Ballico, E. Complements of complete intersections in infinite-dimensional complex projective spaces. (English) Zbl 1054.32013 Bull. Inst. Math., Acad. Sin. 32, No. 2, 127-131 (2004). Summary: Let \(V\) be a Banach space with countable unconditional basis and \({\mathbf P}(V)\) the projective space of all one-dimensional linear subspaces of \(V\). Let \(f_1,\dots,f_s\), \(x>0\), be finitely many continuous homogeneous polynomials on \(V\). Set \(X(f_i):=\{P\in {\mathbf P}(V):f_i(P)\neq 0\}\) and \(X:=X (f_1)\cup\cdots\cup X(f_s)\). Here we prove that \(H^i(X,{\mathcal O}_X)=0\) for every \(i\geq s\) and we give a condition assuring that \(H^{s-1}(X,{\mathcal O}_X)\) is infinite-dimensional. MSC: 32K05 Banach analytic manifolds and spaces 58B99 Infinite-dimensional manifolds 14M10 Complete intersections 32C35 Analytic sheaves and cohomology groups 32L20 Vanishing theorems 32C38 Sheaves of differential operators and their modules, \(D\)-modules Keywords:complex infinite-dimensional projective space; Banach manifold; complex Banach manifold; cohomology group; vanishing theorem; hypersurface; complete intersection; pseudoconvex domain PDFBibTeX XMLCite \textit{E. Ballico}, Bull. Inst. Math., Acad. Sin. 32, No. 2, 127--131 (2004; Zbl 1054.32013)