Singularities are determined by the cohomology of their cotangent complexes (Q1074742)

Let \((X_ 0,0)\) and \((X_ 0',0)\) be singular analytic germs, embedded in affine space \({\mathbb{A}}^ n\). Equivalent are (i) \((X_ 0,0)\) and \((X_ 0',0)\) are isomorphic. (ii) \(T^ i(X_ 0,0)\) and \(T^ i(X_ 0',0)\) are isomorphic as \({\mathbb{C}}\{x_ 1,...,x_ n\}\)-modules, where the isomorphism is induced by a generalized contact group. (iii) \(T^ 1_{(1)}(X_ 0,0)\) and \(T^ 1_{(1)}(X_ 0',0)\) are isomorphic as \({\mathbb{C}}\{x_ 1,...,x_ n\}\)-modules. Here \(T^ i\) denotes the i-th cohomology group of the cotangent complex and \(T^ i_{(r)}\) the i-th cohomology group of the restricted cotangent complex. In fact the theorem is stated in more generality. It applies also to families \(\phi_ 0: X_ 0\to S\). For this purpose a cotangent complex \(T^ i(\phi)\) is defined with the help of the Tjurina resolvent of \(\phi^*_ 0: {\mathcal O}_ s\to {\mathcal O}_{x_ 0}\). The notion of Kodaira-Spencer map in connection with trivial and infinitesimally trivial families is essential in the proof. Special cases of the problem include: a) contact equivalence of complete intersections, b) right equivalence of q-types of functions \(({\mathbb{C}}^ n,0)\to {\mathbb{C}}.\) For functions \({\mathbb{C}}^ n\to {\mathbb{C}}\) with isolated singularities we obtain the so-called Mather-Yau equivalence.