By Ragnar-Olaf Buchweitz
During this memoir, it's proven that the parameter house for the versal deformation of an remoted singularity $(V,O)$ ---whose lifestyles used to be tested by means of Grauert in 1972---is isomorphic to the distance linked to the hyperlink $M$ of $V$ by way of Kuranishi utilizing the CR-geometry of $M$ .
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Proof By induction on n. We have G*+1(M*)/? = LR(Gk(Mm))p 0 = p*aGk(M>)0 ®PaP arC/? M«o- D We can now prove our main formula. 11. )) = Homo v . ). CR-GEOMETRY AND DEFORMATIONS OF ISOLATED SINGULARITIES 41 Proof. (M;N*)). 7. We leave to the reader the task of checking that the face maps and hence the boundary maps on the two sides coincide. 2. -module. 6. Af,Homov. (M„ N+)) is a homotopy equivalence for any N+. We obtain a vanishing theorem for the simplicial cohomology groups H% (sd W, Horno^ (Af*, #*)).
6 we have Ext*,^ (j*M,j*N) = Extj 7x (M, N). 2 will follow from the next five lemmas. The first of these follows immediately from the definition of simplicial cochains with values in H. 7. C n ( s d A f , # ) = f] H o m c V a ( M a o , ( p a 0 ) 0 J , i V a J . 5 and [Go], appendice, associated to an adjoint pair of functors L and R (denoted T and U respectively in [I]). We briefly review the construction. We recall that if C is a category, the underlying discrete category Cd is the category such that Obj Cd = Obj C and for a, b £ Obj Cd we have m» / *x f id, if a = 6.
We consider the category C whose objects are triples M = (M, £", V) where M ( - i = (( ® © is a complex manifold, E = \ . E»,a\ J is E^^s is a cochain complex of holomorphic vector bundles over M and V is a connection on E' of type (1,0) which preserves the grading. We do not assume that the differential) s of E' is parallel. We note that V induces a connection, also denoted V, on the graded symmetric algebra. We require V(l) = 0 where 1 is the constant function with value 1 on M considered as a section of S°(E').