Approximation and interpolation of regular maps from affine varieties to algebraic manifolds
DOI:
https://doi.org/10.7146/math.scand.a-114893Abstract
We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.
References
Arzhantsev, I. V., Flenner, H., Kaliman, S., Kutzschebauch, F., and Zaidenberg, M. G., Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767–823. https://doi.org/10.1215/00127094-2080132
Arzhantsev, I. V., Flenner, H., Kaliman, S., Kutzschebauch, F., and Zaidenberg, M. G., Infinite transitivity on affine varieties, in “Birational geometry, rational curves, and arithmetic”, Simons Symp., Springer, Cham, 2013, pp. 1–13. https://doi.org/10.1007/978-1-4614-6482-2_1
Arzhantsev, I. V., Perepechko, A., and Süß, H., Infinite transitivity on universal torsors, J. Lond. Math. Soc. (2) 89 (2014), no. 3, 762–778. https://doi.org/10.1112/jlms/jdt081
Arzhantsev, I. V., Zaidenberg, M. G., and Kuyumzhiyan, K. G., Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity, Mat. Sb. 203 (2012), no. 7, 3–30. https://doi.org/10.1070/SM2012v203n07ABEH004248
Cox, D. A., Little, J. B., and Schenck, H. K., Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. https://doi.org/10.1090/gsm/124
Flenner, H., Kaliman, S., and Zaidenberg, M., A Gromov-Winkelmann type theorem for flexible varieties, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 11, 2483–2510. https://doi.org/10.4171/JEMS/646
Forstnerič, F., Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006), no. 1, 239–270.
Forstnerič, F., Oka manifolds: from Oka to Stein and back, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 747–809. https://doi.org/10.5802/afst.1388
Forstnerič, F., Stein manifolds and holomorphic mappings: The homotopy principle in complex analysis, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 56, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-61058-0
Forstnerič, F., Surjective holomorphic maps onto Oka manifolds, in “Complex and symplectic geometry”, Springer INdAM Ser., vol. 21, Springer, Cham, 2017, pp. 73–84.
Forstnerič, F. and Lárusson, F., Survey of Oka theory, New York J. Math. 17A (2011), 11–38.
Gromov, M., Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), no. 4, 851–897. https://doi.org/10.2307/1990897
Hamm, H. A. and Mihalache, N., Deformation retracts of Stein spaces, Math. Ann. 308 (1997), no. 2, 333–345. https://doi.org/10.1007/s002080050078
Hironaka, H. and Rossi, H., On the equivalence of imbeddings of exceptional complex spaces, Math. Ann. 156 (1964), 313–333. https://doi.org/10.1007/BF01361027
Iskovskih, V. A. and Manin, J. I., Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86(128) (1971), 140–166.
Kaliman, S., Kutzschebauch, F., and Truong, T. T., On subelliptic manifolds, Israel J. Math. 228 (2018), no. 1, 229–247. https://doi.org/10.1007/s11856-018-1760-7
Lärkäng, R. and Lárusson, F., Extending holomorphic maps from Stein manifolds into affine toric varieties, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4613–4626. https://doi.org/10.1090/proc/13108
Lárusson, F., Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54 (2005), no. 4, 1145–1159. https://doi.org/10.1512/iumj.2005.54.2731
Lárusson, F., Smooth toric varieties are Oka, eprint arXiv:1107.3604 [math.AG], 2011.
Lárusson, F. and Truong, T. T., Algebraic subellipticity and dominability of blow-ups of affine spaces, Doc. Math. 22 (2017), 151–163.
Mo\u ıšezon, B. G., Reducution theorems for compact complex spaces with a sufficiently large field of meromorphic functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1385–1414.
Raynaud, M., Flat modules in algebraic geometry, Compositio Math. 24 (1972), 11–31.
Arzhantsev, I. V., Flenner, H., Kaliman, S., Kutzschebauch, F., and Zaidenberg, M. G., Infinite transitivity on affine varieties, in “Birational geometry, rational curves, and arithmetic”, Simons Symp., Springer, Cham, 2013, pp. 1–13. https://doi.org/10.1007/978-1-4614-6482-2_1
Arzhantsev, I. V., Perepechko, A., and Süß, H., Infinite transitivity on universal torsors, J. Lond. Math. Soc. (2) 89 (2014), no. 3, 762–778. https://doi.org/10.1112/jlms/jdt081
Arzhantsev, I. V., Zaidenberg, M. G., and Kuyumzhiyan, K. G., Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity, Mat. Sb. 203 (2012), no. 7, 3–30. https://doi.org/10.1070/SM2012v203n07ABEH004248
Cox, D. A., Little, J. B., and Schenck, H. K., Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. https://doi.org/10.1090/gsm/124
Flenner, H., Kaliman, S., and Zaidenberg, M., A Gromov-Winkelmann type theorem for flexible varieties, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 11, 2483–2510. https://doi.org/10.4171/JEMS/646
Forstnerič, F., Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006), no. 1, 239–270.
Forstnerič, F., Oka manifolds: from Oka to Stein and back, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 747–809. https://doi.org/10.5802/afst.1388
Forstnerič, F., Stein manifolds and holomorphic mappings: The homotopy principle in complex analysis, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 56, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-61058-0
Forstnerič, F., Surjective holomorphic maps onto Oka manifolds, in “Complex and symplectic geometry”, Springer INdAM Ser., vol. 21, Springer, Cham, 2017, pp. 73–84.
Forstnerič, F. and Lárusson, F., Survey of Oka theory, New York J. Math. 17A (2011), 11–38.
Gromov, M., Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), no. 4, 851–897. https://doi.org/10.2307/1990897
Hamm, H. A. and Mihalache, N., Deformation retracts of Stein spaces, Math. Ann. 308 (1997), no. 2, 333–345. https://doi.org/10.1007/s002080050078
Hironaka, H. and Rossi, H., On the equivalence of imbeddings of exceptional complex spaces, Math. Ann. 156 (1964), 313–333. https://doi.org/10.1007/BF01361027
Iskovskih, V. A. and Manin, J. I., Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86(128) (1971), 140–166.
Kaliman, S., Kutzschebauch, F., and Truong, T. T., On subelliptic manifolds, Israel J. Math. 228 (2018), no. 1, 229–247. https://doi.org/10.1007/s11856-018-1760-7
Lärkäng, R. and Lárusson, F., Extending holomorphic maps from Stein manifolds into affine toric varieties, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4613–4626. https://doi.org/10.1090/proc/13108
Lárusson, F., Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54 (2005), no. 4, 1145–1159. https://doi.org/10.1512/iumj.2005.54.2731
Lárusson, F., Smooth toric varieties are Oka, eprint arXiv:1107.3604 [math.AG], 2011.
Lárusson, F. and Truong, T. T., Algebraic subellipticity and dominability of blow-ups of affine spaces, Doc. Math. 22 (2017), 151–163.
Mo\u ıšezon, B. G., Reducution theorems for compact complex spaces with a sufficiently large field of meromorphic functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1385–1414.
Raynaud, M., Flat modules in algebraic geometry, Compositio Math. 24 (1972), 11–31.
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Published
2019-10-19
How to Cite
Lárusson, F., & Truong, T. T. (2019). Approximation and interpolation of regular maps from affine varieties to algebraic manifolds. MATHEMATICA SCANDINAVICA, 125(2), 199–209. https://doi.org/10.7146/math.scand.a-114893
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