SIGMA 22 (2026), 009, 17 pages      arXiv:2507.20276      https://doi.org/10.3842/SIGMA.2026.009

Joint Deformations of Manifolds, Coherent Sheaves and Sections

Donatella Iacono ^a and Marco Manetti ^b
a) Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy
^b) Dipartimento di Matematica Guido Castelnuovo, Università degli studi di Roma ''La Sapienza'', P. le Aldo Moro 5, 00185 Roma, Italy

Received August 06, 2025, in final form January 14, 2026; Published online February 03, 2026

Abstract
We describe a differential graded Lie algebra controlling infinitesimal deformations of triples $(X,\mathcal{F},\sigma)$, where $\mathcal{F}$ is a coherent sheaf on a smooth variety $X$ over a field of characteristic 0 and $\sigma\in H^0(X,\mathcal{F})$. Then, we apply this result to investigate deformations of pairs (variety, divisor).

Key words: deformation of manifolds; coherent sheaves and sections; differential graded Lie algebras.

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References

1. Artin M., Lectures on deformations of singularities, Tata Institute of Fundamental Research, Bombay, 1976.
2. Borelli M., Some results on ampleness and divisorial schemes, Pacific J. Math. 23 (1967), 217-227.
3. Fiorenza D., Iacono D., Martinengo E., Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves, J. Eur. Math. Soc. (JEMS) 14 (2012), 521-540, arXiv:0904.1301.
4. Fiorenza D., Manetti M., Martinengo E., Cosimplicial DGLAs in deformation theory, Comm. Algebra 40 (2012), 2243-2260, arXiv:0803.0399.
5. Friedman R., On the geometry of anticanonical pairs, arXiv:1502.02560.
6. Hartshorne R., Algebraic geometry, Grad. Texts in Math., Vol. 52, Springer, New York, 1977.
7. Hinich V., Descent of Deligne groupoids, Int. Math. Res. Not. 1997 (1997), 223-239, arXiv:alg-geom/9606010.
8. Iacono D., Deformations of algebraic subvarieties, Rend. Mat. Appl. 30 (2010), 89-109, arXiv:1003.3333.
9. Iacono D., Deformations and obstructions of pairs $(X,D)$, Int. Math. Res. Not. 2015 (2015), 9660-9695, arXiv:1302.1149.
10. Iacono D., Manetti M., On deformations of pairs (manifold, coherent sheaf), Canad. J. Math. 71 (2019), 1209-1241, arXiv:1707.06612.
11. Iacono D., Manetti M., Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle), Internat. J. Math. 32 (2021), 2150086, 7 pages, arXiv:1902.10386.
12. Iacono D., Martinengo E., On the local structure of the Brill-Noether locus of locally free sheaves on a smooth variety, Rend. Semin. Mat. Univ. Padova 153 (2025), 179-209, arXiv:2207.13935.
13. Iacono D., Martinengo E., Deformations of morphisms of sheaves, Canad. J. Math., to appear, arXiv:2312.09677.
14. Katzarkov L., Kontsevich M., Pantev T., Hodge theoretic aspects of mirror symmetry, in From Hodge Theory to Integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., Vol. 78, American Mathematical Society, Providence, RI, 2008, 87-174.
15. Li S., Pan X., Unobstructedness of deformations of Calabi-Yau varieties with a line bundle, arXiv:1310.7162.
16. Manetti M., Differential graded Lie algebras and formal deformation theory, in Algebraic Geometry -- Seattle 2005. Part 2, Proc. Sympos. Pure Math., Vol. 80, American Mathematical Society, Providence, RI, 2009, 785-810.
17. Manetti M., Lie methods in deformation theory, Springer Monogr. Math., Springer, Singapore, 2022.
18. Sernesi E., Deformations of algebraic schemes, Grundlehren Math. Wiss., Vol. 334, Springer, Berlin, 2006.
19. Vistoli A., Notes on Grothendieck topologies, fibered categories and descent theory, arXiv:math.AG/0412512.
20. Weibel C.A., An introduction to homological algebra, Cambridge Stud. Adv. Math., Vol. 38, Cambridge University Press, Cambridge, 1994.
21. Welters G.E., Polarized abelian varieties and the heat equations, Compositio Math. 49 (1983), 173-194.