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 $c⩾N/2$ 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 $d_1,\dots ,d_c⩾N^{N^2}$ works. Lastly, thanks to known results about the Fujita Conjecture, we establish the very-ampleness of ${\mathsf{Sym}}^{\kappa }{\Omega }_X$ for all $\kappa ⩾64({\sum }_{i=1}^c{d}_i{)}^2$.

We modify the effective degree estimates so that they are also valid for the Algorithm in our recent paper arXiv:1601.05133

References in corpus (2)

• Some recent transcendental techniques in algebraic and complex geometry
• Slanted Vector Fields for Jet Spaces

Cited by in corpus (2)

• Effectivity in the Hyperbolicity-related problems
• On the Diverio-Trapani Conjecture