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