Effective bounds on ampleness of cotangent bundles

We prove that a general complete intersection of dimension n, codimension c and type d1, . . . , dc in PN has ample cotangent bundle if c ≥ 2n − 2 and the degrees di are all greater than a bound that is O(1) in N and quadratic in n. This degree bound substantially improves the currently best-known super-exponential bound in N by Deng, although our result does not address the case n ≤ c < 2n − 2.