Abstract
In order to study the deformations of foliations of codimension 1 of a smooth manifold L, de Bartolomeis and Iordan defined the DGLA \( \mathcal {Z}^{*}\left( L\right) \), where \(\mathcal {Z}^{*}\left( L\right) \) is a subset of differential forms on L. In another paper, de Bartolomeis and Iordan studied the deformations of foliations of a smooth manifold L by defining the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations \(\mathcal {D}^{*}\left( L\right) \). Let L be a Levi flat hypersurface in a complex manifold. Then the deformation theories in \(\mathcal {Z}^{*}\left( L\right) \) and \(\mathcal {D }^{*}\left( L\right) \) lead to the moduli space for the Levi flat deformations of L. In this paper we discuss the relationship between the infinitesimal deformations of L defined by the solutions of Maurer–Cartan equation in \(\mathcal {Z}^{*}\left( L\right) \) and the infinitesimal deformations of L obtained by means of the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations \(\mathcal {D}^{*}\left( L\right) \).
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In 2009, at the Conference of Complex Analysis and Geometry—Levico Terme, Paolo de Bartolomeis proposed me to study the deformations of Levi flat hypersurfaces. From 2010, we met several months a year and wrote several papers on this subject. In the first semester of 2016, I visited the Laboratory Fibonacci in Pisa and we had a lot of interesting discussions about the connection between the classical paper of Kodaira and Spencer, where they studied the foliations by means of the DGLA of graded derivations, and our papers. The present paper has its roots in these discussions.
We planned to continue our work in Paris during the spring 2017, but unfortunately, Paolo passed away on 29th of November 2016. The mathematical research of Paolo holds a major position in the great tradition of Italian geometers. I am honoured to have been Paolo’s collaborator and friend.
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Iordan, A. Infinitesimal deformations of Levi flat hypersurfaces. Boll Unione Mat Ital 12, 197–219 (2019). https://doi.org/10.1007/s40574-018-0176-y
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DOI: https://doi.org/10.1007/s40574-018-0176-y
Keywords
- Levi flat hypersurface
- Differential graded Lie Algebras
- Maurer–Cartan equation
- Foliations
- Graded derivations