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In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let ${\displaystyle {\mathcal {O))(x_{1},\ldots ,x_{n})}$ denote the ring of smooth functions in ${\displaystyle n}$ variables and ${\displaystyle f}$ a function in the ring. The Jacobian ideal of ${\displaystyle f}$ is

${\displaystyle J_{f}:=\left\langle {\frac {\partial f}{\partial x_{1))},\ldots ,{\frac {\partial f}{\partial x_{n))}\right\rangle .}$

## Relation to deformation theory

In deformation theory, the deformations of a hypersurface given by a polynomial ${\displaystyle f}$ is classified by the ring${\displaystyle {\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{(f)+J_{f))}.}$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 ${\displaystyle H_{\mathbb {R} ))$ and an increasing filtration ${\displaystyle F^{\bullet ))$ of ${\displaystyle H_{\mathbb {C} }=H_{\mathbb {R} }\otimes _{\mathbb {R} }\mathbb {C} }$ satisfying a list of compatibility structures. For a smooth projective variety ${\displaystyle X}$ there is a canonical Hodge structure.

### Statement for degree d hypersurfaces

In the special case ${\displaystyle X}$ is defined by a homogeneous degree ${\displaystyle d}$ polynomial ${\displaystyle f\in \Gamma (\mathbb {P} ^{n+1},{\mathcal {O))(d))}$ this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map[1]${\displaystyle \mathbb {C} [Z_{0},\ldots ,Z_{n}]^{(d(n-1+p)-(n+2))}\to {\frac {F^{p}H^{n}(X,\mathbb {C} )}{F^{p+1}H^{n}(X,\mathbb {C} )))}$which is surjective on the primitive cohomology, denoted ${\displaystyle {\text{Prim))^{p,n-p}(X)}$ and has the kernel ${\displaystyle J_{f))$. Note the primitive cohomology classes are the classes of ${\displaystyle X}$ which do not come from ${\displaystyle \mathbb {P} ^{n+1))$, which is just the Lefschetz class ${\displaystyle [L]^{n}=c_{1}({\mathcal {O))(1))^{d))$.

### Sketch of proof

#### Reduction to residue map

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

#### Computation of de Rham cohomology group

In turns out the de Rham cohomology group ${\displaystyle H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)}$ is much more tractable and has an explicit description in terms of polynomials. The ${\displaystyle F^{p))$ part is spanned by the meromorphic forms having poles of order ${\displaystyle \leq n-p+1}$ which surjects onto the ${\displaystyle F^{p))$ part of ${\displaystyle {\text{Prim))^{n}(X)}$. This comes from the reduction isomorphism${\displaystyle F^{p+1}H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X;\mathbb {C} )\cong {\frac {\Gamma (\Omega _{\mathbb {P} ^{n+1))(n-p+1))}{d\Gamma (\Omega _{\mathbb {P} ^{n+1))(n-p))))}$Using the canonical ${\displaystyle (n+1)}$-form${\displaystyle \Omega =\sum _{j=0}^{n}(-1)^{j}Z_{j}dZ_{0}\wedge \cdots \wedge {\hat {dZ_{j))}\wedge \cdots \wedge dZ_{n+1))$on ${\displaystyle \mathbb {P} ^{n+1))$ where the ${\displaystyle {\hat {dZ_{j))))$ denotes the deletion from the index, these meromorphic differential forms look like${\displaystyle {\frac {A}{f^{n-p+1))}\Omega }$where{\displaystyle {\begin{aligned}{\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{aligned))}Finally, it turns out the kernel[1] Lemma 8.11 is of all polynomials of the form ${\displaystyle A'+fB}$ where ${\displaystyle A'\in J_{f))$. Note the Euler identity${\displaystyle f=\sum Z_{j}{\frac {\partial f}{\partial Z_{j))))$shows ${\displaystyle f\in J_{f))$.

## References

1. ^ a b José Bertin (2002). Introduction to Hodge theory. Providence, R.I.: American Mathematical Society. pp. 199–205. ISBN 0-8218-2040-0. OCLC 48892689.