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Arakelov Inequalities for a Family of Surfaces Fibered by Curves

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  • Mohammad Reza Rahmati

    (Laboratorio de Percepción y Robótica, Centro de Investigacion en Optica (CIO), Lomas del Bosque 115, Lomas del Campestre, Leon 37150, Mexico
    Centro de Investigacion en Matematicas (CIMAT), Col. Valenciana, Guanajuato 36023, Mexico)

Abstract

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.

Suggested Citation

  • Mohammad Reza Rahmati, 2024. "Arakelov Inequalities for a Family of Surfaces Fibered by Curves," Mathematics, MDPI, vol. 12(13), pages 1-19, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:13:p:1963-:d:1421429
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