Jacobian ideal

In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let ${𝒪}(x_1,\dots ,x_n)$ denote the ring of smooth functions in $n$ variables and $f$ a function in the ring. The Jacobian ideal of $f$ is

$J_f:=⟨\frac{\partial f}{\partial x_1},\dots ,\frac{\partial f}{\partial x_n}⟩.$

Relation to deformation theory

In deformation theory, the deformations of a hypersurface given by a polynomial $f$ is classified by the ring$$\frac{\mathbb{ℂ}[x_1,\dots ,x_n]}{(f)+J_f}.$$ 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 $f\in \mathrm{\Gamma }({\mathbb{\mathbb{P}}}^{n+1},{𝒪}(d))$ this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map$$\mathbb{ℂ}[Z_0,\dots ,Z_n{]}^{(d(n-1+p)-(n+2))}\to \frac{F^p{H}^n(X,\mathbb{ℂ})}{F^{p+1}{H}^n(X,\mathbb{ℂ})}$$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 ${\mathbb{\mathbb{P}}}^{n+1}$, which is just the Lefschetz class $[L{]}^n=c_1({𝒪}(1){)}^d$.

Sketch of proof

Reduction to residue map

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

Computation of de Rham cohomology group

In turns out the de Rham cohomology group $H_{dR}^{n+1}({\mathbb{\mathbb{P}}}^{n+1}-X)$ 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 $\le n-p+1$ which surjects onto the $F^p$ part of ${\text{Prim}}^n(X)$. This comes from the reduction isomorphism$$F^{p+1}{H}_{dR}^{n+1}({\mathbb{\mathbb{P}}}^{n+1}-X;\mathbb{ℂ})\cong \frac{\mathrm{\Gamma }({\mathrm{\Omega }}_{{\mathbb{\mathbb{P}}}^{n+1}}(n-p+1))}{d\mathrm{\Gamma }({\mathrm{\Omega }}_{{\mathbb{\mathbb{P}}}^{n+1}}(n-p))}$$Using the canonical $(n+1)$-form$$\mathrm{\Omega }=\sum _{j=0}^n(-1{)}^j{Z}_{j}d{Z}_0\wedge \cdots \wedge \widehat{d{Z}_j}\wedge \cdots \wedge d{Z}_{n+1}$$on ${\mathbb{\mathbb{P}}}^{n+1}$ where the $\widehat{d{Z}_j}$ denotes the deletion from the index, these meromorphic differential forms look like$$\frac{A}{f^{n-p+1}}\mathrm{\Omega }$$where$$\begin{array}{rl}\text{deg}(A)& =(n-p+1)\cdot \text{deg}(f)-\text{deg}(\mathrm{\Omega })\\ & =(n-p+1)\cdot d-(n+2)\\ & =d(n-p+1)-(n+2)\end{array}$$Finally, it turns out the kernel ^{Lemma 8.11} is of all polynomials of the form $A^{\prime }+fB$ where $A^{\prime }\in J_f$. Note the Euler identity$$f=\sum Z_j\frac{\partial f}{\partial Z_j}$$shows $f\in J_f$.

See also

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