A Frobenius Version of Tians Alpha-Invariant
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909725476519936 |
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| author | Pande, Suchitra |
| author_facet | Pande, Suchitra |
| contents | For a pair (X,L) consisting of a projective variety X over a perfect field of characteristic p>0 and an ample line bundle L on X, we introduce and study a positive characteristic analog of the $α$-invariant introduced by Tian, which we call the $α_F$-invariant. We utilize the theory of F-singularities in positive characteristics, and our approach is based on replacing klt singularities with the closely related notion of global F-regularity. We show that the $α_F$-invariant of a pair (X,L) can be understood in terms of the global Frobenius splittings of the linear systems |mL|, for m>0. We establish inequalities relating the $α_F$-invariant with the F- signature, and use that to prove the positivity of the $α_F$-invariant for all globally F-regular projective varieties (with respect to any ample L on X). When X is a Fano variety and L is $-K_X$, we prove that the $α_F$-invariant of X is always bounded above by 1/2 and establish tighter comparisons with the F-signature. We also show that for toric Fano varieties, the $α_F$-invariant matches with the usual (complex) $α$-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00989 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Frobenius Version of Tians Alpha-Invariant Pande, Suchitra Algebraic Geometry Commutative Algebra 14J45, 14J17, 13A35 For a pair (X,L) consisting of a projective variety X over a perfect field of characteristic p>0 and an ample line bundle L on X, we introduce and study a positive characteristic analog of the $α$-invariant introduced by Tian, which we call the $α_F$-invariant. We utilize the theory of F-singularities in positive characteristics, and our approach is based on replacing klt singularities with the closely related notion of global F-regularity. We show that the $α_F$-invariant of a pair (X,L) can be understood in terms of the global Frobenius splittings of the linear systems |mL|, for m>0. We establish inequalities relating the $α_F$-invariant with the F- signature, and use that to prove the positivity of the $α_F$-invariant for all globally F-regular projective varieties (with respect to any ample L on X). When X is a Fano variety and L is $-K_X$, we prove that the $α_F$-invariant of X is always bounded above by 1/2 and establish tighter comparisons with the F-signature. We also show that for toric Fano varieties, the $α_F$-invariant matches with the usual (complex) $α$-invariant. |
| title | A Frobenius Version of Tians Alpha-Invariant |
| topic | Algebraic Geometry Commutative Algebra 14J45, 14J17, 13A35 |
| url | https://arxiv.org/abs/2311.00989 |