Skip to main content
SpringerLink
Log in

Standard canonical support loci

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

We consider the union of certain irreducible components of cohomological support loci of the canonical bundle, which we call standard. We prove a structure theorem about them and single out some particular cases, recovering and improving results of Beauville and Chen–Jiang. Finally, as an example of application, we extend to compact Kahler manifolds the classification of smooth complex projective varieties with \(p_1(X)=1\), \(p_3(X)=2\) and \(q(X)=\dim X\).

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. Proof: let \(X_g\) be a general fiber of g. Then g induces a map \(a:X_g\rightarrow K:=\ker (\mathrm {Alb}\, X\rightarrow \mathrm {Alb}\, Y)\) whose image spans K. Hence the homomorphism \(\mathrm{Pic}^0 K\rightarrow \mathrm{Pic}^0 F\) has finite kernel. Then the exact sequence (1.5) follows from the dualization of the exact sequence \(0\rightarrow K\rightarrow \mathrm {Alb}\, X\rightarrow \mathrm {Alb}\, Y\rightarrow 0\).

  2. The holomorphic Euler characteristic \(\chi (K_{\widetilde{Y}})\) does not depend on the particular resolution \(\widetilde{Y}\) considered. Since one can choose a \(\widetilde{Y}\) which is Kahler (see e.g. [5, 1.9]), \(\chi (K_{\widetilde{Y}})\ge 0\) by generic vanishing (see below).

  3. If \(V^i(K_X)\) is empty or zero-dimensional for all i we define \(p=-\infty \).

  4. Strictly speaking the proof of this application uses only the well known theorem of Beauville mentioned above, which is now a particular case of Theorem C. However we included it in this paper because it is suggestive about the possible use of Theorems A and C when dealing with this sort of problems.

  5. The surjective homorphisms with connected fibres \(\pi _k:A\rightarrow B_k\) are not uniquely determined. However one can arrange them in such a way that \(\pi _k\) factorizes trough \(\pi _h\) if \(\pi _k^*\mathrm{Pic}^0 B_k\) is contained \(\pi _h^*\mathrm{Pic}^0 B_h\).

  6. In brief: one can define more generally loci \(V^i_{m}(K_X\otimes P_\eta )=\{\alpha \in \mathrm{Pic}^0 X\>|\> h^i(K_X\otimes P_\eta \otimes P_\alpha )\ge m\}\) and the Theorems of Green-Lazarsfeld and Simpson-Wang prove as well that all components \(V^i_{m}(K_X\otimes P_\eta )\) are translates of subtori by points of finite order. Then one proves, using Kollár’s decomposition, that a component of \(V^r(R^i g_*(K_X\otimes P_\eta ))\) is also a component of \(V^{r+i}_m(K_X\otimes P_\eta )\) for some m.

  7. We recall (see the footnote to Remark 3.5) that, unlike the homomorphisms \(\pi _{B}\), the subtori T are uniquely determined by the Chen–Jiang decomposition.

References

  1. Arapura, D.: Higgs line bundles, Green-Lazarsfeld sets, and maps of Kahler manifolds to curves. Bull. Am. Math. Soc. 26, 310–314 (1992)

    Article  MATH  Google Scholar 

  2. Beauville, A.: Annulation du \(H^1\) pour les fibrés en droites plats. In: Complex algebraic varieties LN 1507, pp. 1–15. Springer, Berlin, Heidelberg (1992)

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Catanese, F.: Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations. Invent. Math. 104, 389–407 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Catanese, F.: Compact complex manifolds bimeromorphic to tori. In: Abelian Varieties, pp. 55–62. De Gruyter (1995)

  6. Chen, J.A., Hacon, Ch.: Characterization of abelian varieties. Invent. Math. 143, 435–447 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, J.A., Hacon, Ch.: On algebraic fiber spaces over varieties of maximal Albanese dimension. Duke Math. J. 111, 159–175 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, J.A., Hacon, Ch.: Varieties with \(P_3(X) = 4\) and \(q(X) = dim(X)\). Ann. Sc. Norm. Super. Pisa Cl. Sci. III(2), 399–425 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Chen, J.A., Jiang, Z.: Positivity in varieties of maximal Albanese dimension. J. Reine Angew. Math. (2015). doi:10.1515/crelle-2015-0027 (Published online)

  10. Ein, L., Lazarsfeld, R.: Singularities of theta divisors and the birational geometry of irregular varieties. J. Am. Math. Soc. 10(1), 243–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Green, M., Lazarsfeld, R.: Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90, 389–407 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Green, M., Lazarsfeld, R.: Higher obstructions to deforming cohomology groups of line bundles. J. Am. Math. Soc. 1, 87–103 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hacon, Ch.: A derived category approach to generic vanishing. J. Reine Angew. Math. 575, 173–187 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Hacon, Ch.: Varieties with \(P_3=3\) and \(q(X)=dim (X)\). Math. Nachr. 278, 409–420 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hacon, C., Pardini, R.: On the birational geometry of varieties of maximal Albanese dimension. J. Reine Angew. Math. 546, 177–199 (2002)

    MathSciNet  MATH  Google Scholar 

  16. Hacon, C., Pardini, R.: Birational characterization of products of curves of genus 2. Math. Res. Lett. 12, 129–140 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Clarendon Press, Oxford (2006)

    Book  MATH  Google Scholar 

  18. Jiang, Z.: Varieties with \(q(X)=\dim (X)\) and \(P_2(X)=2\). Ann. Sc. Norm. Super. Pisa Cl. Sci. XI, 243–258 (2012)

  19. Kollár, J.: Higher direct images of dualizing sheaves I. Ann. Math. 123, 11–42 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kollár, J.: Higher direct images of dualizing sheaves II. Ann. Math. 124, 171–202 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lombardi, L., Popa, M.: Derived equivalence and non-vanishing loci II. In: London Math. Soc. Lecture Note Series, Recent Advances in Algebraic Geometry, vol. 417, pp. 291–306. Cambridge University Press, Cambridge (2015)

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Pareschi, G.: Basic results on irregular varieties via Fourier-Mukai methods. In: Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, pp. 379–403. Cambridge Univ. Press, Cambridge (2012)

  24. Pareschi, G., Popa, M.: Regularity on Abelian varieties III: relationship with generic vanishing and applications. In: Grassmannians, Moduli Spaces and Vector Bundles, pp. 141–167, AMS, Providence, RI (2011)

  25. Pareschi, G., Popa, M., Schnell, Ch.: Hodge modules on complex tori and generic vanishing for compact Kähler manifolds. arXiv:1505.00635v1 [math.AG]

  26. Popa, M.: Derived equivalence and non-vanishing loci. In: Clay Mathematical Proceedings, vol. 18, pp. 567–575. AMS, Providence, RI (2013)

  27. Saito, M.: Decomposition theorem for proper Kähler morphisms. Tohoku Math. J. (2) 42, 127–147 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saito, M.: On Kollár’s conjecture. Several complex variables and complex geometry. In: Proc. Sympos. Pure Math., vol. 52, pp. 509–517. AMS, Providence, RI (1991)

  29. Schnell, Ch.: Torsion points on cohomology support loci: from D-modules to Simpson’s theorem. In: Recent advances in algebraic geometry. Lond. Math. Soc. Lect. Note Ser., vol. 417, pp. 405–421 (2015)

  30. Simpson, C.T.: Subspaces of moduli spaces of rank one local systems. Ann. Sci. École Norm. Sup. 4(26), 361–401 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Takegoshi, K.: Higher direct images of canonical sheaves tensorized by semi-positive vector bundles by proper Kahler morphisms. Math. Ann. 303, 389–416 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, B.: Torsion points on the cohomology jump loci of compact Kähler manifolds. Math. Res. Lett. 23, 545–563 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I am very indebted to Zhi Jiang for pointing out some gaps and errors in a previous version of this paper.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma, Tor Vergata, V.le della Ricerca Scientifica, 00133, Rome, Italy

    Giuseppe Pareschi

Authors
  1. Giuseppe Pareschi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Pareschi.

Additional information

Dedicated to Philippe Ellia on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pareschi, G. Standard canonical support loci. Rend. Circ. Mat. Palermo, II. Ser 66, 137–157 (2017). https://doi.org/10.1007/s12215-016-0269-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-016-0269-3

Keywords

Mathematics Subject Classification

Search

Navigation