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Canonical filtrations and stability of direct images by Frobenius morphisms. (English) Zbl 1201.14030
Summary: We study the stability of direct images by Frobenius morphisms. First, we compute the first Chern classes of direct images of vector bundles by Frobenius morphisms modulo rational equivalence up to torsions. Next, introducing the canonical filtrations, we prove that if \(X\) is a nonsingular projective minimal surface of general type with semistable \(\varOmega_X^1\) with respect to the canonical line bundle \(K_X\), then the direct images of line bundles on \(X\) by Frobenius morphisms are semistable with respect to \(K_X\).

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14J29 Surfaces of general type
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[1] P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique. II, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973.
[2] P. Deligne and L. Illusie, Relèvements modulo \(p^2\) et décomposition du complexe de De Rham, Invent. Math. 89 (1987), 247–270. · Zbl 0632.14017
[3] T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes études Sci. Publ. Math. 67 (1988), 97–144. · Zbl 0674.14028
[4] W. Fulton, Intersection theory Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2., Springer-Verlag, Berlin, 1998. · Zbl 0885.14002
[5] D. Gieseker, Stable vector bundles and the Frobenius morphism, Ann. Sci. école Norm. Sup. (4) 6 (1973), 95–101. · Zbl 0281.14013
[6] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects Math. E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. · Zbl 0872.14002
[7] S. Ilangovan, V. B. Mehta and A. J. Parameswaran, Semistability and semisimplicity in representations of low height in positive characteristic, A tribute to C. S. Seshadri (Chennai, 2002), 271–282, Trends Math., Birkhäuser, Basel, 2003. · Zbl 1067.20061
[8] K. Joshi, S. Ramanan, E. Xia and J. K. Yu, On vector bundles destabilized by Frobenius pull-back, Compos. Math. 142 (2006), 616–630. · Zbl 1101.14049
[9] N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes études Sci. Publ. Math. 39 (1970), 175–232. · Zbl 0221.14007
[10] K. Kurano, The singular Riemann-Roch theorem and Hilbert-Kunz functions, J. Algebra 304 (2006), 487–499. · Zbl 1109.13015
[11] H. Lange and C. Pauly, On Frobenius-destabilized rank-2 vector bundles over curves, arXiv.math.AG/0309456 v2, (2005). · Zbl 1157.14017
[12] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276. · Zbl 1080.14014
[13] V. B. Mehta and C. Pauly, Semistability of Frobenius direct images over curves, arXiv.math.AG/0607565 v1, (2006). · Zbl 1201.14021
[14] A. Noma, Stability of Frobenius pull-backs of tangent bundles of weighted complete intersections, Math. Nachr. 221 (2001), 87–93. · Zbl 0992.14016
[15] J. P. Serre, Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math. 116 (1994), 513–530. · Zbl 0816.20014
[16] N. I. Shepherd-Barron, Unstable vector bundles and linear systems on surfaces in characteristic \(p\), Invent. Math. 106 (1991), 243–262. · Zbl 0769.14006
[17] N. I. Shepherd-Barron, Geography for surfaces of general type in positive characteristic, Invent. Math. 106 (1991), 263–274. · Zbl 0813.14025
[18] N. I. Shepherd-Barron, Semi-stability and reduction mod \(p\), Topology 37 (1998), 659–664. · Zbl 0926.14021
[19] X. Sun, Stability of direct images under Frobenius morphism, arXiv.math.AG/0608043 v2, (2006).
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