Jacobian ideal

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

$${\displaystyle \mathbb {H} ^{n+1}\left(\mathbb {P} ^{n+1},\Omega _{\mathbb {P} ^{n+1}}^{\bullet }\right)\to \mathbb {H} ^{n+1}(\mathbb {P} ^{n+1},\Omega _{X}^{\bullet }[-1])}$$

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

$${\displaystyle H_{dR}^{n+1}(\mathbb {P} ^{n+1}-X)\cong \mathbb {H} ^{n+1}\left(\mathbb {P} ^{n+1};\Omega _{\mathbb {P} ^{n+1}}^{\bullet }\right)}$$

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