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Corrigendum to: Holomorphic Morse inequalities on manifolds with boundary

— A statement in the paper “Holomorphic Morse inequalities on manifolds with boundary” saying that the holomorphic Morse inequalities for an hermitian line bundle L over X are sharp as long as L extends as semi-positive bundle over a Stein-filling is corrected, by adding certain assumptions. A more general situation is also treated. Résumé. — Nous corrigeons l’énoncé qui affirme que les inégalités de Morse holomorphes pour un fibré hermitien en droites L sur X sont optimales tant que L s’étend comme un fibré semi-positif sur un remplissage Stein, paru dans l’article “Inégalités de Morse holomorphes sur des variétés à bord”. Nous ajoutons certaines conditions et considérons une situation plus générale. Let (L, φ) be a hermitian holomorphic line bundle over a closed (i.e. compact without boundary) complex hermitian manifold (Y, ω) of dimension n and let X = {ρ 6 0} be a strongly pseudoconcave domain in Y with smooth boundary. Let d := i(−∂+∂)/4π and denote by ddφ(= i∂∂φ/2π) the normalized curvature form of φ. If η is a (1, 1)-form we will write ηp := η/p! and we will denote by {η}y the hermitian linear operator on T (Y )y (or on some specified subbundle) corresponding to η (using a fixed metric form ω). The purpose here is to correct a statement in [1] saying that the holomorphic Morse inequalities on X obtained in [1] are always sharp when the curvature form ddφ is semi-positive on all of Y and Y −X is a Stein manifold. The correct statement, as shown below, is obtained by adding