SOME REMARKS AID PROBLEMS CONCERNING THE GEOGRAPHY OF FANO 4-FOLDS OF INDEX AND PICARDNUMBER ONE

Abstract

Let V be a Fano 4-fold of index one and Picardnumber one, i.e. a 4-dimensional smooth variety such that the anticanonical divisor—K _V is ample and generates Div(V). We consider problems concerning the “geography” of these manifolds: for which values of (—K _V )⁴ and h ⁰(V, O _V (—K _V )) do such V exist? We describe various classes of examples, discuss some of their properties and state several open problems.