Jacobian ideal

In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ.
Let formula_1 denote the ring of smooth functions in formula_2 variables and formula_3 a function in the ring. The Jacobian ideal of formula_3 is
Relation to deformation theory.
In deformation theory, the deformations of a hypersurface given by a polynomial formula_6 is classified by the ringformula_7This is shown using the Kodaira–Spencer map.
Relation to Hodge theory.
In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space formula_8 and an increasing filtration formula_9 of formula_10 satisfying a list of compatibility structures. For a smooth projective variety formula_11 there is a canonical Hodge structure.
Statement for degree d hypersurfaces.
In the special case formula_11 is defined by a homogeneous degree formula_13 polynomial formula_14 this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the mapformula_15which is surjective on the primitive cohomology, denoted formula_16 and has the kernel formula_17. Note the primitive cohomology classes are the classes of formula_11 which do not come from formula_19, which is just the Lefschetz class formula_20.
Sketch of proof.
Reduction to residue map.
For formula_21 there is an associated short exact sequence of complexesformula_22where 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 formula_11, which is formula_24. From the long exact sequence of this short exact sequence, there the induced residue mapformula_25where the right hand side is equal to formula_26, which is isomorphic to formula_27. Also, there is an isomorphism formula_28Through these isomorphisms there is an induced residue mapformula_29which is injective, and surjective on primitive cohomology. Also, there is the Hodge decompositionformula_30and formula_31.
Computation of de Rham cohomology group.
In turns out the cohomology group formula_32 is much more tractable and has an explicit description in terms of polynomials. The formula_33 part is spanned by the meromorphic forms having poles of order formula_34 which surjects onto the formula_33 part of formula_36. This comes from the reduction isomorphismformula_37Using the canonical formula_38-formformula_39on formula_19 where the formula_41 denotes the deletion from the index, these meromorphic differential forms look likeformula_42whereformula_43Finally, it turns out the kernel Lemma 8.11 is of all polynomials of the form formula_44 where formula_45. Note the Euler identityformula_46shows formula_47.