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2 results

Positivity of Hodge bundles of abelian varieties over some function fields
Part of
- Arithmetic problems. Diophantine geometry
Xinyi Yuan
Journal:

Compositio Mathematica / Volume 157 / Issue 9 / September 2021

Published online by Cambridge University Press:

03 August 2021, pp. 1964-2000

Print publication:

September 2021
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The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate–Shafarevich group and the Tate conjecture of surfaces over finite fields.

Some Adjunction Properties of Ample Vector Bundles
Hironobu Ishihara
Journal:

Canadian Mathematical Bulletin / Volume 44 / Issue 4 / 01 December 2001

Published online by Cambridge University Press:

20 November 2018, pp. 452-458

Print publication:

01 December 2001
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Let $\mathcal{E}$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors ${{K}_{X}}\,+\,\left( t-r \right)\,\det \,\mathcal{E}$ when $t\,\ge \,\dim\,X$ and $t\,>\,r$. As an application, we classify pairs $\left( X,\,\mathcal{E} \right)$ with ${{c}_{r}}$-sectional genus zero.