Sections of Hodge bundles II: deformation of $(p,p)$-classes and applications to Kähler geometry
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918339548282880 |
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| author | Liu, Kefeng Shen, Yang |
| author_facet | Liu, Kefeng Shen, Yang |
| contents | Let $(X,ω_0)$ be a compact Kähler manifold and $\mathcal X\to B$ its Kuranishi family, where the base $B$ may be singular with $\dim_{\C} B \ge 1$. Using explicit sections of Hodge bundles obtained from algebraic and geometric constructions, we define an intrinsic period map and a Hodge map that parametrizes nearby $(p,p)$-classes.
For deformations over irreducible analytic bases, we introduce $\nabla^{1,1}$-flat extensions of Kähler cones and obtain explicit positive representatives, leading to an upper semicontinuity property for these extensions. Combined with the characterization of Kähler cones due to Demailly--Paun, this yields a complete local description of Kähler cones in terms of analytic cycles. We further show that this upper semicontinuity persists on a large region of the base determined by a uniform bound on the operator norm of the Beltrami differential.
As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms, and give an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13951 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sections of Hodge bundles II: deformation of $(p,p)$-classes and applications to Kähler geometry Liu, Kefeng Shen, Yang Algebraic Geometry Let $(X,ω_0)$ be a compact Kähler manifold and $\mathcal X\to B$ its Kuranishi family, where the base $B$ may be singular with $\dim_{\C} B \ge 1$. Using explicit sections of Hodge bundles obtained from algebraic and geometric constructions, we define an intrinsic period map and a Hodge map that parametrizes nearby $(p,p)$-classes. For deformations over irreducible analytic bases, we introduce $\nabla^{1,1}$-flat extensions of Kähler cones and obtain explicit positive representatives, leading to an upper semicontinuity property for these extensions. Combined with the characterization of Kähler cones due to Demailly--Paun, this yields a complete local description of Kähler cones in terms of analytic cycles. We further show that this upper semicontinuity persists on a large region of the base determined by a uniform bound on the operator norm of the Beltrami differential. As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms, and give an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture. |
| title | Sections of Hodge bundles II: deformation of $(p,p)$-classes and applications to Kähler geometry |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2602.13951 |