Homogeneous ACM and Ulrich bundles on rational homogeneous spaces
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Xinyi Fang
Abstract
In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles on rational homogeneous spaces. From this result, we see that there are only finitely many irreducible homogeneous ACM bundles (up to twist) and Ulrich bundles on these varieties. Moreover, we give numerical criteria for some special irreducible homogeneous bundles to be ACM bundles.
Funding statement: The research is sponsored by Innovation Action Plan (Basic research projects) of Science and Technology Commission of Shanghai Municipality (Grant No. 21JC1401900) and Jiangsu Funding Program for Excellent Postdoctoral Talent.
References
[1] M. C. Brambilla and D. Faenzi, Moduli spaces of rank-2 ACM bundles on prime Fano threefolds, Michigan Math. J. 60 (2011), no. 1, 113–148. 10.1307/mmj/1301586307Search in Google Scholar
[2] J. P. Brennan, J. Herzog and B. Ulrich, Maximally generated Cohen–Macaulay modules, Math. Scand. 61 (1987), no. 2, 181–203. 10.7146/math.scand.a-12198Search in Google Scholar
[3] R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Cohen–Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), no. 1, 165–182. 10.1007/BF01405096Search in Google Scholar
[4] R. W. Carter, Lie Algebras of Finite and Affine Type, Cambridge Stud. Adv. Math. 96, Cambridge University, Cambridge, 2005. 10.1017/CBO9780511614910Search in Google Scholar
[5] M. Casanellas, E. Drozd and R. Hartshorne, Gorenstein liaison and ACM sheaves, J. Reine Angew. Math. 584 (2005), 149–171. 10.1515/crll.2005.2005.584.149Search in Google Scholar
[6] M. Casanellas and R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 709–731. 10.4171/jems/265Search in Google Scholar
[7] M. Casanellas, R. Hartshorne, F. Geiss and F.-O. Schreyer, Stable Ulrich bundles, Internat. J. Math. 23 (2012), no. 8, Article ID 1250083. 10.1142/S0129167X12500838Search in Google Scholar
[8] I. Coskun, L. Costa, J. Huizenga, R. M. Miró-Roig and M. Woolf, Ulrich Schur bundles on flag varieties, J. Algebra 474 (2017), 49–96. 10.1016/j.jalgebra.2016.11.008Search in Google Scholar
[9]
L. Costa and R. M. Miró-Roig,
[10] L. Costa and R. M. Miró-Roig, Homogeneous ACM bundles on a Grassmannian, Adv. Math. 289 (2016), 95–113. 10.1016/j.aim.2015.11.013Search in Google Scholar
[11] R. Du, X. Fang and P. Ren, Homogeneous ACM bundles on isotropic Grassmannians, Forum Math. 35 (2023), no. 3, 763–782. 10.1515/forum-2022-0157Search in Google Scholar
[12] D. Eisenbud and J. Herzog, The classification of homogeneous Cohen–Macaulay rings of finite representation type, Math. Ann. 280 (1988), no. 2, 347–352. 10.1007/BF01456058Search in Google Scholar
[13] D. Faenzi, Rank 2 arithmetically Cohen–Macaulay bundles on a nonsingular cubic surface, J. Algebra 319 (2008), no. 1, 143–186. 10.1016/j.jalgebra.2007.10.005Search in Google Scholar
[14] D. Faenzi and J. Pons-Llopis, The Cohen–Macaulay representation type of projective arithmetically Cohen–Macaulay varieties, Épijournal Géom. Algébrique 5 (2021), Article ID 8. 10.46298/epiga.2021.volume5.7113Search in Google Scholar
[15] X. Y. Fang, The generalizations of Horrocks criterion on flag varieties, Comm. Anal. Geom., to appear. Search in Google Scholar
[16] X. Y. Fang, Y. Nakayama and P. Ren, Homogeneous ACM bundles on exceptional Grassmannians, preprint (2023), https://arxiv.org/abs/2211.02275. Search in Google Scholar
[17] M. Filip, Rank 2 ACM bundles on complete intersection Calabi–Yau threefolds, Geom. Dedicata 173 (2014), 331–346. 10.1007/s10711-013-9945-zSearch in Google Scholar
[18] A. Fonarev, Irreducible Ulrich bundles on isotropic Grassmannians, Mosc. Math. J. 16 (2016), no. 4, 711–726. 10.17323/1609-4514-2016-16-4-711-726Search in Google Scholar
[19] W. Fulton and J. Harris, Representation Theory, Grad. Texts in Math. 129, Springer, New York, 1991. Search in Google Scholar
[20] G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. Lond. Math. Soc. (3) 14 (1964), 689–713. 10.1112/plms/s3-14.4.689Search in Google Scholar
[21] H. Knörrer, Cohen–Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), no. 1, 153–164. 10.1007/BF01405095Search in Google Scholar
[22] K.-S. Lee and K.-D. Park, Equivariant Ulrich bundles on exceptional homogeneous varieties, Adv. Geom. 21 (2021), no. 2, 187–205. 10.1515/advgeom-2020-0018Search in Google Scholar
[23]
K.-S. Lee and K.-D. Park,
Moduli spaces of Ulrich bundles on the Fano 3-fold
[24] G. Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. (4) 155 (1989), 317–341. 10.1007/BF01765948Search in Google Scholar
[25] G. Ottaviani, Rational homogeneous varieties, Lecture Notes for the Summer school in Algebraic Geometry in Cortona (1995). Search in Google Scholar
[26]
G. V. Ravindra and A. Tripathi,
Rank 3 ACM bundles on general hypersurfaces in
[27] D. M. Snow, Homogeneous vector bundles, Group Actions and Invariant Theory, CMS Conf. Proc. 10, American Mathematical Society, Providence (1989), 193–205. Search in Google Scholar
[28] K. Watanabe, ACM line bundles on polarized K3 surfaces, Geom. Dedicata 203 (2019), 321–335. 10.1007/s10711-019-00436-2Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Some properties of extended frame measure
- Orthogonal separation of variables for spaces of constant curvature
- Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications
- Uniform bounds for Kloosterman sums of half-integral weight with applications
- q-supercongruences from Watson's 8φ7 transformation
- Submodules of normalisers in groupoid C*-algebras and discrete group coactions
- Explicit bounds for the solutions of superelliptic equations over number fields
- The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth
- A Kollár-type vanishing theorem for k-positive vector bundles
- New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators
- On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting
- A p-adic analog of Hasse--Davenport product relation involving ϵ-factors
- On arithmetic quotients of the group SL2 over a quaternion division k-algebra
- Euler’s integral, multiple cosine function and zeta values
- Paley inequality for the Weyl transform and its applications
- Homogeneous ACM and Ulrich bundles on rational homogeneous spaces
- Arithmetic progression in a finite field with prescribed norms
