With a foreword by Shigeru Mukai. MSJ Memoirs, 33. Mathematical Society of Japan, Tokyo, 2016. xi+154 pp., DOI 10.2969/msjmemoirs/033010000, zbl 1357.14017, MR3495489.
  • 9. Moriwaki, Atsushi. A note on Bogomolov-Gieseker's inequality in positive characteristic, Duke Math. J. 64 (1991), 361-375, DOI 10.1215/S0012-7094-91-06418-5, zbl 0769.14005, MR1136381.


Langer, Adrian
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland