Abstract
Given a smooth, projective variety X and an effective divisor \(D\,\subseteq \, X\), it is well-known that the (topological) obstruction to the deformation of the fundamental class of D as a Hodge class, lies in \(H^2({{\,\mathrm{{\mathcal {O}}}\,}}_X)\). In this article, we replace \(H^2({{\,\mathrm{{\mathcal {O}}}\,}}_X)\) by \(H^2_D({{\,\mathrm{{\mathcal {O}}}\,}}_X)\) and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of D as an effective Cartier divisor of a first order infinitesimal deformations of X). We apply this to study the jumping locus of families of linear systems and the Noether–Lefschetz locus. Finally, we give examples of first order deformations \(X_t\) of X for which the cohomology class [D] deforms as a Hodge class but D does not lift as an effective Cartier divisor of \(X_t\).
References
Bloch, S.: Semi-regularity and de-Rham cohomology. Invent. Math. 17, 51–66 (1972)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models, vol. 21. Springer, Berlin (2012)
Brodmann, M.P., Sharp, R.Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications, vol. 136. Cambridge University Press, Cambridge (2012)
Bruns, W., Herzog, H.J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998)
Dan, A., Kaur, I.: Semi-regular varieties and variational Hodge conjecture. Comptes Rendus Math. 354(3), 297–300 (2016)
Grothendieck, A.: Fondements de la géométrie algébrique (extraits du Séminaire Bourbaki, 1957–1962) (1962)
Hartshorne, R.: On the De Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 45, 5–99 (1975)
Hartshorne, R.: Algebraic Geometry. Graduate Text in Mathematics, vol. 52. Springer, Berlin (1977)
Hartshorne, R.: Deformation Theory. Graduate Text in Mathematics. Springer, Berlin (2010)
Hartshorne, R., Grothendieck, A.: Local Cohomology: A Seminar, vol. 41. Springer, Berlin (1967)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Springer, Berlin (2010)
Kodaira, K., Spencer, D.C.: A theorem of completeness of characteristic systems of complete continuous systems. Am. J. Math. 81(2), 477–500 (1959)
Liu, Q.: Algebraic Geometry and Arithmetic Curves, vol. 6. Oxford University Press, Oxford (2002)
Reid, M.: Minimal models of canonical 3-Folds. In: Iitaka, S. (ed.) Algebraic Varieties and Analytic Varieties, pp. 131–180. Mathematical Society of Japan, Tokyo (1983)
Sernesi, E.: Deformaions of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften, vol. 334. Springer, Berlin (2006)
The Stacks Project authors. The Stacks Project. https://stacks.math.columbia.edu/tag/0C4S (2018)
Voisin, C.: Hodge Theory and Complex Algebraic Geometry-I. Cambridge Studies in Advanced Mathematics, vol. 76. Cambridge University Press, Cambridge (2002)
Voisin, C.: Hodge Theory and Complex Algebraic Geometry-II. Cambridge Studies in Advanced Mathematics, vol. 77. Cambridge University Press, Cambridge (2003)
Acknowledgements
The comments of the referee helped us to reformulate the results of § 4 in the previous version, in terms of the local obstruction theory and improve the overall exposition of the article. We are grateful for his/her feedbacks and suggestions. The first-named author acknowledges support of a J. C. Bose Fellowship. The second author is currently supported by ERCEA Consolidator Grant 615655-NMST and also by the Basque Government through the BERC \(2014{-}2017\) program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-\(2013-0323\).
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Biswas, I., Dan, A. Local topological obstruction for divisors. Rev Mat Complut 34, 615–640 (2021). https://doi.org/10.1007/s13163-020-00376-6
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DOI: https://doi.org/10.1007/s13163-020-00376-6
Keywords
- Obstruction theories
- Hodge locus
- Semi-regularity map
- Deformation of linear systems
- Noether–Lefschetz locus