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^[1] $\mathbb {C} [Z_{0},\ldots ,Z_{n}]^{(d(n-1+p)-(n+2))}\to {\frac {F^{p}H^{n}(X,\mathbb {C} )}{F^{p+1}H^{n}(X,\mathbb {C} )}}$ which is surjective on the primitive cohomology, denoted ${\text{Prim}}^{p,n-p}(X)$ 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 $\mathbb {H} ^{n+1}\left(\mathbb {P} ^{n+1},\Omega _{\mathbb {P} ^{n+1}}^{\bullet }(\log X)\right)\to \mathbb {H} ^{n+1}(\mathbb {P} ^{n+1},\Omega _{X}^{\bullet }[-1])$ where the right hand side is equal to $\mathbb {H} ^{n}(\mathbb {P} ^{n+1},\Omega _{X}^{\bullet })$ , which is isomorphic to $\mathbb {H} ^{n}(X;\Omega _{X}^{\bullet })$ . Also, there is an isomorphism $H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)\cong \mathbb {H} ^{n+1}\left(\mathbb {P} ^{n+1};\Omega _{\mathbb {P} ^{n+1}}^{\bullet }(\log X)\right)$ Through these isomorphisms there is an induced residue map $res:H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)\to H^{n}(X;\mathbb {C} )$ which is injective, and surjective on primitive cohomology. Also, there is the Hodge decomposition $H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)\cong \bigoplus _{p+q=n+1}H^{q}(\Omega _{\mathbb {P} }^{p}(\log X))$ 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 $H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)$ 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 $F^{p+1}H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X;\mathbb {C} )\cong {\frac {\Gamma (\Omega _{\mathbb {P} ^{n+1}}(n-p+1))}{d\Gamma (\Omega _{\mathbb {P} ^{n+1}}(n-p))}}$ Using the canonical $(n+1)$ -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 ${\begin{aligned}{\text{deg}}(A)&=(n-p+1)\cdot {\text{deg}}(f)-{\text{deg}}(\Omega )\\&=(n-p+1)\cdot d-(n+2)\\&=d(n-p+1)-(n+2)\end{aligned}}$ Finally, it turns out the kernel^[1] ^{Lemma 8.11} is of all polynomials of the form $A'+fB$ where $A'\in J_{f}$ . Note the Euler identity $f=\sum Z_{j}{\frac {\partial f}{\partial Z_{j}}}$ shows $f\in J_{f}$ .

References

^ ^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.