Jacobian ideal
In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let [math]\displaystyle{ \mathcal{O}(x_1,\ldots,x_n) }[/math] denote the ring of smooth functions in [math]\displaystyle{ n }[/math] variables and [math]\displaystyle{ f }[/math] a function in the ring. The Jacobian ideal of [math]\displaystyle{ f }[/math] is
- [math]\displaystyle{ J_f := \left\langle \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right\rangle. }[/math]
Relation to deformation theory
In deformation theory, the deformations of a hypersurface given by a polynomial [math]\displaystyle{ f }[/math] is classified by the ring[math]\displaystyle{ \frac{\mathbb{C}[x_1,\ldots, x_n]} {(f) + J_f} }[/math]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 [math]\displaystyle{ H_\mathbb{R} }[/math] and an increasing filtration [math]\displaystyle{ F^\bullet }[/math] of [math]\displaystyle{ H_\mathbb{C} = H_\mathbb{R}\otimes_{\mathbb{R}}\mathbb{C} }[/math] satisfying a list of compatibility structures. For a smooth projective variety [math]\displaystyle{ X }[/math] there is a canonical Hodge structure.
Statement for degree d hypersurfaces
In the special case [math]\displaystyle{ X }[/math] is defined by a homogeneous degree [math]\displaystyle{ d }[/math] polynomial [math]\displaystyle{ f \in \Gamma(\mathbb{P}^{n+1},\mathcal{O}(d)) }[/math] this Hodge structure can be understood completely from the Jacobian ideal. For its graded-pieces, this is given by the map[1][math]\displaystyle{ \mathbb{C}[Z_0,\ldots, Z_n]^{(d(n-1+p) - (n+2))} \to \frac{F^pH^n(X,\mathbb{C})}{F^{p+1}H^n(X,\mathbb{C})} }[/math]which is surjective on the primitive cohomology, denoted [math]\displaystyle{ \text{Prim}^{p,n-p}(X) }[/math] and has the kernel [math]\displaystyle{ J_f }[/math]. Note the primitive cohomology classes are the classes of [math]\displaystyle{ X }[/math] which do not come from [math]\displaystyle{ \mathbb{P}^{n+1} }[/math], which is just the Lefschetz class [math]\displaystyle{ [L]^n = c_1(\mathcal{O}(1))^d }[/math].
Sketch of proof
Reduction to residue map
For [math]\displaystyle{ X \subset \mathbb{P}^{n+1} }[/math] there is an associated short exact sequence of complexes[math]\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 }[/math]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 [math]\displaystyle{ X }[/math], which is [math]\displaystyle{ H^n(X;\mathbb{C}) = \mathbb{H}^n(X;\Omega_X^\bullet) }[/math]. From the long exact sequence of this short exact sequence, there the induced residue map[math]\displaystyle{ \mathbb{H}^{n+1}\left(\mathbb{P}^{n+1}, \Omega^\bullet_{\mathbb{P}^{n+1}}\right) \to \mathbb{H}^{n+1}(\mathbb{P}^{n+1},\Omega^\bullet_X[-1]) }[/math]where the right hand side is equal to [math]\displaystyle{ \mathbb{H}^{n}(\mathbb{P}^{n+1},\Omega^\bullet_X) }[/math], which is isomorphic to [math]\displaystyle{ \mathbb{H}^n(X;\Omega_X^\bullet) }[/math]. Also, there is an isomorphism [math]\displaystyle{ H^{n+1}_{dR}(\mathbb{P}^{n+1}-X) \cong \mathbb{H}^{n+1}\left(\mathbb{P}^{n+1};\Omega_{\mathbb{P}^{n+1}}^\bullet\right) }[/math]Through these isomorphisms there is an induced residue map[math]\displaystyle{ res: H^{n+1}_{dR}(\mathbb{P}^{n+1}-X) \to H^n(X;\mathbb{C}) }[/math]which is injective, and surjective on primitive cohomology. Also, there is the Hodge decomposition[math]\displaystyle{ H^{n+1}_{dR}(\mathbb{P}^{n+1}-X) \cong \bigoplus_{p+q = n+1}H^q(\Omega_{\mathbb{P}}^p(\log X)) }[/math]and [math]\displaystyle{ H^q(\Omega_{\mathbb{P}}^p(\log X)) \cong \text{Prim}^{p-1,q}(X) }[/math].
Computation of de Rham cohomology group
In turns out the cohomology group [math]\displaystyle{ H^{n+1}_{dR}(\mathbb{P}^{n+1}-X) }[/math] is much more tractable and has an explicit description in terms of polynomials. The [math]\displaystyle{ F^p }[/math] part is spanned by the meromorphic forms having poles of order [math]\displaystyle{ \leq n - p + 1 }[/math] which surjects onto the [math]\displaystyle{ F^p }[/math] part of [math]\displaystyle{ \text{Prim}^n(X) }[/math]. This comes from the reduction isomorphism[math]\displaystyle{ F^{p+1}H^{n+1}_{dR}(\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)) } }[/math]Using the canonical [math]\displaystyle{ (n+1) }[/math]-form[math]\displaystyle{ \Omega = \sum_{j=0}^n (-1)^j Z_j dZ_0\wedge \cdots \wedge \hat{dZ_j}\wedge \cdots \wedge dZ_{n+1} }[/math]on [math]\displaystyle{ \mathbb{P}^{n+1} }[/math] where the [math]\displaystyle{ \hat{dZ_j} }[/math] denotes the deletion from the index, these meromorphic differential forms look like[math]\displaystyle{ \frac{A}{f^{n-p+1}}\Omega }[/math]where[math]\displaystyle{ \begin{align} \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{align} }[/math]Finally, it turns out the kernel[1] Lemma 8.11 is of all polynomials of the form [math]\displaystyle{ A' + fB }[/math] where [math]\displaystyle{ A' \in J_f }[/math]. Note the Euler identity[math]\displaystyle{ f = \sum Z_j \frac{\partial f}{\partial Z_j} }[/math]shows [math]\displaystyle{ f \in J_f }[/math].
References
See also
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