Geometric criteria for $\mathbb {A}^1$-connectedness and applications to norm varieties
Authors:
Chetan Balwe, Amit Hogadi and Anand Sawant
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/790
Published electronically:
August 29, 2022
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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 $k$.
References
- 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 $\Bbb A^1$-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 $\Bbb A^1$-homotopy and algebraic $h$-cobordisms, Adv. Math. 227 (2011), no. 5, 1990–2058. MR 2803793, DOI 10.1016/j.aim.2011.04.009
- Chetan Balwe, Amit Hogadi, and Anand Sawant, $\Bbb {A}^1$-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
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
- Marvin J. Greenberg, Schemata over local rings, Ann. of Math. (2) 73 (1961), 624–648. MR 126449, DOI 10.2307/1970321
- 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
- Bruno Kahn and R. Sujatha, Birational geometry and localisation of categories, Doc. Math. Extra vol.: Alexander S. Merkurjev’s sixtieth birthday (2015), 277–334. With appendices by Jean-Louis Colliot-Thélène and Ofer Gabber. MR 3404383
- Nikita A. Karpenko and Alexander S. Merkurjev, 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
- Arthur Mattuck, The field of multisymmetric functions, Proc. Amer. Math. Soc. 19 (1968), 764–765. MR 225774, DOI 10.1090/S0002-9939-1968-0225774-6
- Fabien Morel, The stable ${\Bbb A}^1$-connectivity theorems, $K$-Theory 35 (2005), no. 1-2, 1–68. MR 2240215, DOI 10.1007/s10977-005-1562-7
- Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
- Dinh Huu Nguyen, Standard norm varieties for Milnor symbols $\textrm {mod}\, p$, Ann. K-Theory 1 (2016), no. 4, 457–475. MR 3536435, DOI 10.2140/akt.2016.1.457
- Markus Rost, Norm varieties and algebraic cobordism, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 77–85. MR 1957022
- Rodney Y. Sharp, The dimension of the tensor product of two field extensions, Bull. London Math. Soc. 9 (1977), no. 1, 42–48. MR 437510, DOI 10.1112/blms/9.1.42
- Andrei Suslin and Seva Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276. MR 2220090, DOI 10.1016/j.jpaa.2005.12.012
- 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
References
- 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
- Chetan Balwe, Amit Hogadi, and Anand Sawant, $\mathbb {A}^1$-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
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
- Marvin J. Greenberg, Schemata over local rings, Ann. of Math. (2) 73 (1961), 624–648. MR 126449, DOI 10.2307/1970321
- 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
- Bruno Kahn and R. Sujatha, Birational geometry and localisation of categories, Doc. Math. Extra vol.: Alexander S. Merkurjev’s sixtieth birthday (2015), 277–334. With appendices by Jean-Louis Colliot-Thélène and Ofer Gabber. MR 3404383
- Nikita A. Karpenko and Alexander S. Merkurjev, 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
- Arthur Mattuck, The field of multisymmetric functions, Proc. Amer. Math. Soc. 19 (1968), 764–765. MR 225774, DOI 10.2307/2035879
- Fabien Morel, The stable ${\mathbb {A}}^1$-connectivity theorems, $K$-Theory 35 (2005), no. 1-2, 1–68. MR 2240215, DOI 10.1007/s10977-005-1562-7
- Fabien Morel and Vladimir Voevodsky, ${\mathbf {A}}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
- Dinh Huu Nguyen, Standard norm varieties for Milnor symbols ${\mathrm {mod}}\, p$, Ann. K-Theory 1 (2016), no. 4, 457–475. MR 3536435, DOI 10.2140/akt.2016.1.457
- Markus Rost, Norm varieties and algebraic cobordism, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 77–85. MR 1957022
- Rodney Y. Sharp, The dimension of the tensor product of two field extensions, Bull. London Math. Soc. 9 (1977), no. 1, 42–48. MR 437510, DOI 10.1112/blms/9.1.42
- Andrei Suslin and Seva Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276. MR 2220090, DOI 10.1016/j.jpaa.2005.12.012
- 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.
