**Stable rationality and diagonal decomposition**

July 04-July 15, 2016 | Location: BICMR, Peking University

Topics:

Stable rationality problem, integral Chow and cohomological diagonal decomposition

Organizers:

Zhiyuan Li (Fudan), Zhiyu Tian (CNRS) and Chenyang Xu (BICMR)

Classroom: 82J12

Schedule:

- A schedule is here.

Participants:

Qile Chen (Boston College)

Lie Fu (Lyon)

Xiaowen Hu (BICMR)

Zhi Jiang (Fudan)

Junpeng Jiao (BICMR)

Jun Li (Fudan and Stanford)

Zhiyuan Li (Fudan)

Yuchen Liu (Princeton)

Mingmin Shen (Amsterdam)

Peng Sun (BICMR)

Zhiyu Tian (CNRS)

Xiaowei Wang (Rutgers)

Chenyang Xu (BICMR)

Qizheng Yin (ETH)

Tong Zhang (Durham)

Qiang Zhao (BICMR)

Chuyu Zhou (BICMR)

Yi Zhu (Waterloo)

Reference

Warm up

- Hassett: Some rational cubic fourfolds/ Special cubic fourfolds
- Kollar: Nonrational hypersurface

- Nonrationality

- Voisin: Unirational threefolds with no universal codimension $2$ cycle
- Totaro: Hypersurfaces that are not stably rational
- Colliot-Thelene and Pirutka: Hypersurfaces quartiques de dimension 3: non rationalite stable; Cyclic covers that are not stably rational

- Beauville: A very general sextic double solid is not stably rational
- Hassett-Tschinkel: Stable rationality and conic bundles, On stable rationality of Fano threefolds and del Pezzo fibrations

- Hassett-Pirutka-Tschinkel: Stable rationality of quadratic surface bundle over surfaces

- Cohomological diagonal decomposition

- Voisin:Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, On the universal $CH_0$ group of cubic hypersurfaces

- Shen: Hyperkähler manifolds of jacobian type, Rationality, universal generation and the integral Hodge conjecture

Further questions and discussion

- Relation between Chow DDC and cohomological DDC, i.e
- (i). find an example (dimension is expected >4) which admits cohomological DDC but does not admit Chow DDC
- (ii). find other examples where cohomological DDC is equivalent to Chow DDC

- Possible improvement of Voisin’s criterion
- Stably rationality of conic bundles (threefolds), quadric bundles (fourfolds), cubic fourfolds

Acknowledgement

The activity is supported by BICMR, funding from NSFC.