Fano Hypersurfaces in Weighted Projective 4-Spaces

A Fano variety is a projective variety whose anticanonical class is ample. A 2–dimensional Fano variety is called a Del Pezzo surface. In higher dimensions, attention originally centered on smooth Fano 3–folds, but singular Fano varieties are also of considerable interest in connection with the minimal model program. The existence of Kähler–Einstein metrics on Fano varieties has also been explored, see [Bourguignon97] for a summary of the main results. Here again the smooth case is of primary interest, but Fano varieties with quotient singularities and their orbifold metrics have also been studied. In any given dimension there are only finitely many families of smooth Fano varieties [Campana91, Nadel91, KoMiMo92], but very little is known about them in dimensions 4 and up. By allowing singularities, infinitely many families appear and their distribution is very poorly understood. A natural experimental testing ground is given by hypersurfaces and complete intersections in weighted projective spaces. These can be written down rather explicitly, but they still provide many more examples than ordinary projective spaces. Experimental lists of certain 3–dimensional complete intersections were compiled by [Fletcher89]. In connection with Kähler–Einstein metrics, the 2–dimensional cases were first investigated in [Demailly-Kollár99] and later in [Johnson-Kollár00]. It is also of interest to study Calabi–Yau hypersurfaces and hypersurfaces of general type in weighted projective spaces. [Fletcher89] contains some lists with terminal singularities. The aim of this paper is threefold. First, we determine the complete list of anticanonically embedded quasi smooth Fano hypersurfaces in weighted projective 4-spaces. There are 48 infinite series and 4442 sporadic examples (7). As a consequence we obtain that the Reid–Fletcher list (cf. [Fletcher89, II.6.6]) of 95 types of anticanonically embedded quasi smooth terminal Fano threefolds in weighted projective 4-spaces is complete (11). Second, we prove that many of these Fano hypersurfaces admit a Kähler–Einstein metric (15). We also study the nonexistence of tigers on these Fano 3–folds (in the colorful terminology of [Keel-McKernan99]).