The conjecture on specialization of 1-cycles

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Let XX be as in the cited theorem of Kollár--Tian, with generic fiber XKX_K and integer nn as in that theorem. Write C ⁣H2(X)C\!H_2(X) and C ⁣H1(XK)C\!H_1(X_K) for the relevant Chow groups, and A1(XK)A_1(X_K) for the group of algebraically trivial 1-cycles. The specialization conjecture for 1-cycles. The natural homomorphisms

C ⁣H2(X)/n→C ⁣H1(XK)/n→A1(XK)/nC\!H_2(X)/n\to C\!H_1(X_K)/n\to A_1(X_K)/n

are isomorphisms.

This concerns specialization and algebraic equivalence of 1-cycles in the degeneration considered by the paper. The source gives no resolution status for the assertion.

References

Primary source

Morten Lüders, “On analogues of the Kato conjectures and proper base change for 1-cycles on rationally connected varieties”, arXiv:2409.14497 (2024).

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