On the ampleness of the cotangent bundles of complete intersections

Abstract

Based on a geometric interpretation of Brotbek's symmetric differential forms, for the intersection family $\mathcal{X}$ of generalized Fermat-type hypersurfaces in $\mathbb{P}_{\mathbb{K}}^N$ defined over any field $\mathbb K$, we reconstruct explicit symmetric differential forms by applying Cramer's rule, skipping cohomology arguments, and we further exhibit unveiled families of lower degree symmetric differential forms on all possible intersections of $\mathcal{X}$ with coordinate hyperplanes. Thereafter, we develop what we call the `moving coefficients method' to prove a conjecture made by Olivier Debarre: for generic $c\geqslant N/2$ hypersurfaces $H_1,\dots,H_c\subset \mathbb{P}_{\mathbb C}^N$ of degrees $d_1,\dots,d_c$ sufficiently large, the intersection $X:=H_1 \cap \cdots \cap H_c $ has ample cotangent bundle $\Omega_X$, and concerning effectiveness, the lower bound $ d_1,\dots,d_c\geqslant N^{N^2} $ works. Lastly, thanks to known results about the Fujita Conjecture, we establish the very-ampleness of $\mathsf{Sym}^{\kappa}\,\Omega_X$ for all $\kappa\geqslant 64\, \Big( \sum_{i=1}^c\, d_i \Big)^2 $.