Generalized Brotbek's symmetric differential forms and applications

TL;DR

This paper advances the understanding of the Debarre Ampleness Conjecture by developing new symmetric differential forms and applying the moving coefficients method to prove the conjecture under certain conditions.

Contribution

It introduces formal matrices and a dividing device for constructing negatively twisted symmetric differential forms, extending previous methods and proving the conjecture when line bundles are nearly proportional.

Findings

Established the conjecture for almost proportional line bundles.

Provided explicit degree bounds in special cases.

Extended Brotbek's constructions with new algebraic tools.

Abstract

Over an algebraically closed field $\mathbb{K}$ with any characteristic, on an $N$-dimensional smooth projective $\mathbb{K}$-variety $\mathbf{P}$ equipped with $c\geqslant N/2$ very ample line bundles $\mathcal{L}_1,\dots,\mathcal{L}_c$, we study the General Debarre Ampleness Conjecture, which expects that for all large degrees $d_1,\dots,d_c\geqslant d_0\gg 1$, for generic $c$ hypersurfaces $ H_1\in \big|\mathcal{L}_1^{\,\otimes\,d_1}\big|$, $\dots$, $H_c\in \big|\mathcal{L}_c^{\,\otimes\,d_c}\big|$, the complete intersection $X:=H_1 \cap \cdots \cap H_c$ has ample cotangent bundle $\Omega_X$. First, we introduce a notion of formal matrices and a dividing device to produce negatively twisted symmetric differential forms, which extend the previous constructions of Brotbek and the author. Next, we adapt the moving coefficients method (MCM), and we establish that, if $\mathcal{L}_1,\dots,\mathcal{L}_c$ are almost proportional to each other, then the above conjecture holds true. Our method is effective: for instance, in the simple case $\mathcal{L}_1=\cdots=\mathcal{L}_c$, we provide an explicit lower degree bound $d_0=N^{N^2}$.