Skip to main content
SpringerLink
Log in

DG-resolutions of NC-smooth thickenings and NC-Fourier–Mukai transforms

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We give a construction of NC-smooth thickenings [a notion defined by Kapranov (J Reine Angew Math 505:73–118, 1998)] of a smooth variety equipped with a torsion free connection. We show that a twisted version of this construction realizes all NC-smooth thickenings as \(0\)th cohomology of a differential graded sheaf of algebras, similarly to Fedosov’s construction in (J Differ Geom 40:213–238, 1994). We use this dg resolution to construct and study sheaves on NC-smooth thickenings. In particular, we construct an NC version of the Fourier–Mukai transform from coherent sheaves on a (commutative) curve to perfect complexes on the canonical NC-smooth thickening of its Jacobian. We also define and study analytic NC-manifolds. We prove NC-versions of some of GAGA theorems, and give a \(C^\infty \)-construction of analytic NC-thickenings that can be used in particular for Kähler manifolds with constant holomorphic sectional curvature. Finally, we describe an analytic NC-thickening of the Poincaré line bundle for the Jacobian of a curve, and the corresponding Fourier–Mukai functor, in terms of \(A_\infty \)-structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. This grading is different from the one considered in [17], cf. Remark 2.3.27.

  2. More precisely, we use the analytic version of Theorem 2.3.23 which is proved similarly.

  3. Even though \(\mathrm{Ext }^*(L,L)\) has trivial \(A_\infty \)-algebra structure, \(A_\infty \)-modules over it are not the same as ordinary modules over an associative algebra.

References

  1. Allcock, D., Carlson, J., Toledo, D.: Complex hyperbolic structure for moduli of cubic surfaces. C. R. Acad. Sci. Paris Sér. I(326), 49–54 (1998)

    Article  MathSciNet  Google Scholar 

  2. Arinkin, D., Bloch, J., Pantev, T.: *-Quantizations of Fourier–Mukai transforms. Geom. Funct. Anal. 23, 1403–1482 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atiyah, M.: Complex analytic connections in fibre bundles. Trans. AMS 85, 181–207 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baranovsky, V., Ginzburg, V., Pecharich, J.: Deformation of line bundles on coisotropic subvarieties. Q. J. Math. 63(3), 525–537(13) (2012)

    Google Scholar 

  5. Ben-Bassat, O., Bloch, J., Pantev, T.: Non-commutative tori and Fourier–Mukai duality. Compos. Math. 143, 423–475 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, X.-Z., Getzler, E.: Transferring homotopy commutative algebraic structures. J. Pure Appl. Algebra 212, 2535–2542 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cortiñas, G.: De Rham and infinitesimal cohomology in Kapranov’s model for noncommutative algebraic geometry. Compos. Math. 136, 171–208 (2003)

    Article  MATH  Google Scholar 

  8. Cortiñas, G.: The structure of smooth algebras in Kapranov’s framework for noncommutative geometry. J. Algebra 281, 679–694 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Costello, K.: A geometric construction of Witten genus, II. arXiv:1112.0816

  10. Cuntz, J., Quillen, D.: Algebra extensions and nonsingularity. J. AMS 8, 251–289 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29, 245–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Fukaya, K.: Deformation Theory, Homological Algebra, and Mirror Symmetry. Geometry and Physics of Branes. Como 2001, pp. 121–209. IOP, Bristol (2003)

  14. Getzler, E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111, 535–560 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9, 119–221 (1957)

    MathSciNet  MATH  Google Scholar 

  16. Huebschmann, J., Kadeishvili, T.: Small models for chain algebras. Math. Z. 207, 245–280 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kapranov, M.: Noncommutative geometry based on commutator expansions. J. Reine Angew. Math. 505, 73–118 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kapranov, M.: Rozansky–Witten invariants via Atiyah classes. Compos. Math. 115, 71–113 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kashiwara, M., Schapira, P.: Deformation quantization modules. Astérisque 345 (2012)

  20. Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibration. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 203–263. World Scientific, River Edge, NJ (2001)

  21. Kriz, I. May, J.P.: Operads, Algebras, Modules, and Motives. Astérisque (1995), no. 233, iv+145 pp

  22. Laumon, G.: Transformation de Fourier Generalisée. alg-geom/9603004 (preprint)

  23. Le Bruyn, L., Van de Weyer, G.: Formal structures and representation spaces. J. Algebra 247, 616–635 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Markl, M.: Transferring \(A_\infty \) (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo 2(Suppl 79), 139–151 (2006)

    MathSciNet  Google Scholar 

  25. Mukai, S.: Duality between \(D(X)\) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)

    MathSciNet  MATH  Google Scholar 

  26. Pecharich, J.: Deformations of vector bundles on coisotropic subvarieties via the Atiyah class. arXiv:1010.3671

  27. Polishchuk, A.: \(A_\infty \)-structures, Brill–Noether loci and the Fourier–Mukai transform. Compos. Math. 140, 459–481 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Serre, J.-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble 6, 1–42 (1955–1956)

    Google Scholar 

  29. Serre, J.-P.: Galois Cohomology, Springer Monographs in Mathematics, Translated from the French by Patrick Ion, Berlin, New York

Download references

Acknowledgments

Part of this work was done while the first author was visiting the Mathematical Sciences Research Institute and the Institut des Hautes Études Scientifiques, and the second author was visiting the Chinese University of Hong Kong. We would like to thank these institutions for excellent working conditions. Also, we are grateful to Kevin Costello, Weiyong He and Dmitry Tamarkin for useful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, USA

    Alexander Polishchuk & Junwu Tu

Authors
  1. Alexander Polishchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junwu Tu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Polishchuk.

Additional information

A.P. is partially supported by the NSF Grant DMS-1001364.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Polishchuk, A., Tu, J. DG-resolutions of NC-smooth thickenings and NC-Fourier–Mukai transforms. Math. Ann. 360, 79–156 (2014). https://doi.org/10.1007/s00208-014-1030-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-014-1030-x

Search

Navigation