Restrictions of semistable bundles on projective varieties. (English) Zbl 0599.14015
Let E be a semistable vector bundle of rank r on an n-dimensional normal projective variety X over an algebraically closed field of characteristic 0. A restriction theorem is a theorem implying the semistableness (or stableness if E is stable) of the restriction of E to a general hypersurface of a certain degree (or more generally, to a general complete intersection of a certain type). There are several such theorems in the literature [see e.g., the papers by O. Forster, A. Hirschowitz and M. Schneider in Vector bundles and differential equations, Proc., Nice 1979, Prog. Math. 7, 65-81 (1980; Zbl 0441.14007); M. Maruyama, Nagoya Math. J. 78, 65-94 (1978; Zbl 0456.14011) and V. B. Mehta and A. Ramanathan, Invent. Math. 77, 163-172 (1984; Zbl 0525.55012)]. However, all these results need hypotheses either on the rank of E or on the variety X, or the bounds depends on invariants of E, which are difficult to determine.
The paper under review now gives a restriction theorem valid for an arbitrary normal projective variety, arbitrary rank of E, and the bound only depends on the degrees of X and E. To be more precise: Let E be a semistable torsion-free \({\mathcal O}_ X\)-module of rank r and d and c integers (1\(\leq c\leq n-1)\), such that \[ (\binom {n+d}{d})-cd-1)/d>\deg X\cdot \max ((r^ 2-1)/4,1). \] Then for a general complete intersection \(Y=H_ 1\cap...\cap H_ c\), where the \(H_ i\) are restrictions of hyperplanes of degree d to X, the restriction \(E| Y\) is semistable of Y. The bound is not optimal, but it does imply some of the old restriction theorems, however not all of them. It is a consequence of the following theorem: Let E be semistable and torsion-free on X, \(Y\subseteq X\) a general complete intersection, and let \(0=E_ 0\subseteq E_ 1\subseteq...\subseteq E\otimes {\mathcal O}_ Y\) be the Harder-Narasimhan filtration of \(E\times {\mathcal O}_ Y\). Then \[ 0<\mu (E_ i/E_{i-1})- \mu (E_{i+1}/E_ i)\leq d^{c+1} \deg Y/(\binom {n+d}{d}-cd-1). \] Here \(\mu(F)\) denotes the number \(\deg(F)/rk(F)\) for a nonzero torsion- free \({\mathcal O}_ Y\)-module F. The proof uses the associated flag manifolds and, in particular, a result for its relative tangent sheaf.
The paper under review now gives a restriction theorem valid for an arbitrary normal projective variety, arbitrary rank of E, and the bound only depends on the degrees of X and E. To be more precise: Let E be a semistable torsion-free \({\mathcal O}_ X\)-module of rank r and d and c integers (1\(\leq c\leq n-1)\), such that \[ (\binom {n+d}{d})-cd-1)/d>\deg X\cdot \max ((r^ 2-1)/4,1). \] Then for a general complete intersection \(Y=H_ 1\cap...\cap H_ c\), where the \(H_ i\) are restrictions of hyperplanes of degree d to X, the restriction \(E| Y\) is semistable of Y. The bound is not optimal, but it does imply some of the old restriction theorems, however not all of them. It is a consequence of the following theorem: Let E be semistable and torsion-free on X, \(Y\subseteq X\) a general complete intersection, and let \(0=E_ 0\subseteq E_ 1\subseteq...\subseteq E\otimes {\mathcal O}_ Y\) be the Harder-Narasimhan filtration of \(E\times {\mathcal O}_ Y\). Then \[ 0<\mu (E_ i/E_{i-1})- \mu (E_{i+1}/E_ i)\leq d^{c+1} \deg Y/(\binom {n+d}{d}-cd-1). \] Here \(\mu(F)\) denotes the number \(\deg(F)/rk(F)\) for a nonzero torsion- free \({\mathcal O}_ Y\)-module F. The proof uses the associated flag manifolds and, in particular, a result for its relative tangent sheaf.
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
