Higher multiplier ideals are introduced as a two-parameter family of ideal sheaves via mixed Hodge modules, with vanishing/restriction theorems proved and new cases of theta divisor conjectures established.
Arakelov-type inequalities for Hodge bundles
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abstract
We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy around the singular fibers is unipotent. This relative canonical bundle is the (canonical extension of) the Hodge bundle and the inequality is generalized to the degrees of the Hodge bundles of a complex variation of Hodge structures.
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Proves algebraic hyperbolicity and big Picard theorems for Kähler manifolds with zero-dimensional period maps from polarized VHS, plus hyperbolicity and general type properties for their compactifications.
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Higher multiplier ideals
Higher multiplier ideals are introduced as a two-parameter family of ideal sheaves via mixed Hodge modules, with vanishing/restriction theorems proved and new cases of theta divisor conjectures established.
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Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures
Proves algebraic hyperbolicity and big Picard theorems for Kähler manifolds with zero-dimensional period maps from polarized VHS, plus hyperbolicity and general type properties for their compactifications.