SUMMARY

The mini-course discusses a recent work establishing Colliot-Thélène’s conjecture on the local-global principle for 0-cycles on geometrically rational surfaces over global function fields. Consisting of an arithmetic part and a geometric, and based on an analysis of spaces of 1-cycles, the work relies on proving the integral Tate conjecture for 1-cycles on certain 3-folds over finite fields.

Schedule: January 13, 15, 20, 22, 2026; 2–4 p.m. each day

Session Breakdown:

January 13 : Main theorem and deformations of stable maps of curves (Speaker: Suzuki)

● Colliot-Thélène's conjecture on local-global principle for 0-cycles

● Strategy for addressing arithmetic and geometric parts of the cycle map

January 15 : Descent of algebraic equivalence of 1-cycles (Speaker: Suzuki)

● Smoothing a curve by attaching to it a comb of certain rational curves

● Technical versions of Kollár–Tian theorems

January 20 : Coniveau and strong coniveau filtrations I (Speaker: Paranjape)

● Proof of the geometric part

● Lawson homology for complex varieties

January 22 : Coniveau and strong coniveau filtrations II (Speaker: Suzuki)

● Chow sheaves and motivic cohomology theory

● Conclusion of the proof of the main theorem

Mini-Course Abstract on Recent Work of Kollár–Tian and Tian

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SPEAKERS

Fumiaki Suzuki

Kapil Paranjape