Vector Bundles on Products of Projective Spaces and Hyperquadrics

Here we consider the space $\mathbf P^{n_1}\times\dots \times\mathbf P^{n_s}\times Q_{m_1}\times\dots \times\mathbf Q_{m_q}$. We introduce a notion of Castelnuovo-Mumford regularity in order to prove two splitting criteria for vector bundles with arbitrary rank.