Complex projective hypersurfaces of general type: toward a conjecture of Green and Griffiths

Abstract

Let X be a geometrically smooth n-dimensional projective algebraic complex hypersurface in P^{n+1}(C). Using Green-Griffiths jets, we establish the existence of nonzero global algebraic differential equations that must be satisfied by every nonconstant entire holomorphic curve C -> X if X is of general type, namely if its degree d satisfies the optimal possible lower bound: d >= n + 3. The case n = 2 dates back to Green-Griffiths 1979, while according to very recent advances (Invent. Math. 180, pp. 161-223, February 2010), the best (and only) lower degree bound known previously in arbitrary dimension n was, using instead Demailly-Semple jets, something like d >= 2^{n^4} . n^{5n^3}, which, visibly, was far from the conjectured n + 3.