Let X be a smooth complex variety of dimension n and let E be an ample vector bundle of rank ron X. Then we calculate the i_th sectional Euler number ei(PX(E),H(E))for i ≥ 2_n - 3 or i = 1, and the i_th sectional Hodge numberof type (j,i - j) hi i-j(PX(E),H(E)) for i ≥ 2_n - 1 and 0 ≤ j ≤ i, where PX(E) is the projective space bundle associated withE and H(E) is its tautological line bundle. Moreover we define a new invariant v(X,E)for r ≥ n - 1. This invariant is thought to be a generalization of curve genus. We will investigate some properties of this invariant.
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Yoshiaki Fukuma, Sectional Invariants of Scroll over a Smooth Projective Variety. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 93–119DOI 10.4171/RSMUP/121-6