The Euclidean elliptic complex
Nicholas Hanges, Howard Jacobowitz · Indiana University Mathematics Journal · 1997
Let Ω ⊂ C m × R n be an open subset with smooth boundary.The de Rham complex of R n and the Dolbeaut complex of C m induce a natural elliptic complex on Ω D : Λ p (Ω) -→ Λ p+1 (Ω), p = 0, . . ., m + n.D induces a natural involutive (formally integrable) structure, D b , on the boundary of Ω.Let Σ ⊂ ∂Ω be the set of points where D b is not elliptic.We assume that the Levi form is positive definite at each point of Σ. Away from Σ we make no assumption.Under these conditions we show that the cohomology for D vanishes in dimension p ≥ 1.