Chern classes of vector bundles on arithmetic varieties

Nakashima, Tohru; Takeda, Yoshifumi

doi:10.2140/pjm.1996.176.205

Cited by 2 publications

(1 citation statement)

References 12 publications

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“…Incidentally, the upper bound above is related to a question posed in [59], asking whether (−K X ) n+1 is bounded from above by a universal constant C n , under the assumption that X be non-singular and −K X be relatively ample. This is a stronger condition than having positive curvature, as we assume.…”

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confidence: 99%

Sharp bounds on the height of K-semistable toric Fano varieties

Andreasson¹,

Berman²

2022

Preprint

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Inspired by Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety X of relative dimension n is maximal when X is the projective space over the integers, endowed with the Fubini-Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when n ≤ 6 (the extension to higher dimensions is conditioned on a conjectural "gap hypothesis" for the degree). Translated into toric Kähler geometry this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. We furthermore reformulate our conjecture as an optimal lower bound on Odaka's modular height. Along the way we show how to control the height of the canonical toric model X , with respect to the Kähler-Einstein metric, by the algebrogeometric degree of X in any dimension n. Logarithmic generalizations of the results are also given.

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confidence: 99%

Sharp bounds on the height of K-semistable toric Fano varieties

Andreasson¹,

Berman²

2022

Preprint

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Sharp bounds on the height of K-semistable Fano varieties I, the toric case

Andreasson,

Berman

2024

Compositio Math.

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Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\mathcal {X}$ of relative dimension $n$ is maximal when $\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\leq 6$ (the extension to higher dimensions is conditioned on a conjectural ‘gap hypothesis’ for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\mathcal {X}$ . In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.

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Chern classes of vector bundles on arithmetic varieties

Cited by 2 publications

References 12 publications

Sharp bounds on the height of K-semistable toric Fano varieties

Sharp bounds on the height of K-semistable toric Fano varieties

Sharp bounds on the height of K-semistable Fano varieties I, the toric case

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