Harder–Narasimhan Filtrations which are not split by the Frobenius maps
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- 作者：SAURAV BHAUMIK ; VIKRAM MEHTA
- 关键词：Frobenius splitting ; Borel–Weil–Bott theorem ; strong Harder–Narasimhan Filtrations
- 刊名：Proceedings Mathematical Sciences
- 出版年：2013
- 出版时间：August 2013
- 年：2013
- 卷：123
- 期：3
- 页码：361-363
- 全文大小：168KB
- 参考文献：1. Biswas Indranil, Holla Yogish I, Parameswaran A J and Subramanian S, Construction of a Frobenius nonsplit Harder–Narasimhan filtration, / Comptes Rendus Mathematique 346(9-0) (2008) 545-48
2. Biswas Indranil and Parameswaran A J, On the ample vector bundles over curves in positive characteristic, / C.R. Acad. Sci. Paris, Ser. I 339 (2004) 355-58 CrossRef
3. Borel Armand, Linear representations of semi-simple algebraic groups, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I. 19, pp. 421-40
4. Haboush W J, A short proof of the Kempf vanishing theorem, / Inventiones Math. 56 (1980) 109-12 CrossRef
5. Langer Adrian, Semistable principal / G-bundles in positive characteristic, / Duke Math. J. 128(3) (2005) 511-40 CrossRef
6. Mehta Vikram and Ramanathan A, Frobenius splitting and cohomology vanishing for schubert varieties, / Ann. Math., Second Series 122(1) (1985) 27-0 CrossRef - 作者单位：SAURAV BHAUMIK (1)
VIKRAM MEHTA (2)
1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India
2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India - ISSN：0973-7685
文摘
We will produce a smooth projective scheme X over ? a rank 2 vector bundle V on X with a line subbundle L having the following property. For a prime p, let F p be the absolute Fobenius of X p , and let L p ??-em class="a-plus-plus">V p be the restriction of L??-em class="a-plus-plus">V. Then for almost all primes p, and for all t?≥-, $(F_p^*)^t L_P \subset (F_p^*)^t V_p$ is a non-split Harder-Narasimhan filtration. In particular, $(F_p^*)^t V_p$ is not a direct sum of strongly semistable bundles for any t. This construction works for any full flag veriety G/B, with semisimple rank of G?≥-. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic p.