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Hard Lefschetz theorems for free line bundles

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arxiv 2305.19085 v1 pith:2PLHJG2C submitted 2023-05-30 math.AG

Jiajun Hu , Shijie Shang , Jian Xiao This is my paper · ORCID
classification math.AG
keywords positivityalgebraicbundlesfreehardlefschetzlinemaps
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abstract

We introduce a partial positivity notion for algebraic maps via the defect of semismallness. This positivity notion is modeled on $m$-positivity in the analytic setting and $m$-ampleness in the geometric setting. Using this positivity condition for algebraic maps, we establish K\"ahler packages, that is, Hard Lefschetz theorems and Hodge-Riemann bilinear relations, for the complete intersections of Chern classes of free line bundles.

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  1. Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes

    math.AG 2025-05 conditional novelty 7.0 of 10

    For a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the product operator on (1,1)-classes is exactly the span of the prime divisors annihilated by the collection.

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