Effective bounds of linear series on algebraic varieties and arithmetic varieties
-
Xinyi Yuan
and
Tong Zhang
Abstract
In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert–Samuel formula and the arithmetic Hilbert–Samuel formula. The treatments are high-dimensional generalizations of [25] and [26]. Similar results are obtained independently by Huayi Chen [7] with less explicit error terms.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1330987
Funding statement: The first author is supported by the NSF grant DMS-1330987. The second author is supported by an NSERC discovery grant.
Acknowledgements
The authors would like to thank Huayi Chen and the anonymous referees of our article [Duke. Math. J. 162 (2013), 1723–1770], who bring insights of the current high-dimensional setting. We are also indebted to the referees of the current article for their invaluable comments.
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Articles in the same Issue
- Frontmatter
- Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities
- The exponential map at a cuspidal singularity
-
Independence of
- Pro unitality and pro excision in algebraic K-theory and cyclic homology
- Kinematic formulas for tensor valuations
- Linear independence of dilogarithmic values
- Positivity in varieties of maximal Albanese dimension
- Effective bounds of linear series on algebraic varieties and arithmetic varieties
- Spectral asymptotics for canonical systems
Articles in the same Issue
- Frontmatter
- Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities
- The exponential map at a cuspidal singularity
-
Independence of
- Pro unitality and pro excision in algebraic K-theory and cyclic homology
- Kinematic formulas for tensor valuations
- Linear independence of dilogarithmic values
- Positivity in varieties of maximal Albanese dimension
- Effective bounds of linear series on algebraic varieties and arithmetic varieties
- Spectral asymptotics for canonical systems
