Jacobian ideal


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

Relation to deformation theory

In deformation theory, the deformations of a hypersurface given by a polynomial is classified by the ring
This 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 and an increasing filtration of satisfying a list of compatibility structures. For a smooth projective variety there is a canonical Hodge structure.

Statement for degree d hypersurfaces

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
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.

Sketch of proof

Reduction to residue map

For there is an associated short exact sequence of complexes
where 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 map
where the right hand side is equal to, which is isomorphic to. Also, there is an isomorphism
Through these isomorphisms there is an induced residue map
which is injective, and surjective on primitive cohomology. Also, there is the Hodge decomposition
and.

Computation of de Rham cohomology group

In turns out the 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 isomorphism
Using the canonical -form
on where the denotes the deletion from the index, these meromorphic differential forms look like
where
Finally, it turns out the kernel Lemma 8.11 is of all polynomials of the form where. Note the Euler identity
shows.