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We investigate the deformation of an $L_\infty$ structure that encodes thegeometric information about a smooth vector bundle. We establish that thisspecific type of deformation endows the bundle with a Lie algebroid structure.Additionally, we compute the Atiyah class and Atiyah-Chern classes in thegeometrically deformed case. The precise relationship between the firstAtiyah-Chern class and the modular class is shown. In the case of action Liealgebroids, the Atiyah-Chern classes at the leading order are given by theequivariant Chern characters.

The geometric deformation of curved $L_\infty$ algebras and Lie algebroids