In this article the work is done over a ringed site over a field \(k\) of characteristic zero. However, this review will state the results for a commutatively ringed space \((X,\mathcal O_X)\). A Lie algebroid on \(X\) is then a sheaf of Lie algebras \(\mathcal L\) which is an \(\mathcal O_X\)-module and is equipped with an action \(\mathcal L\times\mathcal O_X\rightarrow\mathcal O_X\) with properties mimicking those of the tangent bundle. Throughout this review \(\mathcal L\) denotes a locally free Lie algebroid over \((X,\mathcal O_X)\) of constant rank \(d\).
Lie algebroids are a means of algebraizing differential geometry. E.g., they allow us to treat the algebraic/complex analytic and \(C^\infty\)-case in a uniform way. Examples (of Lie algebroids) are the sheaf of vector fields on a \(C^\infty\)-manifold, the sheaf of holomorphic vector fields on a complex analytic variety, the sheaf of algebriaic vector fields on a smooth algebraic variety, \(\mathcal O_X\otimes\mathfrak g\) where \(\mathfrak g\) is the Lie algebra of an algebraic group acting on a smooth algebraic variety \(X\). The authors setting also applies to some extent to the singular case as well.
The Atiyah class \(A(\mathcal L)\) of \(\mathcal L\) is the element of \(\text{Ext}^1_{\mathcal O_X}(\mathcal L, \mathcal L^\ast\otimes_{\mathcal O_X}\mathcal L)\) which is the obstruction against the existence of an \(\mathcal L\)-connection on \(\mathcal L\). The \(i\)th \((i>0)\) scalar Atiyah class \(a_i(\mathcal L)\) is defined as \(a_i(\mathcal L)=\text{AltTr}(A(\mathcal L)^i)\in H^i(X,(\bigwedge^i\mathcal L)^\ast)\). In the \(C^\infty\) or affine case \(a_i(\mathcal L)=0\) as the cohomology groups \(H^i(X,(\bigwedge^i\mathcal L^\ast)\) vanish. If \(X\) is a Kähler manifold and \(\mathcal T_X\) is the sheaf of holomorphic vector fields then \(a_i(\mathcal T_X)\) coincides with the \(i\)th Chern class of \(\mathcal T_X\).
The Todd class of \(\mathcal L\) is defined as \(\text{td}(\mathcal L)=\det(q(A(\mathcal L)))\) where \(q(x)=x/(1-e^{-x})\). Then \(\text{td}(\mathcal L)\) can be expanded formally in terms of \(a_i(\mathcal L).\)
The sheaf of \(\mathcal L\)-poly-vector fields on \(X\) is defined as \(T^{\mathcal L}_{\text{poly}}(\mathcal O_X)=\bigoplus_i\bigwedge^i\mathcal L\). Then \(T^{\mathcal L}_{\text{poly}}(\mathcal O_X)\) is a sheaf of Gerstenhaber algebras on \(X\). In the case that \(X\) is a \(C^\infty\)-manifold Kontsevich introduced the sheaf of poly-differential operators on \(X\), and it is possible to construct a Lie algebroid generalization \(D^\mathcal L_{\text{poly}}(\mathcal O_X)\) of this concept as well. Like \(T^\mathcal L_{\text{poly}}(\mathcal O_X)\), \(D^\mathcal L_{\text{poly}}(\mathcal O_X)\) is equipped with a Lie bracket and an associative cupproduct but these operations satisfy the Gerstenhaber axioms only up to globally defined homotopies.
The Hochschild-Kostant-Rosenberg map is a quasi-isomorphism between \(T^{\mathcal L}_{\text{poly}}(\mathcal O_X)\) and \(D^{\mathcal L}_{\text{poly}}(\mathcal O_X)\). This article is concerned with the failure of the HKR-map to be compatible with the Lie brackets and cupproducts on \(T^{\mathcal L}_{\text{poly}}(\mathcal O_X)\) and \(D^{\mathcal L}_{\text{poly}}(\mathcal O_X)\).
Let \(\text{D}(X)\) be the derived category of sheaves of \(k\)-vector spaces. This category is equipped with a symmetric monoidal structure given by the derived tensor product.
The articles first main result is that the map in \(\text{D}(X)\), \[ T^{\mathcal L}_{\text{poly}}(\mathcal O_X)\overset{\text{HKR}\circ(\text{td}(\mathcal L)^{1/2}\wedge-)}\longrightarrow D^{\mathcal L}_{\text{poly}}(\mathcal O_X) \] is an isomorphisms of Gerstenhaber algebras in \(\text{D}(X)\). Applying the hypercohomology functor \(\mathbb H^\ast(X,-)\) the authors immediately get also that the map \[ \bigoplus_{i,j}H^j(X,\bigwedge^i\mathcal L)\overset{\text{HKR}\circ(\text{td}(\mathcal L)^{1/2}\wedge-)}\longrightarrow\mathbb{H}^\ast(X,D^{\mathcal L}_{\text{poly}}(\mathcal O_X)) \] is an isomorphism of Gerstenhaber algebras.
Restricting to the setting where \(X\) is a smooth algebraic variety and \(\mathcal L=\mathcal T_X\), the above right hand side can be viewed as the Hochschild cohomology \(\text{HH}^\ast\) of \(X\). Then the above result can be rephrased as saying that there is an isomorphism of Gerstenhaber algebras \[ \bigoplus_{i,j}H^j(X,\bigwedge^i\mathcal L)\overset{\text{HKR}\circ(\text{td}(\mathcal T_X)^{1/2}\wedge-)}\longrightarrow \text{HH}^\ast(X). \]
Looking only at the Lie algebra structure the authors actually prove a stronger result: Let \(\text{HoLieAlg}(X)\) be the category of sheaves of DG-Lie algebras on \(X\) with quasi-isomorphisms inverted. Then the isomorphism between \(T^{\mathcal L}_{\text{poly}}(\mathcal O_X)\) and \(D^{\mathcal L}_{\text{poly}}(\mathcal O_X)\) is obtained from an isomorphism in \(\text{HoLieAlg}(X)\).
The article is very well written, though not self contained. However, with the more advanced basic knowledge of the field, it is easy to understand the important results, their applications and proofs.