Jacobian ideal

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

Let $\mathcal{O}(x_1,\ldots,x_n)$ denote the ring of smooth functions in $n$ variables and $f$ a function in the ring. The Jacobian ideal of $f$ is

: $J_f := \left\langle \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right\rangle.$

Relation to deformation theory

In deformation theory, the deformations of a hypersurface given by a polynomial $f$ is classified by the ring $\frac{\mathbb{C}[x_1, \ldots, x_n]}{(f) + J_f}.$

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 $H_\mathbb{R}$ and an increasing filtration $F^\bullet$ of $H_\mathbb{C} = H_\mathbb{R}\otimes_{\mathbb{R}}\mathbb{C}$ 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 $f \in \Gamma(\mathbb{P}^{n+1},\mathcal{O}(d))$ this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map{{Cite book|title=Introduction to Hodge theory|date=2002|publisher=American Mathematical Society|author=José Bertin|isbn=0-8218-2040-0|location=Providence, R.I.|pages=199–205|oclc=48892689}} $\mathbb{C}[Z_0,\ldots, Z_n]^{(d(n-1+p) - (n+2))} \to \frac{F^pH^n(X,\mathbb{C})}{F^{p+1}H^n(X,\mathbb{C})}$ which is surjective on the primitive cohomology, denoted $\text{Prim}^{p,n-p}(X)$ and has the kernel $J_f$ . Note the primitive cohomology classes are the classes of $X$ which do not come from $\mathbb{P}^{n+1}$ , which is just the Lefschetz class $[L]^n = c_1(\mathcal{O}(1))^d$ .

= Sketch of proof =

== Reduction to residue map ==

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

== Computation of de Rham cohomology group ==

In turns out the de Rham cohomology group $H^{n+1}_{dR}(\mathbb{P}^{n+1}-X)$ 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 $\text{Prim}^n(X)$ . This comes from the reduction isomorphism $F^{p+1}H^{n+1}_{dR}(\mathbb{P}^{n+1}-X;\mathbb{C}) \cong \frac{
\Gamma(\Omega_{\mathbb{P}^{n+1}}(n-p+1))
}{
d\Gamma(\Omega_{\mathbb{P}^{n+1}}(n-p))
}$ Using the canonical $(n+1)$ -form $\Omega = \sum_{j=0}^n (-1)^j Z_j dZ_0\wedge \cdots \wedge \hat{dZ_j}\wedge \cdots \wedge dZ_{n+1}$ on $\mathbb{P}^{n+1}$ where the $\hat{dZ_j}$ denotes the deletion from the index, these meromorphic differential forms look like $\frac{A}{f^{n-p+1}}\Omega$ where $\begin{align}
\text{deg}(A) &= (n-p+1)\cdot\text{deg}(f) - \text{deg}(\Omega) \\
&= (n-p+1)\cdot d - (n + 2) \\
&= d(n-p+1) - (n+2)
\end{align}$ 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 $f = \sum Z_j \frac{\partial f}{\partial Z_j}$ shows $f \in J_f$ .