• The Journal of Natural Sciences
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• v.10 no.1
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• pp.13-16
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• 1998

Every complex vector bundle over $S^1$ splits sum of line bundle and the first Chern class classify complex line bundle. This implies every complex vector bundle over $S^1$ is trivial. In this paper, we show the existence of some nontrivial complex vector bundle over $S^1$ in the equivariant case.

• East Asian mathematical journal
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• v.14 no.2
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• pp.371-375
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• 1998

• Journal of the Korean Mathematical Society
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• v.47 no.6
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• pp.1147-1165
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• 2010

In this paper we compare a torsion free sheaf F on $P^N$ and the free vector bundle $\oplus^n_{i=1}O_{P^N}(b_i)$ having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of F. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes $c_i$(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove for rank n torsion free sheaves on $P^N$, whose splitting type has no gap (i.e., $b_i{\geq}b_{i+1}{\geq}b_i-1$ 1 for every i = 1,$\ldots$,n-1), the following formula for the discriminant: $$\Delta(F):=2_{nc_2}-(n-1)c^2_1\geq-\frac{1}{12}n^2(n^2-1)$$. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes $c_3$(F(t)),$\ldots$,$c_n$(F(t)) for the dimension of the cohomology modules $H^iF(t)$ and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend only on $c_1(F)$, $c_2(F)$, the splitting type of F and t.