0 Ja n 20 06 Cayley submanifolds of Calabi - Yau 4 - folds

Our main results are: (1) The complex and Lagrangian points of a non-complex and nonLagrangian 2n-dimensional submanifold F :M →N , immersed with parallel mean curvature and with equal Kähler angles into a Kähler-Einstein manifold (N, J, g) of complex dimension 2n, are zeros of finite order of sin θ and cos θ respectively, where θ is the common J-Kähler angle. (2) If M is a Cayley submanifold of a Calabi-Yau (CY) manifold N of complex dimension 4, then ∧2 + NM is naturally isomorphic to ∧2 + TM . (3) If N is Ricci-flat (not necessarily CY) and M is a Cayley submanifold, then p1( ∧2 + NM) = p1( ∧2 + TM) still holds, but p1( ∧2 − NM) − p1( ∧2 − TM) may describe a residue on the J-complex points, in the sense of Harvey and Lawson. We describe this residue by a PDE on a natural morphism Φ : TM → NM , Φ(X) = (JX), with singularities at the complex points. We give an explicit formula of this residue in a particular case. When (N, I, J,K, g) is a hyper-Kähler manifold and M is an I-complex closed 4-submanifold, the first Weyl curvature invariant of M may be described as a residue on the J-Kähler angle at the J-Lagrangian points by a Lelong-Poincaré type formula. We study the almost complex structure Jω on M induced by F .