Generalized Brotbek's symmetric differential forms and applications
Song-Yan Xie

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 with any characteristic, on an -dimensional smooth projective -variety equipped with very ample line bundles , we study the General Debarre Ampleness Conjecture, which expects that for all large degrees , for generic hypersurfaces , , , the complete intersection has ample cotangent bundle . 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…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Nonlinear Waves and Solitons
