Abstract
A co-Higgs sheaf on a smooth complex projective variety X is a pair of a torsion-free coherent sheaf \(\mathcal {E}\) and a global section of \(\mathcal {E}nd(\mathcal {E})\otimes T_X\) with \(T_X\) the tangent bundle. We construct 2-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for \(\mathbb {P}^3\), using the Hartshorne-Serre correspondence. Then we investigate the non-existence, especially over projective spaces.
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References
Ballico, E., Huh, S.: A note on co-Higgs bundles. Taiwanese J. Math. 2, 267–281 (2017)
Barth, W.: Moduli of vector bundles on the projective plane. Invent. Math. 42, 63–91 (1977)
Beorchia, V.: Bounds for the genus of space curves. Math. Nachr. 184, 59–71 (1997)
Bolondi, G.: Smoothing curves by reflexive sheaves. Proc. Am. Math. Soc. 102(4), 797–803 (1988)
Brun, J.: Les fibrés de rang deux sur \(\mathbf{P}^2\) et leurs sections. Bull. Soc. Math. Fr. 108(4), 457–473 (1980)
Colmenares, A.V.: Semistable rank 2 co-Higgs bundles over Hirzebruch surfaces, Ph.D. thesis, Waterloo, Ontario (2015)
Colmenares, A.V.: Moduli spaces of semistable rank-2 co-Higgs bundles over P1*P1. Q. J. Math. 68(4), 1139–1162 (2017)
Corrêa, M.: Rank two nilpotent co-Higgs sheaves on complex surface. Geom. Dedicata 183(1), 25–31 (2016)
Gotzmann, G.: Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z. 158, 61–70 (1978)
Gotzmann, G.: Einige einfach-zusammenhängende Hilbertschemata. Math. Z. 180, 291–305 (1982)
Gualtieri, M.: Generalized complex geometry. Ann. Math. 174(1), 75–123 (2011)
Hartshorne, R.: Stable vector bundles of rank \(2\) on \(\mathbb{P}^3\). Math. Ann. 238, 229–280 (1978)
Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254(2), 121–176 (1980)
Hitchin, N.: Generalized Calabi–Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)
Hulek, K.: Stable rank-\(2\) vector bundles on \(\mathbf{P}_2\) with \(c_1\) odd. Math. Ann. 242(3), 241–266 (1979)
Iarrobino, A., Kanev, V.: With an appendix by A. Iarrobino and S. L. Kleiman, Power sums, Gorenstein algebras, and determinantal loci. In: Lecture Notes in Mathematics, vol. 1721. Appendix C by A. Iarrobino and S. L. Kleiman. Springer, Berlin (1999)
Le Potier, J.: Fibrés stables de rang 2 sur \(\mathbf{P}_2(\mathbb{C})\). Math. Ann. 241(3), 217–256 (1979)
Maruyama, M.: Stable vector bundles on an algebraic surface. Nagoya Math. J. 58, 25–68 (1975)
Rayan, S.: Geometry of co-Higgs bundles, Ph.D. thesis (2011)
Rayan, S.: Co-Higgs bundles on \(\mathbb{P}^1\). N. Y. J. Math. 19, 925–945 (2013)
Rayan, S.: Constructing co-Higgs bundles on \(\mathbb{C}\mathbb{P}^2\). Q. J. Math. 65(4), 1437–1460 (2014)
Simpson, C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1(4), 867–918 (1988)
Wahl, J.: A cohomological characterization of \(\mathbb{P}_n\). Invent. Math. 72, 315–322 (1983)
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The first author is partially supported by MIUR, GNSAGA of INDAM (Italy) and PRIN 2015 “Geometria delle varietà algebriche”, cofinanced by MIUR. The second author is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2018R1C1A6004285 and No. 2016R1A5A1008055).
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Ballico, E., Huh, S. 2-Nilpotent co-Higgs structures. manuscripta math. 159, 39–56 (2019). https://doi.org/10.1007/s00229-018-1045-9
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DOI: https://doi.org/10.1007/s00229-018-1045-9