If \(A\) is a \(\mathbb Z\)-graded commutative algebra and \(\ell\) a differential operator of second order, of odd degree \(k\), such that \(\ell(1)=0\) and \(\ell^2=0\), J.-L. Koszul [Astérisque, No. Hors Sér. 1985, 257–271 (1985; Zbl 0615.58029)] proved that the bracket, defined by \([a,b]_{\ell}=(-1)^{|a|}(\ell(ab)-\ell(a)b)-a\ell(b)\), induces what is nowadays called a structure of Batalin-Vilkovisky algebra. In the case of the de Rham complex of a differential manifold and \(\ell\) the Lie derivative with respect to a Poisson bivector, G. Sharygin and D. Talalaev [J. K-Theory 2, No. 2, 361–384 (2008; Zbl 1149.17013)] show that the associated differential graded Lie algebra (DGLA) on \(A\) is quasi-isomorphic to an abelian DGLA.
In the paper under review, the authors generalize this last situation by considering a differential graded commutative algebra, \((A,d)\), endowed with a degree -\(2k\) second order differential operator, \(\iota\), such that \(\iota(1)=0\). They prove that \((A,d)\) carries a natural Gerstenhaber algebra structure whose underlying DGLA is formal, as soon as the differential operator \(\ell=[\iota,d]\) is such that \([\ell,\iota]\) is a second order differential operator. Geometric examples are given, as the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrangian submanifold. As a corollary, a generalization of a recent result by D. Hitchin [Mosc. Math. J. 12, No. 3, 567–591 (2012; Zbl 1267.32010)] is given.