Geometric criteria for $\mathbb {A}^1$-connectedness and applications to norm varieties

Authors:
Chetan Balwe, Amit Hogadi and Anand Sawant

Journal:
J. Algebraic Geom. **32** (2023), 677-696

DOI:
https://doi.org/10.1090/jag/790

Published electronically:
August 29, 2022

Full-text PDF

Abstract |
References |
Additional Information

Abstract: We show that $\mathbb {A}^1$-connectedness of a large class of varieties over a field $k$ can be characterized as the condition that their generic point can be connected to a $k$-rational point using (not necessarily naive) $\mathbb {A}^1$-homotopies. We also show that symmetric powers of $\mathbb {A}^1$-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic $0$) are $\mathbb {A}^1$-connected. As an application of these results, we show that the standard norm varieties over a field $k$ of characteristic $0$ become $\mathbb {A}^1$-connected (and consequently, universally $R$-trivial) after base change to an algebraic closure of $k$.

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- Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jarosław Włodarczyk,
*Torification and factorization of birational maps*, J. Amer. Math. Soc. **15** (2002), no. 3, 531–572. MR **1896232**, DOI 10.1090/S0894-0347-02-00396-X
- Aravind Asok,
*Birational invariants and $\mathbb {A}^1$-connectedness*, J. Reine Angew. Math. **681** (2013), 39–64. MR **3181489**, DOI 10.1515/crelle-2012-0034
- Aravind Asok,
*Rationality problems and conjectures of Milnor and Bloch-Kato*, Compos. Math. **149** (2013), no. 8, 1312–1326. MR **3103066**, DOI 10.1112/S0010437X13007021
- Aravind Asok and Fabien Morel,
*Smooth varieties up to $\mathbb {A}^1$-homotopy and algebraic $h$-cobordisms*, Adv. Math. **227** (2011), no. 5, 1990–2058. MR **2803793**, DOI 10.1016/j.aim.2011.04.009
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*$\mathbb {A}^1$-connected components of schemes*, Adv. Math. **282** (2015), 335–361. MR **3374529**, DOI 10.1016/j.aim.2015.07.003
- C. Balwe and A. Sawant,
*$\mathbb A^1$-connected components of ruled surfaces*, Preprint, arXiv:1911.05549 [math.AG], 2019, Geom. Topol. (2021) to appear.
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc,
*La $R$-équivalence sur les tores*, Ann. Sci. École Norm. Sup. (4) **10** (1977), no. 2, 175–229 (French). MR **450280**
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc,
*The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)*, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 113–186. MR **2348904**
- Shizuo Endo,
*The rationality problem for norm one tori*, Nagoya Math. J. **202** (2011), 83–106. MR **2804547**, DOI 10.1215/00277630-1260459
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- A. Grothendieck,
*Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV*, Inst. Hautes Études Sci. Publ. Math. **32** (1967), 361 (French). MR **238860**
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*On standard norm varieties*, Ann. Sci. Éc. Norm. Supér. (4) **46** (2013), no. 1, 175–214 (2013) (English, with English and French summaries). MR **3087392**, DOI 10.24033/asens.2187
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- Vladimir Voevodsky,
*On motivic cohomology with $\mathbf Z/l$-coefficients*, Ann. of Math. (2) **174** (2011), no. 1, 401–438. MR **2811603**, DOI 10.4007/annals.2011.174.1.11

Additional Information

**Chetan Balwe**

Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, Knowledge City, Sector-81, Mohali 140306, India

MR Author ID:
677361

ORCID:
0000-0001-9989-5606

Email:
cbalwe@iisermohali.ac.in

**Amit Hogadi**

Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India

MR Author ID:
788736

Email:
amit@iiserpune.ac.in

**Anand Sawant**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

MR Author ID:
1118734

Email:
asawant@math.tifr.res.in

Received by editor(s):
February 16, 2021

Received by editor(s) in revised form:
March 16, 2021, July 12, 2021, and July 13, 2021

Published electronically:
August 29, 2022

Additional Notes:
The first author was supported by SERB-DST MATRICS Grant: MTR/2017/000690. The third author was supported by SERB Start-up Research Grant SRG/2020/000237 and the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.01-0500

Article copyright:
© Copyright 2022
University Press, Inc.