Arakelov Inequalities for a Family of Surfaces Fibered by Curves

The numerical invariants of a variation of Hodge structure over a smooth quasi-projective variety are a measure of complexity for the global twisting of the limit-mixed Hodge structure when it degenerates. These invariants appear in formulas that may have correction terms called Arakelov inequalitie...

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Bibliographic Details
Published in:Mathematics (Basel) 2024-07, Vol.12 (13), p.1963
Main Author: Rahmati, Mohammad Reza
Format: Article
Language:English
Subjects:
Algebra
Decomposition
degeneration of Hodge structure
Hodge–Arakelov identities
Inequalities
Invariants
variation of Hodge structure
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Summary:The numerical invariants of a variation of Hodge structure over a smooth quasi-projective variety are a measure of complexity for the global twisting of the limit-mixed Hodge structure when it degenerates. These invariants appear in formulas that may have correction terms called Arakelov inequalities. We investigate numerical Arakelov-type equalities for a family of surfaces fibered by curves. Our method uses Arakelov identities in weight-one and weight-two variations of Hodge structure in a commutative triangle of two-step fibrations. Our results also involve the Fujita decomposition of Hodge bundles in these fibrations. We prove various identities and relationships between Hodge numbers and degrees of the Hodge bundles in a two-step fibration of surfaces by curves.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12131963