Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives
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2025
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| _version_ | 1866908442366574592 |
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| author | Lin, Xun Zhang, Shizhuo Zhang, Zheng |
| author_facet | Lin, Xun Zhang, Shizhuo Zhang, Zheng |
| contents | We investigate infinitesimal Torelli problems for some of the Fano threefolds of the following two types: (a) those which can be described as zero loci of sections of vector bundles on Grassmannians (for instance, ordinary Gushel-Mukai threefolds), and (b) double covers of rigid Fano threefolds branched along a $K3$ surface (such as, special Gushel-Mukai threefolds). The differential of the period map for ordinary Gushel-Mukai threefolds has been studied by Debarre, Iliev and Manivel; in particular, it has a $2$-dimensional kernel. The main result of this paper is that the invariant part of the infinitesimal period map for a special Gushel-Mukai threefold is injective. We prove this result using a Hodge theoretical argument as well as a categorical method. Through similar approaches, we also study infinitesimal Torelli problems for prime Fano threefolds with genus $7$, $8$, $9$, $10$, $12$ (type (a)) and for special Verra threefolds (type (b)). Furthermore, a geometric description of the kernel of the differential of the period maps for Gushel-Mukai threefolds (and for prime Fano threefolds of genus $8$) is given via a Bridgeland moduli space in the Kuznetsov components. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives Lin, Xun Zhang, Shizhuo Zhang, Zheng Algebraic Geometry Primary 14F08, secondary 14J45, 14J10, 14C34 We investigate infinitesimal Torelli problems for some of the Fano threefolds of the following two types: (a) those which can be described as zero loci of sections of vector bundles on Grassmannians (for instance, ordinary Gushel-Mukai threefolds), and (b) double covers of rigid Fano threefolds branched along a $K3$ surface (such as, special Gushel-Mukai threefolds). The differential of the period map for ordinary Gushel-Mukai threefolds has been studied by Debarre, Iliev and Manivel; in particular, it has a $2$-dimensional kernel. The main result of this paper is that the invariant part of the infinitesimal period map for a special Gushel-Mukai threefold is injective. We prove this result using a Hodge theoretical argument as well as a categorical method. Through similar approaches, we also study infinitesimal Torelli problems for prime Fano threefolds with genus $7$, $8$, $9$, $10$, $12$ (type (a)) and for special Verra threefolds (type (b)). Furthermore, a geometric description of the kernel of the differential of the period maps for Gushel-Mukai threefolds (and for prime Fano threefolds of genus $8$) is given via a Bridgeland moduli space in the Kuznetsov components. |
| title | Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives |
| topic | Algebraic Geometry Primary 14F08, secondary 14J45, 14J10, 14C34 |
| url | https://arxiv.org/abs/2507.06995 |