STABLE MAPS OF CURVES AND ALGEBRAIC EQUIVALENCE OF 1-CYCLES

János Kollár; Zhiyu Tian

doi:10.1215/00127094-2024-0045

STABLE MAPS OF CURVES AND ALGEBRAIC EQUIVALENCE OF 1-CYCLES

János Kollár, Zhiyu Tian

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern A₁.X/, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If K=k is an algebraic extension, then the kernel of A₁.X_k/ ! A₁.X_K/ is at most Z=2Z. If k is finite, then the image equals the subgroup of Galois invariant cycles.

Original language	English (US)
Pages (from-to)	911-947
Number of pages	37
Journal	Duke Mathematical Journal
Volume	174
Issue number	5
DOIs	https://doi.org/10.1215/00127094-2024-0045
State	Published - Apr 1 2025

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1215/00127094-2024-0045

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abstract = "We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern A1.X/, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If K=k is an algebraic extension, then the kernel of A1.Xk/ ! A1.XK/ is at most Z=2Z. If k is finite, then the image equals the subgroup of Galois invariant cycles.",

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AB - We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern A1.X/, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If K=k is an algebraic extension, then the kernel of A1.Xk/ ! A1.XK/ is at most Z=2Z. If k is finite, then the image equals the subgroup of Galois invariant cycles.

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STABLE MAPS OF CURVES AND ALGEBRAIC EQUIVALENCE OF 1-CYCLES

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