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Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ of two curves over$$\mathbb {Q} $$ with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ is finite, where$$J_i$$ is the Jacobian variety of$$C_i$$ . Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ for which the analogous map$$\varepsilon $$ has finite image.
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An analytic proof of the stable reduction theorem
Abstract The stable reduction theorem says that a family of curves of genus$$g\ge 2$$ over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new this result for curves defined over$${\mathbb {C}}$$ , using the Kähler–Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.
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- Award ID(s):
- 2203607
- PAR ID:
- 10500847
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 390
- Issue:
- 3
- ISSN:
- 0025-5831
- Format(s):
- Medium: X Size: p. 4245-4263
- Size(s):
- p. 4245-4263
- Sponsoring Org:
- National Science Foundation
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