Jacobian Ideal

In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ.
Let $\mathcal(x_1,\ldots,x_n)$ denote the ring of smooth functions in $n$ variables and $f$ a function in the ring. The Jacobian ideal of $f$ is
:$J_f := \left\langle \frac, \ldots, \frac \right\rangle.$

Relation to deformation theory

In deformation theory, the deformations of a hypersurface given by a polynomial $f$ is classified by the ring$\frac$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 $H_\mathbb$ and an increasing filtration $F^\bullet$ of $H_\mathbb = H_\mathbb\otimes_\mathbb$ satisfying a list of compatibility structures. For a smooth projective variety $X$ there is a canonical Hodge structure.

Statement for degree d hypersurfaces

In the special case $X$ is defined by a homogeneous degree $d$ polynomial $f \in \Gamma(\mathbb^,\mathcal(d))$ this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map\mathbb _0,\ldots, Z_n \to \fracwhich is surjective on the primitive cohomology, denoted $\text^(X)$ and has the kernel $J_f$. Note the primitive cohomology classes are the classes of $X$ which do not come from $\mathbb^$, which is just the Lefschetz class n = c_1(\mathcal(1))^d.

Sketch of proof

Reduction to residue map

For $X \subset \mathbb^$ there is an associated short exact sequence of complexes0 \to \Omega_^\bullet \to \Omega_^\bullet(\log X) \xrightarrow \Omega_X^\bullet 1\to 0where 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 $X$, which is $H^n(X;\mathbb) = \mathbb^n(X;\Omega_X^\bullet)$. From the long exact sequence of this short exact sequence, there the induced residue map\mathbb^\left(\mathbb^, \Omega^\bullet_\right) \to
\mathbb^(\mathbb^,\Omega^\bullet_X 1where the right hand side is equal to $\mathbb^(\mathbb^,\Omega^\bullet_X)$, which is isomorphic to $\mathbb^n(X;\Omega_X^\bullet)$. Also, there is an isomorphism $H^_(\mathbb^-X) \cong \mathbb^\left(\mathbb^;\Omega_^\bullet\right)$Through these isomorphisms there is an induced residue map$res: H^_(\mathbb^-X) \to H^n(X;\mathbb)$which is injective, and surjective on primitive cohomology. Also, there is the Hodge decomposition$H^_(\mathbb^-X) \cong \bigoplus_H^q(\Omega_^p(\log X))$and $H^q(\Omega_^p(\log X)) \cong \text^(X)$.

Computation of de Rham cohomology group

In turns out the cohomology group $H^_(\mathbb^-X)$ is much more tractable and has an explicit description in terms of polynomials. The $F^p$ part is spanned by the meromorphic forms having poles of order $\leq n - p + 1$ which surjects onto the $F^p$ part of $\text^n(X)$. This comes from the reduction isomorphism$F^H^_(\mathbb^-X;\mathbb) \cong \frac$Using the canonical $(n+1)$-form$\Omega = \sum_^n (-1)^j Z_j dZ_0\wedge \cdots \wedge \hat\wedge \cdots \wedge dZ_$on $\mathbb^$ where the $\hat$ denotes the deletion from the index, these meromorphic differential forms look like$\frac\Omega$where$\begin \text(A) &= (n-p+1)\cdot\text(f) - \text(\Omega) \\ &= (n-p+1)\cdot d - (n + 2) \\ &= d(n-p+1) - (n+2) \end$Finally, it turns out the kernel ^{Lemma 8.11} is of all polynomials of the form $A' + fB$ where $A' \in J_f$. Note the Euler identity$f = \sum Z_j \frac{\partial Z_j}$shows $f \in J_f$.

References

See also

*Milnor number
*Hodge structure
*Kodaira–Spencer map
*Gauss–Manin connection
* Unfolding

Singularity theory
Ideals (ring theory)