Jacobian ideal

In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ.

denote the ring of smooth functions in

is In deformation theory, the deformations of a hypersurface given by a polynomial

This 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

satisfying a list of compatibility structures.

For a smooth projective variety

there is a canonical Hodge structure.

is defined by a homogeneous degree

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

{\displaystyle {\text{Prim}}^{p,n-p}(X)}

Note the primitive cohomology classes are the classes of

, which is just the Lefschetz class

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

From the long exact sequence of this short exact sequence, there the induced residue map

where the right hand side is equal to

Through these isomorphisms there is an induced residue map

{\displaystyle res:H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)\to H^{n}(X;\mathbb {C} )}

which is injective, and surjective on primitive cohomology.

In turns out the de Rham cohomology group

is much more tractable and has an explicit description in terms of polynomials.

part is spanned by the meromorphic forms having poles of order

denotes the deletion from the index, these meromorphic differential forms look like

Finally, it turns out the kernel[1] Lemma 8.11 is of all polynomials of the form

Note the Euler identity