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Equivariant Ulrich bundles on exceptional homogeneous varieties

  • Kyoung-Seog Lee and Kyeong-Dong Park EMAIL logo
From the journal Advances in Geometry


We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane E6/P1 and the E7-adjoint variety E7/P1. From this result,we see that a general hyperplane section F4/P4 of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.

Funding statement: The authors were supported by IBS-R003-Y1.


The authors would like to thank Rosa Maria Miró-Roig and Laurent Manivel for useful discussions and comments. Also, they are grateful to the referee for several helpful suggestions to improve the first draft.

  1. Communicated by: I. Coskun


[1] M. Aprodu, G. Farkas, A. Ortega, Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces. J. Reine Angew. Math. 730 (2017), 225–249. MR3692019 Zbl 1387.1408810.1515/crelle-2014-0124Search in Google Scholar

[2] R. J. Baston, M. G. Eastwood, The Penrose transform. Oxford Univ. Press 1989. MR1038279 Zbl 0726.58004Search in Google Scholar

[3] A. Beauville, Ulrich bundles on abelian surfaces. Proc. Amer. Math. Soc. 144 (2016), 4609–4611. MR3544513 Zbl 1375.1414810.1090/proc/13091Search in Google Scholar

[4] A. Beauville, An introduction to Ulrich bundles. Eur. J. Math. 4 (2018), 26–36. MR3782216 Zbl 1390.1413010.1007/s40879-017-0154-4Search in Google Scholar

[5] R. Bott, Homogeneous vector bundles. Ann. of Math. (2) 66 (1957), 203–248. MR89473 Zbl 0094.3570110.2307/1969996Search in Google Scholar

[6] R. Bott, On induced representations. In: The mathematical heritage of Hermann Weyl (Durham, NC, 1987), volume 48 of Proc. Sympos. Pure Math., 1–13, Amer. Math. Soc. 1988. MR974328 Zbl 0657.2202010.1090/pspum/048/974328Search in Google Scholar

[7] R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math. 88 (1987), 165–182. MR877011 Zbl 0617.1403410.1007/BF01405096Search in Google Scholar

[8] R. W. Carter, Lie algebras of finite and affine type. Cambridge Univ. Press 2005. MR2188930 Zbl 1110.1700110.1017/CBO9780511614910Search in Google Scholar

[9] M. Casanellas, R. Hartshorne, ACM bundles on cubic surfaces. J. Eur. Math. Soc. 13 (2011), 709–731. MR2781930 Zbl 1245.1404410.4171/JEMS/265Search in Google Scholar

[10] M. Casanellas, R. Hartshorne, F. Geiss, F.-O. Schreyer, Stable Ulrich bundles. Internat. J. Math. 23 (2012), 1250083, 50 pages. MR2949221 Zbl 1255.1403410.1142/S0129167X12500838Search in Google Scholar

[11] G. Casnati, Special Ulrich bundles on non-special surfaces with pg = q = 0. Internat. J. Math. 28 (2017), 1750061, 18 pages. MR3681122 Zbl 0677434410.1142/S0129167X17500616Search in Google Scholar

[12] Y. Cho, Y. Kim, K.-S. Lee, Ulrich bundles on intersections of two 4-dimensional quadrics. Int. Math. Res. Not. IMRN (2019) in Google Scholar

[13] I. Coskun, L. Costa, J. Huizenga, R. M. Miró-Roig, M. Woolf, Ulrich Schur bundles on flag varieties. J. Algebra 474 (2017), 49–96. MR3595784 Zbl 1358.1403210.1016/j.jalgebra.2016.11.008Search in Google Scholar

[14] L. Costa, R. M. Miró-Roig, GL(V)-invariant Ulrich bundles on Grassmannians. Math. Ann. 361 (2015), 443–457. MR3302625 Zbl 1330.1402110.1007/s00208-014-1076-9Search in Google Scholar

[15] L. Costa, R. M. Miró-Roig, Homogeneous ACM bundles on a Grassmannian. Adv. Math. 289 (2016), 95–113. MR3439681 Zbl 1421.1400610.1016/j.aim.2015.11.013Search in Google Scholar

[16] D. Eisenbud, J. Herzog, The classification of homogeneous Cohen-Macaulay rings of finite representation type. Math. Ann. 280 (1988), 347–352. MR929541 Zbl 0616.1301110.1007/BF01456058Search in Google Scholar

[17] D. Eisenbud, F.-O. Schreyer, J. Weyman, Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16 (2003), 537–579. MR1969204 Zbl 1069.1401910.1090/S0894-0347-03-00423-5Search in Google Scholar

[18] A. Fonarev, Irreducible Ulrich bundles on isotropic Grassmannians. Mosc. Math. J. 16 (2016), 711–726. MR3598504 Zbl 1386.1418010.17323/1609-4514-2016-16-4-711-726Search in Google Scholar

[19] W. Fulton, J. Harris, Representation theory. Springer 1991. MR1153249 Zbl 0744.22001Search in Google Scholar

[20] J. Herzog, B. Ulrich, J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71 (1991), 187–202. MR1117634 Zbl 0734.1300710.1016/0022-4049(91)90147-TSearch in Google Scholar

[21] J. Hong, Classification of smooth Schubert varieties in the symplectic Grassmannians.J. Korean Math. Soc. 52 (2015), 1109–1122. MR3393121 Zbl 1353.1405910.4134/JKMS.2015.52.5.1109Search in Google Scholar

[22] J. Hong, N. Mok, Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1. J. Algebraic Geom. 22 (2013), 333–362. MR3019452 Zbl 1274.1406110.1090/S1056-3911-2012-00611-0Search in Google Scholar

[23] J. E. Humphreys, Introduction to Lie algebras and representation theory. Springer 1972. MR0323842 Zbl 0254.1700410.1007/978-1-4612-6398-2Search in Google Scholar

[24] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), 479–508. MR939472 Zbl 0651.1800810.1007/BF01393744Search in Google Scholar

[25] M. Lahoz, E. Macrì, P. Stellari, Arithmetically Cohen-Macaulay bundles on cubic threefolds. Algebr. Geom. 2 (2015), 231–269. MR3350158 Zbl 1322.1403910.14231/AG-2015-011Search in Google Scholar

[26] M. Lahoz, E. Macrì, P. Stellari, Arithmetically Cohen-Macaulay bundles on cubic fourfolds containing a plane. In: Brauer groups and obstruction problems, 155–175, Springer 2017. MR3616010 Zbl 1396.1401910.1007/978-3-319-46852-5_8Search in Google Scholar

[27] J. M. Landsberg, L. Manivel, On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78 (2003), 65–100. MR1966752 Zbl 1048.1403210.1007/s000140300003Search in Google Scholar

[28] K.-S. Lee, K.-D. Park, Moduli spaces of Ulrich bundles on the Fano 3-fold V5. Preprint 2017, arXiv:1711.08305 [math.AG]10.1016/j.jalgebra.2021.01.015Search in Google Scholar

[29] I. A. Mihai, Odd symplectic flag manifolds. Transform. Groups 12 (2007), 573–599. MR2356323 Zbl 1135.1403910.1007/s00031-006-0053-0Search in Google Scholar

[30] B. Pasquier, On some smooth projective two-orbit varieties with Picard number 1. Math. Ann. 344 (2009), 963–987. MR2507635 Zbl 1173.1402810.1007/s00208-009-0341-9Search in Google Scholar

[31] D. M. Snow, Homogeneous vector bundles. In: Group actions and invariant theory (Montreal, PQ, 1988), volume 10 of CMS Conf. Proc., 193–205, Amer. Math. Soc. 1989. MR1021290 Zbl 0701.14017Search in Google Scholar

[32] J. Weyman, Cohomology of vector bundles and syzygies, volume 149 of Cambridge Tracts in Mathematics. Cambridge Univ. Press 2003. MR1988690 Zbl 1075.1300710.1017/CBO9780511546556Search in Google Scholar

[33] K. Yamaguchi, Differential systems associated with simple graded Lie algebras. In: Progress in differential geometry, volume 22 of Adv. Stud. Pure Math., 413–494, Math. Soc. Japan, Tokyo 1993. MR1274961 Zbl 0812.17018Search in Google Scholar

Received: 2018-09-18
Revised: 2019-06-20
Published Online: 2021-04-10
Published in Print: 2021-04-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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