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In mathematics the **Jacobian ideal** or **gradient ideal** is the ideal generated by the Jacobian of a function or function germ.
Let denote the ring of smooth functions in variables and a function in the ring. The Jacobian ideal of is

In deformation theory, the deformations of a hypersurface given by a polynomial is classified by the ringThis is shown using the Kodaira–Spencer map.

In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space and an increasing filtration of satisfying a list of compatibility structures. For a smooth projective variety there is a canonical Hodge structure.

In the special case is defined by a homogeneous degree polynomial this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map^{[1]}which is surjective on the primitive cohomology, denoted and has the kernel . Note the primitive cohomology classes are the classes of which do not come from , which is just the Lefschetz class .

For there is an associated short exact sequence of complexeswhere the middle complex is the complex of sheaves of logarithmic forms and the right-hand map is the residue map. This has an associated long exact sequence in cohomology. From the Lefschetz hyperplane theorem there is only one interesting cohomology group of , which is . From the long exact sequence of this short exact sequence, there the induced residue mapwhere the right hand side is equal to , which is isomorphic to . Also, there is an isomorphism Through these isomorphisms there is an induced residue mapwhich is injective, and surjective on primitive cohomology. Also, there is the Hodge decompositionand .

In turns out the de Rham cohomology group is much more tractable and has an explicit description in terms of polynomials. The part is spanned by the meromorphic forms having poles of order which surjects onto the part of . This comes from the reduction isomorphismUsing the canonical -formon where the denotes the deletion from the index, these meromorphic differential forms look likewhereFinally, it turns out the kernel^{[1]} ^{Lemma 8.11} is of all polynomials of the form where . Note the Euler identityshows .

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^{a}^{b}José Bertin (2002).*Introduction to Hodge theory*. Providence, R.I.: American Mathematical Society. pp. 199–205. ISBN 0-8218-2040-0. OCLC 48892689.