On the structure of polarized manifolds with total deficiency one, I

A polarized variety is a pair (V, L ) of a variety V and an ample line bundle L on it.In this paper, for the sake of simplicity, we consider only the complex analytic cases.But many of our results hold to be true also in posi- tive characteristic cases after slight modifications (see [8]).In [5] we defined the following three invariants ofL)$ which might be called the total deficiency or the $Δ$ -genus, and $g(V, L)$ which might be called the sectional genus because of the following properties:\ l a n g l e$KV+(n-1)L)Lⁿ⁻¹$ for any smooth $V$ .In [5] we showed $Δ(V, L) 0$ for any polarized variety (V, $L$ ) and we gave a complete enumeration of polarized varieties with $Δ=0$ .Here we study polarized manifold $(M, L)$ with $Δ=1$ .$Δ(M, L)$ , together with $g(M, L)$ , might be regarded as a higher dimensional version of the genus of curves (see [6]).In particular, when $Δ(M, L)=1$ , we have: $L$ is very $ample⇐ d(M, L) 3⊂>g(M, L)=1$ .Moreover $ KM+(n-1)L=0⇔$ $Δ(M, L)=g(M, L)=1$ .So, to consider polarized surfaces with $Δ=g=1$ is equiv- alent to consider surfaces with negative canonical bundles, which are known classically as Del Pezzo surfaces.Generalizing this fact to higher dimensional cases, we will give a complete enumeration of all the Del Pezzo manifolds, that are, polarized manifolds with Δ=g=1 .However, in this part I, we treat only the cases in which $d(M, L)=2,3,4$ and 6 .For the study of the cases $d=1,5$ we need more complicated techniques.The result in case $d=1$ was announced in [4II] .The cases $d=5$ will be treated in the second part.Finally we give the classification table of Del Pezzo manifolds with dim M 3, $d(M, L)=2,3,4$ or 6 .