## 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.

### MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

### Keywords:

semistable vector bundle; restriction theorem

### Citations:

Zbl 0534.55011; Zbl 0441.14007; Zbl 0456.14011; Zbl 0525.55012
Full Text: