- Source: Jacobian ideal
In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ.
Let
O
(
x
1
,
…
,
x
n
)
{\displaystyle {\mathcal {O}}(x_{1},\ldots ,x_{n})}
denote the ring of smooth functions in
n
{\displaystyle n}
variables and
f
{\displaystyle f}
a function in the ring. The Jacobian ideal of
f
{\displaystyle f}
is
J
f
:=
⟨
∂
f
∂
x
1
,
…
,
∂
f
∂
x
n
⟩
.
{\displaystyle 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
{\displaystyle f}
is classified by the ring
C
[
x
1
,
…
,
x
n
]
(
f
)
+
J
f
.
{\displaystyle {\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
R
{\displaystyle H_{\mathbb {R} }}
and an increasing filtration
F
∙
{\displaystyle F^{\bullet }}
of
H
C
=
H
R
⊗
R
C
{\displaystyle H_{\mathbb {C} }=H_{\mathbb {R} }\otimes _{\mathbb {R} }\mathbb {C} }
satisfying a list of compatibility structures. For a smooth projective variety
X
{\displaystyle X}
there is a canonical Hodge structure.
= Statement for degree d hypersurfaces
=In the special case
X
{\displaystyle X}
is defined by a homogeneous degree
d
{\displaystyle d}
polynomial
f
∈
Γ
(
P
n
+
1
,
O
(
d
)
)
{\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
C
[
Z
0
,
…
,
Z
n
]
(
d
(
n
−
1
+
p
)
−
(
n
+
2
)
)
→
F
p
H
n
(
X
,
C
)
F
p
+
1
H
n
(
X
,
C
)
{\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
Prim
p
,
n
−
p
(
X
)
{\displaystyle {\text{Prim}}^{p,n-p}(X)}
and has the kernel
J
f
{\displaystyle J_{f}}
. Note the primitive cohomology classes are the classes of
X
{\displaystyle X}
which do not come from
P
n
+
1
{\displaystyle \mathbb {P} ^{n+1}}
, which is just the Lefschetz class
[
L
]
n
=
c
1
(
O
(
1
)
)
d
{\displaystyle [L]^{n}=c_{1}({\mathcal {O}}(1))^{d}}
.
= Sketch of proof
=Reduction to residue map
For
X
⊂
P
n
+
1
{\displaystyle X\subset \mathbb {P} ^{n+1}}
there is an associated short exact sequence of complexes
0
→
Ω
P
n
+
1
∙
→
Ω
P
n
+
1
∙
(
log
X
)
→
r
e
s
Ω
X
∙
[
−
1
]
→
0
{\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
X
{\displaystyle X}
, which is
H
n
(
X
;
C
)
=
H
n
(
X
;
Ω
X
∙
)
{\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
H
n
+
1
(
P
n
+
1
,
Ω
P
n
+
1
∙
(
log
X
)
)
→
H
n
+
1
(
P
n
+
1
,
Ω
X
∙
[
−
1
]
)
{\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
H
n
(
P
n
+
1
,
Ω
X
∙
)
{\displaystyle \mathbb {H} ^{n}(\mathbb {P} ^{n+1},\Omega _{X}^{\bullet })}
, which is isomorphic to
H
n
(
X
;
Ω
X
∙
)
{\displaystyle \mathbb {H} ^{n}(X;\Omega _{X}^{\bullet })}
. Also, there is an isomorphism
H
d
R
n
+
1
(
P
n
+
1
−
X
)
≅
H
n
+
1
(
P
n
+
1
;
Ω
P
n
+
1
∙
(
log
X
)
)
{\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
r
e
s
:
H
d
R
n
+
1
(
P
n
+
1
−
X
)
→
H
n
(
X
;
C
)
{\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
H
d
R
n
+
1
(
P
n
+
1
−
X
)
≅
⨁
p
+
q
=
n
+
1
H
q
(
Ω
P
p
(
log
X
)
)
{\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
H
q
(
Ω
P
p
(
log
X
)
)
≅
Prim
p
−
1
,
q
(
X
)
{\displaystyle 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
d
R
n
+
1
(
P
n
+
1
−
X
)
{\displaystyle 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
{\displaystyle F^{p}}
part is spanned by the meromorphic forms having poles of order
≤
n
−
p
+
1
{\displaystyle \leq n-p+1}
which surjects onto the
F
p
{\displaystyle F^{p}}
part of
Prim
n
(
X
)
{\displaystyle {\text{Prim}}^{n}(X)}
. This comes from the reduction isomorphism
F
p
+
1
H
d
R
n
+
1
(
P
n
+
1
−
X
;
C
)
≅
Γ
(
Ω
P
n
+
1
(
n
−
p
+
1
)
)
d
Γ
(
Ω
P
n
+
1
(
n
−
p
)
)
{\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
(
n
+
1
)
{\displaystyle (n+1)}
-form
Ω
=
∑
j
=
0
n
(
−
1
)
j
Z
j
d
Z
0
∧
⋯
∧
d
Z
j
^
∧
⋯
∧
d
Z
n
+
1
{\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
P
n
+
1
{\displaystyle \mathbb {P} ^{n+1}}
where the
d
Z
j
^
{\displaystyle {\hat {dZ_{j}}}}
denotes the deletion from the index, these meromorphic differential forms look like
A
f
n
−
p
+
1
Ω
{\displaystyle {\frac {A}{f^{n-p+1}}}\Omega }
where
deg
(
A
)
=
(
n
−
p
+
1
)
⋅
deg
(
f
)
−
deg
(
Ω
)
=
(
n
−
p
+
1
)
⋅
d
−
(
n
+
2
)
=
d
(
n
−
p
+
1
)
−
(
n
+
2
)
{\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 Lemma 8.11 is of all polynomials of the form
A
′
+
f
B
{\displaystyle A'+fB}
where
A
′
∈
J
f
{\displaystyle A'\in J_{f}}
. Note the Euler identity
f
=
∑
Z
j
∂
f
∂
Z
j
{\displaystyle f=\sum Z_{j}{\frac {\partial f}{\partial Z_{j}}}}
shows
f
∈
J
f
{\displaystyle f\in J_{f}}
.
References
See also
Milnor number
Hodge structure
Kodaira–Spencer map
Gauss–Manin connection
Unfolding