BOCHNER–MARTINELLI FORMULAS ON SINGULAR COMPLEX SPACES

Abstract

Let U be a complex space of complex dimension n ≥ 2, P a point of U, π : Ũ → U a modification such that Ũ is nonsingular and D = π-1 (P) is a divisor with normal crossings. A Bochner–Martinelli form on U\{P} is a [Formula: see text]-closed differential form ω on Ũ\D, of pure type (n,n - 1), logarithmic along D. Such form detects a cohomology class of H2n - 1 (U\{P},ℂ) on the singular space U\{P}. Thanks to a general residue formula we prove that the forms ω give rise to an integral formula of Bochner–Martinelli type for holomorphic functions. If U satisfies the following assumption that {there exists a compact complex space X bimeromorphic to a Kähler manifold, and a closed subspace T ⊂ X, such that X\T = U (an affine, or a quasi-projective variety satisfies the above property), we relate Bochner–Martinelli forms to the mixed Hodge structure carried by H2n-1 (U\{P},ℂ). Most of our results hold for complex spaces which are not Stein.