A general vanishing theorem (Q2253666)

Let \(X\) be a smooth projective variety over the complex numbers \({\mathbb C}\) of dimension \(n=\dim{X}\), \(E\) a vector bundle on \(X\) with \(e=\text{rank}{E}\) and \(L\) a line bundle on \(X\). Assume the tensor of \(L\) with a symmetric product \(S^{\alpha + \beta}E\otimes L\) is ample for non-negative integers \(\alpha, \beta\). In the paper under review, the authors show that \(H^q(X, S^\alpha{E} \otimes \bigwedge^\beta{E} \otimes L \otimes \Omega_X^p)=0\) for \(p+q - n> (r_0+\alpha)(e+\alpha -\beta ) - \alpha(\alpha +1)\) where \(r_0 := \min\{\beta, \delta(n-p), \delta(n-q)\}\) and \(\delta : {\mathbb N}\cup \{0\} \to {\mathbb N}\) is the numerical function such that \(\binom{\delta(x)}{2}\leq x< \binom{\delta(x)+1}{2}\).