NUMERICAL CONSTRAINTS FOR EMBEDDED PROJECTIVE MANIFOLDS GIAN MARIO BESANA AND ALDO BIANCOFIORE Abstract. General formulas giving numerical constraints for projec- tive invariants of embedded, complex, projective manifolds are explicitly worked out in the framework of adjunction theory. 1. Introduction The constraints imposed on a complex manifold by its being embedded in a projective space with a low codimension have been studied intensely. An n- fold X can always be embedded in P 2n+1 . The possibility of embedding an n- fold X in P 2n is related to the number of double points of a generic projection of X from P 2n+1 . Double point formulas, expressing these constraints in terms of Chern classes of the manifolds and its normal bundle, can be traced back to Severi, [45], (see also Catanese, [16]). They were rediscovered and generalized by Holme, [29], and independently by Peters and Simonis, [44]. They are now essentially a special case of the Laksov-Todd double point formula, [37]. An excellent general reference is due to Kleiman, [36]. In the recent past, numerical constraints for embedded projective mani- folds have been extensively utilized in the classification of complex projective varieties of given degree. This clasification was accomplished by Weil, [49], Swinnerton Dyer, [48], Ionescu, [30], [31], [33], Okonek, [43], [42], Alexander, [2], Abo, Decker and Sasakura, [1], up to degree eight. With a successful application of adjunction theory, Beltrametti, Schneider and Sommese, [8], [9], classified threefolds in P 5 of degree up to twelve, while Fania and Livorni, [19], [20], classified manifolds of degree nine and ten, of dimension n ≥ 3, regardless of the codimension. Recently Bertolini, [13], classified threefolds of degree twelve in P 6 . The general approach implemented by most of the above authors is a two-step process. The first step consists of taking full advantage of classical adjunction theory and endow the manifold with a special geometric struc- ture, typically a fibration. The peculiar properties of the acquired geometric structure are then exploited in the second step in searching for constraints for the numerical characters of the manifold. 2000 Mathematics Subject Classification. Primary (14N25, 14N30, 14J) ; Secondary (14J30, 14J35, 14J40). Key words and phrases. classification, adjunction, numerical constraints. 1