| Title:
|
Geometry of universal embedding spaces for almost complex manifolds (English) |
| Author:
|
Clemente, Gabriella |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
35-60 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the geometry of universal embedding spaces for compact almost-complex manifolds of a given dimension, and related constructions that allow for an extrinsic study of the integrability of almost-complex structures. These embedding spaces were introduced by J-P. Demailly and H. Gaussier, and are complex algebraic analogues of twistor spaces. Their goal was to study a conjecture made by F. Bogomolov asserting the “transverse embeddability” of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of related bundles, such as the integrability locus characterizing integrable almost-complex structures. Our approach could potentially lead to finding new obstructions to the existence of a complex structure, which may be useful for tackling Yau’s Challenge. (English) |
| Keyword:
|
almost-complex manifolds |
| Keyword:
|
complex structures |
| Keyword:
|
integrability |
| Keyword:
|
Nijenhuis tensor |
| Keyword:
|
obstruction theory |
| Keyword:
|
transverse embeddings |
| Keyword:
|
fiber bundles |
| Keyword:
|
vector bundles |
| MSC:
|
32L05 |
| MSC:
|
32Q40 |
| MSC:
|
32Q60 |
| DOI:
|
10.5817/AM2024-1-35 |
| . |
| Date available:
|
2024-02-07T14:13:12Z |
| Last updated:
|
2024-02-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152026 |
| . |
| Reference:
|
[1] Arnol’d, V.I.: The Sturm theorems and symplectic geometry.Funct. Anal. Appl. 19 (4) (1985), 251–259. MR 0820079, 10.1007/BF01077289 |
| Reference:
|
[2] Audin, M.: Torus actions on symplectic manifolds.progress in mathematics ed., vol. 93, Birkhäuser Verlag, Basel, 2004, Second revised edition. Zbl 1062.57040, MR 2091310 |
| Reference:
|
[3] Bogomolov, F.: Complex manifolds and algebraic foliations.RIMS-1084 Kyoto Univ. (1996), 1–5. |
| Reference:
|
[4] Bryant, R.: S-S. Chern’s study of almost complex structures on the six-sphere.Asian J. Math. 10 (3) (2006), 561–605. MR 2253159 |
| Reference:
|
[5] Ciliberto, C., Pedrini, C.: Real abelian varieties and real algebraic curves.Lectures in Real Geometry (Broglia, F., ed.), 1994, pp. 167–256. MR 1440212 |
| Reference:
|
[6] Crabb, M., James, I.: Fiberwise Homotopy Theory.Springer Monogr. Math., Springer-Verlag, London, 1998. MR 1646248 |
| Reference:
|
[7] Demailly, J-P.: Algebraic embeddings of complex and almost complex structures.Slides of a talk given at the CIME School on “Non-Kähler geometry” (Cetraro, Italy, July 12, 2018), https://www-fourier.ujf-grenoble.fr/$\sim $demailly/manuscripts/cetraro$\_$2018-print.pdf. |
| Reference:
|
[8] Demailly, J-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials.Algebraic geometry: Santa Cruz 1995, Proc. Sympos. Pure Math. 62. Part 2, Amer. Math. Soc., Providence, RI, 1997, pp. 285–360. MR 1492539 |
| Reference:
|
[9] Demailly, J-P., Gaussier, H.: Algebraic embeddings of smooth almost complex structures.J. Eur. Math. Soc. 19 (2017), 3391–3419. MR 3713044, 10.4171/jems/742 |
| Reference:
|
[10] Diaz, L.O.: A note on Kirchoff’s theorem for almost complex spheres, I.arXiv:1804.05794. |
| Reference:
|
[11] Newlander, A., Niremberg, L.: Complex analytic coordinates in almost complex manifolds.Ann. Math. 65 (3) (1957), 391–404. MR 0088770, 10.2307/1970051 |
| Reference:
|
[12] Yau, S-T.: Open problems in geometry.Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math. 54, Part 1, Amer. Math. Soc., Providence, RI, 1993. MR 1216573 |
| . |
