Derived invariance of the number of holomorphic 1 -forms and vector fields

Mihnea Popa; Christian Schnell

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 3, page 527-536
  • ISSN: 0012-9593

Abstract

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We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.

How to cite

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Popa, Mihnea, and Schnell, Christian. "Derived invariance of the number of holomorphic $1$-forms and vector fields." Annales scientifiques de l'École Normale Supérieure 44.3 (2011): 527-536. <http://eudml.org/doc/272111>.

@article{Popa2011,
abstract = {We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.},
author = {Popa, Mihnea, Schnell, Christian},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {derived categories; Picard variety; Hodge numbers},
language = {eng},
number = {3},
pages = {527-536},
publisher = {Société mathématique de France},
title = {Derived invariance of the number of holomorphic $1$-forms and vector fields},
url = {http://eudml.org/doc/272111},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Popa, Mihnea
AU - Schnell, Christian
TI - Derived invariance of the number of holomorphic $1$-forms and vector fields
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 3
SP - 527
EP - 536
AB - We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.
LA - eng
KW - derived categories; Picard variety; Hodge numbers
UR - http://eudml.org/doc/272111
ER -

References

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  1. [1] S. Barannikov & M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Int. Math. Res. Not.1998 (1998), 201–215. Zbl0914.58004MR1609624
  2. [2] V. V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 1–32. Zbl0963.14015MR1672108
  3. [3] T. Bridgeland & A. Maciocia, Complex surfaces with equivalent derived categories, Math. Z.236 (2001), 677–697. Zbl1081.14023
  4. [4] M. Brion, On the geometry of algebraic groups and homogeneous spaces, preprint arXiv:math/09095014. Zbl1220.14029
  5. [5] M. Brion, Some basic results on actions of non-affine algebraic groups, preprint arXiv:math/0702518. Zbl1217.14029
  6. [6] J. A. Chen & C. D. Hacon, Characterization of abelian varieties, Invent. Math.143 (2001), 435–447. Zbl0996.14020
  7. [7] A. Căldăraru, The Mukai pairing, I: The Hochschild structure, preprint arXiv:math/0308079. Zbl1098.14011
  8. [8] J. Denef & F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math.135 (1999), 201–232. Zbl0928.14004
  9. [9] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Oxford Univ. Press, 2006. Zbl1095.14002
  10. [10] D. Huybrechts & M. Nieper-Wisskirchen, Remarks on derived equivalences of Ricci-flat manifolds, preprint arXiv:0801.4747, to appear in Math. Z. Zbl1213.32012
  11. [11] Y. Kawamata, D -equivalence and K -equivalence, J. Differential Geom.61 (2002), 147–171. Zbl1056.14021
  12. [12] M. Kontsevich, Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, 1995, 120–139. Zbl0846.53021
  13. [13] H. Matsumura, On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.34 (1963), 151–155. Zbl0134.16601
  14. [14] S. Mukai, Duality between D ( X ) and D ( X ^ ) with its application to Picard sheaves, Nagoya Math. J.81 (1981), 153–175. Zbl0417.14036MR607081
  15. [15] D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys58 (2003), 511–591. Zbl1118.14021MR1998775
  16. [16] T. Pham, in preparation. 
  17. [17] R. Rouquier, Automorphismes, graduations et catégories triangulées, preprint http://people.maths.ox.ac.uk/~rouquier/papers/autograd.pdf, 2009. Zbl1244.16009
  18. [18] J-P. Serre, Espaces fibrés algébriques, in Séminaire C. Chevalley, 1958, Exposé 1, Documents mathématiques 1, Soc. Math. France, 2001. MR177011
  19. [19] J-P. Serre, Morphismes universels et variété d’Albanese, in Séminaire C. Chevalley, 1958/59, Exposé 10, Documents mathématiques 1, Soc. Math. France, 2001. Zbl0123.13903MR1795548

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