Jacobian ideal explained

In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ.Let

l{O}(x_1,\ldots,x_n)

denote the ring of smooth functions in

variables and

a function in the ring. The Jacobian ideal of

J_f:=\left\langle

	\partialf
	\partialx₁

,\ldots,

	\partialf
	\partialx_n

\right\rangle.

Relation to deformation theory

In deformation theory, the deformations of a hypersurface given by a polynomial

is classified by the ring

\frac.

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_R

and an increasing filtration

F^\bullet

H_C=H_{R ⊗}_RC

satisfying a list of compatibility structures. For a smooth projective variety

there is a canonical Hodge structure.

Statement for degree d hypersurfaces

In the special case

is defined by a homogeneous degree

polynomial

f\in\Gamma(Pⁿ⁺¹,l{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[Z_0,\ldots, Z_n]^ \to \frac

which is surjective on the primitive cohomology, denoted

Prim^p,n-p(X)

and has the kernel

J_f

. Note the primitive cohomology classes are the classes of

which do not come from

Pⁿ⁺¹

, which is just the Lefschetz class

[L]ⁿ=

	d
c
	1(l{O}(1))

Sketch of proof

Reduction to residue map

For

X\subsetPⁿ⁺¹

there is an associated short exact sequence of complexes

0 \to \Omega_^\bullet \to \Omega_^\bullet(\log X) \xrightarrow \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

, which is

H^n(X;C)=

	\bullet)
H
	X

. From the long exact sequence of this short exact sequence, there the induced residue map

\mathbb^\left(\mathbb^, \Omega^\bullet_(\log X)\right) \to \mathbb^(\mathbb^,\Omega^\bullet_X[-1])

where the right hand side is equal to

Hⁿ(Pⁿ⁺¹

	\bullet
,\Omega
	X)

, which is isomorphic to

	\bullet)
H
	X

. Also, there is an isomorphism

H^_(\mathbb^-X) \cong \mathbb^\left(\mathbb^;\Omega_^\bullet(\log X)\right)

Through these isomorphisms there is an induced residue map

res: H^_(\mathbb^-X) \to H^n(X;\mathbb)

which is injective, and surjective on primitive cohomology. Also, there is the Hodge decomposition

H^_(\mathbb^-X) \cong\bigoplus_H^q(\Omega_^p(\log X))

and

	p(log
H
	P

X))\congPrim^p-1,q(X)

Computation of de Rham cohomology group

In turns out the de Rham cohomology group

	n+1
H
	dR

(Pⁿ⁺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

\leqn-p+1

which surjects onto the

F^p

part of

Prim^n(X)

. This comes from the reduction isomorphism

F^H^_(\mathbb^-X;\mathbb) \cong \frac

Using the canonical

(n+1)

-form

\Omega = \sum_^n (-1)^j Z_j dZ_0\wedge \cdots \wedge \hat\wedge \cdots \wedge dZ_

Pⁿ⁺¹

where the

\hat{dZ_j}

denotes the deletion from the index, these meromorphic differential forms look like

\frac\Omega

where

\begin\text(A) &= (n-p+1)\cdot\text(f) - \text(\Omega) \\&= (n-p+1)\cdot d - (n + 2) \\&= d(n-p+1) - (n+2)\end

Finally, it turns out the kernel ^{Lemma 8.11} is of all polynomials of the form

A'+fB

where

A'\inJ_f

. Note the Euler identity

f = \sum Z_j \frac

shows

f\inJ_f

References

Book: Introduction to Hodge theory. 2002. American Mathematical Society. José Bertin. 0-8218-2040-0. Providence, R.I.. 199–205. 48892689.