Jacobian ideal

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

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

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.

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.

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 ${\displaystyle [L]^{n}=c_{1}({\mathcal {O}}(1))^{d}}$.

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 ${\displaystyle H^{q}(\Omega _{\mathbb {P} }^{p}(\log X))\cong {\text{Prim}}^{p-1,q}(X)}$.

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 ${\displaystyle f\in J_{f}}$.

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