Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Fiammetta Battaglia [a] and Elisa Prato [a]

Nonrational polytopes and fans in toric geometry

Pages: 67-86
Received: 2 May 2022
Accepted in revised form: 16 January 2023
Mathematics Subject Classification: 14M25, 52B20, 53D20.
Keywords: Toric variety, nonrational convex polytope, nonrational fan.
Authors address:
[a]: Università degli Studi di Firenze, Dipartimento di Matematica e Informatica "U. Dini", Firenze, 50134, Italy

This research was partially supported by the PRIN Project "Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics" (MIUR, Italy) and by GNSAGA (INdAM, Italy).

Full Text (PDF)

Abstract: First, we examine the notion of nonrational convex polytope and nonrational fan in the context of toric geometry. We then discuss and interrelate some recent developments in the subject.

References
[1]
M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. MR0642416
[2]
M. Audin, The topology of torus actions on symplectic manifolds, Progr. Math., 93, Birkhäuser Verlag, Basel, 1991. MR1106194
[3]
F. Battaglia, Convex polytopes and quasilattices from the symplectic viewpoint, Comm. Math. Phys. 269 (2007), 283-310. MR2274549
[4]
F. Battaglia, Geometric spaces from arbitrary convex polytopes, Internat. J. Math. 23 (2012), 1250013, 39 pp. MR2888940
[5]
F. Battaglia and E. Prato, Generalized toric varieties for simple nonrational convex polytopes, Internat. Math. Res. Notices (2001), no. 24, 1315-1337. MR1866747
[6]
F. Battaglia and E. Prato, The symplectic geometry of Penrose rhombus tilings, J. Symplectic Geom. 6 (2008), 139-158. MR2434438
[7]
F. Battaglia and E. Prato, The symplectic Penrose kite, Comm. Math. Phys. 299 (2010), 577-601. MR2718924
[8]
F. Battaglia and E. Prato, Ammann tilings in symplectic geometry, SIGMA Symmetry Integrability Geom. Methods Appl. 9 (2013), Paper 021, 13 pp. MR3034408
[9]
F. Battaglia and E. Prato, Toric geometry of the regular convex polyhedra, J. Math. (2017), Art. ID 2542796, 15 pp. MR3630682
[10]
F. Battaglia and E. Prato, Nonrational symplectic toric cuts, Internat. J. Math. 29 (2018), no. 10, 1850063, 19 pp. MR3861905
[11]
F. Battaglia and E. Prato, Nonrational symplectic toric reduction, J. Geom. Phys. 135 (2019), 98-105. MR3872625
[12]
F. Battaglia, E. Prato and D. Zaffran, Hirzebruch surfaces in a one-parameter family, Boll. Unione Mat. Ital. 12 (2019), 293-305. MR3936308
[13]
F. Battaglia and D. Zaffran, Foliations modeling nonrational simplicial toric varieties, Int. Math. Res. Not. IMRN (2015), no. 22, 11785-11815. MR3456702
[14]
F. Battaglia and D. Zaffran, Simplicial toric varieties as leaf spaces, in "Special metrics and group actions in geometry", Springer INdAM Ser. 23, Springer, Cham, 2017, 1-21. MR3751960
[15]
A. Boivin, Non--simplicial quantum toric varieties, arXiv:2006.16715, preprint, 2020. DOI
[16]
F. Bosio, Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano-Verjovsky, Ann. Inst. Fourier (Grenoble) 51 (2001), 1259-1297. MR1860666
[17]
P. Bressler and V. A. Lunts, Intersection cohomology on nonrational polytopes, Compositio Math. 135 (2003), 245-278. MR1956814
[18]
P. Bressler and V. A. Lunts, Hard Lefschetz theorem and Hodge--Riemann relations for intersection cohomology of nonrational polytopes, Indiana Univ. Math. J. 54 (2005), 263-307. MR2126725
[19]
A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Math., 1764, Springer-Verlag, Berlin, 2001 MR1853077
[20]
D. A. Cox, J. B. Little and H. K. Schenck, Toric varieties, Grad. Stud. Math., 124, American Mathematical Society, Providence, 2011. MR2810322
[21]
V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97-154. DOI
[22]
J. A. De Loera, J. Rambau and F. Santos, Triangulations, Structures for algorithms and applications, Algorithms Comput. Math., 25, Springer-Verlag, Berlin, 2010. MR2743368
[23]
T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. MR0984900
[24]
M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. 3 (1970), 507-588. MR0284446
[25]
W. Fulton, Introduction to toric varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, 1993. MR1234037
[26]
B. Grünbaum, Convex polytopes, Grad. Texts in Math., 221, Springer-Verlag, New York, 2003. MR1976856
[27]
V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian \(T^n\)-spaces, Progr. Math., 122, Birkhäuser, Boston, 1994. MR1301331
[28]
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513. MR0664117
[29]
B. Hoffman, Toric symplectic stacks, Adv. Math. 368 (2020), 107135, 43 pp. MR4082991
[30]
B. Hoffman and R. Sjamaar, Stacky Hamiltonian actions and symplectic reduction, Int. Math. Res. Not. IMRN (2020), no. 20, 15209-15300. MR4329869
[31]
P. Iglesias--Zemmour and E. Prato, Quasifolds, diffeology and noncommutative geometry, J. Noncommut. Geom. 15 (2021), 735-759. MR4325720
[32]
H. Ishida, Torus invariant transverse Kähler foliations, Trans. Amer. Math. Soc. 369 (2017), 5137-5155. MR3632563
[33]
H. Ishida, Complex manifolds with maximal torus actions, J. Reine Angew. Math. 751 (2019), 121-184. MR3956693
[34]
H. Ishida , R. Krutowski and T. Panov, Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds, Int. Math. Res. Not. IMRN (2022), no. 7, 5541-5563. MR4403969
[35]
K. Karu, Hard Lefschetz theorem for nonrational polytopes, Invent. Math. 157 (2004), 419-447. MR2076929
[36]
L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky, Quantum (non-commutative) toric geometry: foundations, Adv. Math. 391 (2021), Paper No. 107945, 110 pp. MR4300912
[37]
R. Krutowski and T. Panov, Dolbeault cohomology of complex manifolds with torus action, in "Topology, geometry, and dynamics-V. A. Rokhlin-Memorial", Contemp. Math., 772, American Mathematical Society, Providence, 2021, 73-187. MR4305539
[38]
E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201-4230. MR1401525
[39]
Y. Lin and R. Sjamaar, Convexity properties of presymplectic moment maps, J. Symplectic Geom. 17 (2019), 1159-1200. MR4031537
[40]
J. J. Loeb and M. Nicolau, On the complex geometry of a class of non-Kählerian manifolds, Israel J. Math. 110 (1999), 371-379. MR1750427
[41]
S. López de Medrano and A. Verjovsky, A new family of complex, compact, non-symplectic manifolds, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), 253-269. MR1479504
[42]
A. L. Mackay, De nive quinquangula: on the pentagonal snowflake, Soviet Phys. Cryst. 26 (1981), 517-522. MR0651500
[43]
L. Meersseman, A new geometric construction of compact complex manifolds in any dimension, Math. Ann. 317 (2000), 79-115. MR1760670
[44]
L. Meersseman and A. Verjovsky, Holomorphic principal bundles over projective toric varieties, J. Reine Angew. Math. 572 (2004), 57-96. MR2076120
[45]
R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Math. Intelligencer 2 (1979/80/1980), no. 1, 32-37. MR0558670
[46]
A. F. Pir and F. Sottile, Irrational toric varieties and secondary polytopes, Discrete Comput. Geom. 67 (2022), 1053-1079. MR4419618
[47]
E. Prato, Sur une généralisation de la notion de V-variété, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 887-890. MR1689861
[48]
E. Prato, Simple non-rational convex polytopes via symplectic geometry, Topology 40 (2001), 961-975. MR1860537
[49]
E. Prato, Symplectic toric geometry and the regular dodecahedron, J. Math. (2015), Art. ID 967417, 5 pp. MR3426654
[50]
E. Prato, Toric quasifolds, Math. Intelligencer 45 (2023), 133-138. MR4600262 %%doi:10.1007/s00283-022-10212-y
[51]
T. Ratiu and N. T. Zung, Presymplectic convexity and (ir)rational polytopes, J. Symplectic Geom. 17 (2019), 1479-1511. MR4039815
[52]
M. Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995. MR1340198
[53]
Yu. M. Ustinovsky, Geometry of compact complex manifolds with maximal torus action, Proc. Steklov Inst. Math. 286 (2014), 198-208. MR3482597
[54]
G. M. Ziegler, Lectures on polytopes, Grad. Texts in Math., 152, Springer-Verlag, New York, 1995. MR1311028
[55]
G. M. Ziegler, Nonrational configurations, polytopes, and surfaces, Math. Intelligencer 30 (2008), 36-42. MR2437198

Home Riv.Mat.Univ.Parma