Small bandwidth $\mathbb {C}^*$-actions and birational geometry
Authors:
Gianluca Occhetta, Eleonora A. Romano, Luis E. Solá Conde and Jarosław A. Wiśniewski
Journal:
J. Algebraic Geom. 32 (2023), 1-57
DOI:
https://doi.org/10.1090/jag/808
Published electronically:
June 8, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity—measured in terms of two invariants of the action, called bandwidth and bordism rank—and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.
References
- Dmitri N. Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995. MR 1334091, DOI 10.1007/978-3-322-80267-5
- Marco Andreatta and Gianluca Occhetta, Special rays in the Mori cone of a projective variety, Nagoya Math. J. 168 (2002), 127–137. MR 1942399, DOI 10.1017/S0027763000008400
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- Lorenzo Barban and Eleonora A. Romano, Toric nonequalized flips associated to $\mathbb {C}^*$-actions, arXiv:2104.14442, 2021.
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687, DOI 10.1515/9783110871746
- A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
- Andrzej Białynicki-Birula and Joanna Święcicka, Complete quotients by algebraic torus actions, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 10–22. MR 704983, DOI 10.1007/BFb0101505
- Jarosław Buczyński, Jarosław A. Wiśniewski, and Andrzej Weber, Algebraic torus actions on contact manifolds, arXiv:1802.05002, 2018; J. Differential Geom. (to appear).
- James B. Carrell, Torus actions and cohomology, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia Math. Sci., vol. 131, Springer, Berlin, 2002, pp. 83–158. MR 1925830, DOI 10.1007/978-3-662-05071-2_{2}
- James B. Carrell and Andrew John Sommese, Some topological aspects of $\textbf {C}^{\ast }$ actions on compact Kaehler manifolds, Comment. Math. Helv. 54 (1979), no. 4, 567–582. MR 552677, DOI 10.1007/BF02566293
- Paolo Cascini and Vladimir Lazić, New outlook on the minimal model program, I, Duke Math. J. 161 (2012), no. 12, 2415–2467. MR 2972461, DOI 10.1215/00127094-1723755
- Alessio Corti and Vladimir Lazić, New outlook on the minimal model program, II, Math. Ann. 356 (2013), no. 2, 617–633. MR 3048609, DOI 10.1007/s00208-012-0858-1
- David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
- Lawrence Ein and Nicholas Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), no. 5, 783–800. MR 1020829, DOI 10.2307/2374881
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
- Atsushi Ito, Algebro-geometric characterization of Cayley polytopes, Adv. Math. 270 (2015), 598–608. MR 3286545, DOI 10.1016/j.aim.2014.11.010
- Birger Iversen, A fixed point formula for action of tori on algebraic varieties, Invent. Math. 16 (1972), 229–236. MR 299608, DOI 10.1007/BF01425495
- Akihiro Kanemitsu, Extremal rays and nefness of tangent bundles, Michigan Math. J. 68 (2019), no. 2, 301–322. MR 3961218, DOI 10.1307/mmj/1549681299
- Friedrich Knop, Hanspeter Kraft, Domingo Luna, and Thierry Vust, Local properties of algebraic group actions, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, Birkhäuser, Basel, 1989, pp. 63–75. MR 1044585
- Robert Morelli, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), no. 4, 751–782. MR 1486987
- Gianluca Occhetta, Eleonora A. Romano, Luis E. Solá Conde, and Jarosław A. Wiśniewski, High rank torus actions on contact manifolds, Selecta Math. (N.S.) 27 (2021), no. 1, Paper No. 10, 33. MR 4215378, DOI 10.1007/s00029-021-00621-w
- Boris Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344 (2009), no. 4, 963–987. MR 2507635, DOI 10.1007/s00208-009-0341-9
- V. V. Przhiyalkovskiĭ, I. A. Chel′tsov, and K. A. Shramov, Fano threefolds with infinite automorphism groups, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 4, 226–280 (Russian, with Russian summary); English transl., Izv. Math. 83 (2019), no. 4, 860–907. MR 3985696, DOI 10.4213/im8834
- Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395–418. MR 717617
- Miles Reid, What is a flip?, http://homepages.warwick.ac.uk/staff/Miles.Reid/3folds, 1992.
- Eleonora A. Romano and Jarosław A. Wiśniewski, Adjunction for varieties with a $\mathbb {C}^*$ action, Transf. Groups, https://doi.org/10.1007/s00031-020-09627-8, 2020.
- E. A. Tevelev, Projective duality and homogeneous spaces, Encyclopaedia of Mathematical Sciences, vol. 133, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, IV. MR 2113135
- Michael Thaddeus, Toric quotients and flips, Topology, geometry and field theory, World Sci. Publ., River Edge, NJ, 1994, pp. 193–213. MR 1312182
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
- Jarosław A. Wiśniewski, Toric Mori theory and Fano manifolds, Geometry of toric varieties, Sémin. Congr., vol. 6, Soc. Math. France, Paris, 2002, pp. 249–272. MR 2063740
- Jarosław Włodarczyk, Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425–449. MR 1752010
- F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494, DOI 10.1090/mmono/127
References
- Dmitri N. Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995. MR 1334091, DOI 10.1007/978-3-322-80267-5
- Marco Andreatta and Gianluca Occhetta, Special rays in the Mori cone of a projective variety, Nagoya Math. J. 168 (2002), 127–137. MR 1942399, DOI 10.1017/S0027763000008400
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- Lorenzo Barban and Eleonora A. Romano, Toric nonequalized flips associated to $\mathbb {C}^*$-actions, arXiv:2104.14442, 2021.
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687, DOI 10.1515/9783110871746
- A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
- Andrzej Białynicki-Birula and Joanna Święcicka, Complete quotients by algebraic torus actions, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 10–22. MR 704983, DOI 10.1007/BFb0101505
- Jarosław Buczyński, Jarosław A. Wiśniewski, and Andrzej Weber, Algebraic torus actions on contact manifolds, arXiv:1802.05002, 2018; J. Differential Geom. (to appear).
- James B. Carrell, Torus actions and cohomology, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia Math. Sci., vol. 131, Springer, Berlin, 2002, pp. 83–158. MR 1925830, DOI 10.1007/978-3-662-05071-2_2
- James B. Carrell and Andrew John Sommese, Some topological aspects of $\mathbf {C}^{\ast }$ actions on compact Kaehler manifolds, Comment. Math. Helv. 54 (1979), no. 4, 567–582. MR 552677, DOI 10.1007/BF02566293
- Paolo Cascini and Vladimir Lazić, New outlook on the minimal model program, I, Duke Math. J. 161 (2012), no. 12, 2415–2467. MR 2972461, DOI 10.1215/00127094-1723755
- Alessio Corti and Vladimir Lazić, New outlook on the minimal model program, II, Math. Ann. 356 (2013), no. 2, 617–633. MR 3048609, DOI 10.1007/s00208-012-0858-1
- David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
- Lawrence Ein and Nicholas Shepherd-Barron, Some special Cremona transformations, Amer. J. Math. 111 (1989), no. 5, 783–800. MR 1020829, DOI 10.2307/2374881
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
- Atsushi Ito, Algebro-geometric characterization of Cayley polytopes, Adv. Math. 270 (2015), 598–608. MR 3286545, DOI 10.1016/j.aim.2014.11.010
- Birger Iversen, A fixed point formula for action of tori on algebraic varieties, Invent. Math. 16 (1972), 229–236. MR 299608, DOI 10.1007/BF01425495
- Akihiro Kanemitsu, Extremal rays and nefness of tangent bundles, Michigan Math. J. 68 (2019), no. 2, 301–322. MR 3961218, DOI 10.1307/mmj/1549681299
- Friedrich Knop, Hanspeter Kraft, Domingo Luna, and Thierry Vust, Local properties of algebraic group actions, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, Birkhäuser, Basel, 1989, pp. 63–75. MR 1044585
- Robert Morelli, The birational geometry of toric varieties, J. Algebraic Geom. 5 (1996), no. 4, 751–782. MR 1486987
- Gianluca Occhetta, Eleonora A. Romano, Luis E. Solá Conde, and Jarosław A. Wiśniewski, High rank torus actions on contact manifolds, Selecta Math. (N.S.) 27 (2021), no. 1, Paper No. 10, 33. MR 4215378, DOI 10.1007/s00029-021-00621-w
- Boris Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344 (2009), no. 4, 963–987. MR 2507635, DOI 10.1007/s00208-009-0341-9
- V. V. Przhiyalkovskiĭ, I. A. Chel′tsov, and K. A. Shramov, Fano threefolds with infinite automorphism groups, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 4, 226–280 (Russian, with Russian summary); English transl., Izv. Math. 83 (2019), no. 4, 860–907. MR 3985696, DOI 10.4213/im8834
- Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395–418. MR 717617
- Miles Reid, What is a flip?, http://homepages.warwick.ac.uk/staff/Miles.Reid/3folds, 1992.
- Eleonora A. Romano and Jarosław A. Wiśniewski, Adjunction for varieties with a $\mathbb {C}^*$ action, Transf. Groups, https://doi.org/10.1007/s00031-020-09627-8, 2020.
- E. A. Tevelev, Projective duality and homogeneous spaces, Encyclopaedia of Mathematical Sciences, vol. 133, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, IV. MR 2113135
- Michael Thaddeus, Toric quotients and flips, Topology, geometry and field theory, World Sci. Publ., River Edge, NJ, 1994, pp. 193–213. MR 1312182
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
- Jarosław A. Wiśniewski, Toric Mori theory and Fano manifolds, Geometry of toric varieties, Sémin. Congr., vol. 6, Soc. Math. France, Paris, 2002, pp. 249–272. MR 2063740
- Jarosław Włodarczyk, Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425–449. MR 1752010
- F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494, DOI 10.1090/mmono/127
Additional Information
Gianluca Occhetta
Affiliation:
Dipartimento di Matematica, Università di Trento, via Sommarive 14, I-38123 Trento, Italy
MR Author ID:
637085
ORCID:
0000-0002-4651-3245
Email:
gianluca.occhetta@unitn.it
Eleonora A. Romano
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova, Italy
MR Author ID:
1304597
ORCID:
0000-0003-1090-2588
Email:
eleonoraanna.romano@unige.it
Luis E. Solá Conde
Affiliation:
Dipartimento di Matematica, Università di Trento, via Sommarive 14, I-38123 Trento, Italy
ORCID:
0000-0003-0522-6909
Email:
eduardo.solaconde@unitn.it
Jarosław A. Wiśniewski
Affiliation:
Instytut Matematyki UW, Banacha 2, 02-097 Warszawa, Poland
ORCID:
0000-0003-1022-4254
Email:
J.Wisniewski@uw.edu.pl
Received by editor(s):
December 3, 2019
Published electronically:
June 8, 2022
Additional Notes:
The first and third authors were supported by PRIN project “Geometria delle varietà algebriche” and by MIUR project FFABR. The first, second and fourth authors were supported by the Department of Mathematics of the University of Trento. The first, third and fourth authors were supported by the Polish National Science Center project 2013/08/A/ST1/00804. The second author was supported by the Grant HA4383/-1 “ATAG” by the German Science Agency (DFG) and Thematic Einstein Semester “Varieties, Polyhedra, Computation” by the Berlin Einstein Foundation. The second and fourth authors were supported by the Polish National Science Center project 2016/23/G/ST1/04282.
Article copyright:
© Copyright 2022
University Press, Inc.
