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 $$\mathbb{C}[Z_0,\ldots, Z_n]^{(d(n-1+p) - (n+2))} \to \frac{F^pH^n(X,\mathbb{C})}{F^{p+1}H^n(X,\mathbb{C})}$$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 $$[L]^n = c_1(\mathcal{O}(1))^d$$.

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 $$f \in J_f$$.