A note on homotopy complex surfaces with negative tangent bundles (Q1183100)

The following results are proven: ``Let \(X\) be a compact two-fold with a negative tangent bundle and \(M\) be a minimal compact complex two-fold which is not diffeomorphic with the standard \(S^ 2\times S^ 2\) (respectively \(M\) be a minimal non-spin compact complex two-fold). Suppose \(X\) and \(M\) satisfy one of the conditions: a) \(X\) and \(M\) are homotopic and their fundamental group is finite. b) \(X\) and \(M\) are homeomorphic. Then the cotangent dimension of \(M\) is equal to two. The cotangent dimension is a bimeromorphic invariant; and a compact complex two-fold with cotangent dimension two is algebraic and of general type. The proofs of the results are based on a lemma due to Bogomolov (estimating the growth of dimension of \(H^ 0(V,S^ nE)\) or \(H^ 0(V,S^ nE^*)\) for certain vector bundles \(E\) on a compact twofold \(V)\).