Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Logarithmic differential forms and questions of
residues.
Michel Granger
University of Angers, France
Differential and combinatorial aspects of singularities.
Kaiserslautern, August 3-7, 2015
Joint work with Mathias Schulze and part of Phd work of
Delphine Pol.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
References
Michel Granger and Mathias Schulze, Normal crossing
properties of complex hypersurfaces via logarithmic residues.
Compositio Math. 150 (2014) 1607-1622.
Delphine Pol, Logarithmic residues along plane curves. C. R.
Math. Acad. Sci. Paris, 353(4):345349, 2015, and Module
des résidus logarithmiques des courbes planes,
arXiv:1410.2126 (english version in in preparation).
Delphine Pol, Logarithmic differential forms along complete
intersection, work in progress.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Contents
1
Basic definitions.
2
Logarithmic residues and duality.
3
Normal crossing conditions
4
Residues along plane curves
5
The complete intersection case.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
I. Basic definitions.
Let D ⊂ S = (Cn , 0) be a reduced effective divisor,
ID = OS · h its defining ideal. We set :
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
I. Basic definitions.
Let D ⊂ S = (Cn , 0) be a reduced effective divisor,
ID = OS · h its defining ideal. We set :
ΘS := DerC (OS ) = HomOS (Ω1S , OS ),
Michel Granger University of Angers, France
ΩpS (D) = ΩpS ·
1
h
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
I. Basic definitions.
Let D ⊂ S = (Cn , 0) be a reduced effective divisor,
ID = OS · h its defining ideal. We set :
ΘS := DerC (OS ) = HomOS (Ω1S , OS ),
ΩpS (D) = ΩpS ·
1
h
Definition (K. Saito)
Ωp (log D) := {ω ∈ ΩpS (D) | dω ∈ Ωp+1
S (D)}
Der(− log D) := {δ ∈ ΘS | dh(δ) ∈ ID }
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
I. Basic definitions.
Let D ⊂ S = (Cn , 0) be a reduced effective divisor,
ID = OS · h its defining ideal. We set :
ΘS := DerC (OS ) = HomOS (Ω1S , OS ),
ΩpS (D) = ΩpS ·
1
h
Definition (K. Saito)
Ωp (log D) := {ω ∈ ΩpS (D) | dω ∈ Ωp+1
S (D)}
Der(− log D) := {δ ∈ ΘS | dh(δ) ∈ ID }
All these modules are coherent and reflexive. In particular
Ω1 (log D) = HomOS (Der(− log D), OS ), and
Der(− log D) = HomOS (Ω1 (log D), OS )
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Let Σ = Sing (D), with OΣ = OS /(J(h), h) = OD /JD . We have
the following exact sequences :
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Let Σ = Sing (D), with OΣ = OS /(J(h), h) = OD /JD . We have
the following exact sequences :
(1.1)
0
/ Der(− log D) 
Michel Granger University of Angers, France
/ ΘS
dh
/ JD
/0
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Let Σ = Sing (D), with OΣ = OS /(J(h), h) = OD /JD . We have
the following exact sequences :
(1.1)
0
/ Der(− log D) 
/ ΘS
dh
/ JD
/0
(1.2)
0
/ Der(− log D)
/ ΘS ⊕ OS dh,−h / OS
/ OΣ
/0
In sequence (1.2) the first arrow is : δ 7→ (δ, δ(h)/h)
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Let Σ = Sing (D), with OΣ = OS /(J(h), h) = OD /JD . We have
the following exact sequences :
(1.1)
0
/ Der(− log D) 
/ ΘS
dh
/ JD
/0
(1.2)
0
/ Der(− log D)
/ ΘS ⊕ OS dh,−h / OS
/ OΣ
/0
In sequence (1.2) the first arrow is : δ 7→ (δ, δ(h)/h)
Definition
The divisor D is free iff Der(− log D) or alternatively Ω1 (log D) is
a free module.
Here are two characterisations of freeness.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Theorem (Saito criterion)
The divisor D is free iff there are δ1 , · · · , δn ∈ Der(− log D) such
that
det(δ1 , · · · , δn ) = uh
with u a unit. The n-uples δ1 , · · · , δn with this property are the
generating families of Der(− log D).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Theorem (Saito criterion)
The divisor D is free iff there are δ1 , · · · , δn ∈ Der(− log D) such
that
det(δ1 , · · · , δn ) = uh
with u a unit. The n-uples δ1 , · · · , δn with this property are the
generating families of Der(− log D).
Theorem (Terao in qh case, Aleksandrov.)
The following three conditions are equivalent :
1
The divisor D is free.
2
JD is Cohen Macaulay (of dimension n − 1.)
3
OΣ is Cohen Macaulay (of dimension n − 2.)
The proof essentially uses the Auslander-Buchsbaum formula.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
II. Logarithmic residues
Saito proves that ω ∈ Ωp (log D) iff there is g ∈ OS non zero
divisor in OD and there are holomorphic forms ξ, η such that :
gω =
dh
∧ ξ + η,
h
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
II. Logarithmic residues
Saito proves that ω ∈ Ωp (log D) iff there is g ∈ OS non zero
divisor in OD and there are holomorphic forms ξ, η such that :
gω =
dh
∧ ξ + η,
h
Definition
The residue of ω is the meromorphic (q − 1)-form on D or
equivalently on the normalization D̃ :
ρpD (ω) :=
ξ
p−1
⊗ Q(OD̃ )
|D ∈ Ωp−1
⊗ Q(OD ) = ΩD̃
D
g
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
II. Logarithmic residues
Saito proves that ω ∈ Ωp (log D) iff there is g ∈ OS non zero
divisor in OD and there are holomorphic forms ξ, η such that :
gω =
dh
∧ ξ + η,
h
Definition
The residue of ω is the meromorphic (q − 1)-form on D or
equivalently on the normalization D̃ :
ρpD (ω) :=
ξ
p−1
⊗ Q(OD̃ )
|D ∈ Ωp−1
⊗ Q(OD ) = ΩD̃
D
g
We set ρ1D = ρD , RD := ρD (Ω1 (log D)) ⊂ Q(OD )
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Properties of RD .
Proposition
We have OD̃ ⊂ RD and there is an exact sequence :
(2.1)
0
/ Ω1
S
/ Ω1 (log D)
Michel Granger University of Angers, France
ρD
/ RD
/ 0.
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Properties of RD .
Proposition
We have OD̃ ⊂ RD and there is an exact sequence :
(2.1)
0
/ Ω1
S
/ Ω1 (log D)
ρD
/ RD
/ 0.
By dualizing over OS we obtain the following result :
Proposition (G, M. Schulze)
1) There is an exact sequence
0
/ Der(− log D)
/ ΘS
σD
/ R∨
D
/ Ext1 (Ω1 (log D), OS )
OS
2) The image of σD is JD ⊂ RD∨ and we always have RD = JD∨ .
3) When D is free JD = RD∨
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Ideas for the proof. The presence of RD∨ comes from the change of
ring formula :
RD∨ := HomOD (RD , OD ) = HomOD (RD , ExtO1 S (OD , OS ))
=ExtO1 S (RD , OS )
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Ideas for the proof. The presence of RD∨ comes from the change of
ring formula :
RD∨ := HomOD (RD , OD ) = HomOD (RD , ExtO1 S (OD , OS ))
=ExtO1 S (RD , OS )
The equality σD (δ)(ρ) = hδ, hi · ρ is obtained by studying a
diagram built on the complex
HomOS (Ω1S ,→ Ω1 (log D), h : OS → OS ).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The OD submodules JD , RD , OD̃ of Q(OD ) are fractional ideals,
i.e. contain a non zero divisor.
By a result of De Jong and Van Straten, duality I → I ∨
preserves fractional ideals and is an involution on maximal CM
ones.
This is the case for the conductor CD := O ∨
e
D
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The OD submodules JD , RD , OD̃ of Q(OD ) are fractional ideals,
i.e. contain a non zero divisor.
By a result of De Jong and Van Straten, duality I → I ∨
preserves fractional ideals and is an involution on maximal CM
ones.
This is the case for the conductor CD := O ∨
e
D
We may summarize the situation as follows
We obtain a chain of fractional ideals
JD ⊆ RD∨ ⊆ CD ⊆ OD ⊆ ODe ⊆ RD
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The OD submodules JD , RD , OD̃ of Q(OD ) are fractional ideals,
i.e. contain a non zero divisor.
By a result of De Jong and Van Straten, duality I → I ∨
preserves fractional ideals and is an involution on maximal CM
ones.
This is the case for the conductor CD := O ∨
e
D
We may summarize the situation as follows
We obtain a chain of fractional ideals
JD ⊆ RD∨ ⊆ CD ⊆ OD ⊆ ODe ⊆ RD
If D is free, then JD = RD∨ as fractional ideals. In that case :
RD = ODe ⇐⇒ JD = CD .
We call the condition RD = ODe the normal crossing condition.
The starting point is a result of K. Saito :
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Normal crossing condition.
Theorem (Saito)
For a divisor D in a complex manifold S, consider the following
conditions:
(A) the local fundamental groups of the complement S\D are
Abelian;
(B) in codimension one, that is, outside of an analytic subset of
codimension at least 2 in D, D is a normal crossing;
(C) the residue of any logarithmic 1-form along D is a weakly
holomorphic function on D.
Then the implications (A) ⇒ (B) ⇒ (C) hold true.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
In his 1980 paper Saito asked for the converse implications:
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
In his 1980 paper Saito asked for the converse implications:
Theorem (Lê Dung Tràng and K.Saito 1984)
The implication (A) ⇐ (B) in Theorem 3.1 holds true.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
In his 1980 paper Saito asked for the converse implications:
Theorem (Lê Dung Tràng and K.Saito 1984)
The implication (A) ⇐ (B) in Theorem 3.1 holds true.
Theorem (G, Mathias Schulze)
The implication (B) ⇐ (C) in Theorem 3.1 holds true: if the
residue of any logarithmic 1-form along D is a weakly holomorphic
function on D then D is a normal crossing in codimension one.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Outline of the proof :
Let ϕ : Y → X be a morphism. Then
ΩY /X = 0 ⇔ ϕ is an immersion.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Outline of the proof :
Let ϕ : Y → X be a morphism. Then
ΩY /X = 0 ⇔ ϕ is an immersion.
In codimension 1, D is free, D̃ is smooth.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Outline of the proof :
Let ϕ : Y → X be a morphism. Then
ΩY /X = 0 ⇔ ϕ is an immersion.
In codimension 1, D is free, D̃ is smooth.
Let Rπ := F 0 (ΩD̃/D ) be the ramification ideal.
By a formula of Ragni Piene : CD Rπ = JD OD̃
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Outline of the proof :
Let ϕ : Y → X be a morphism. Then
ΩY /X = 0 ⇔ ϕ is an immersion.
In codimension 1, D is free, D̃ is smooth.
Let Rπ := F 0 (ΩD̃/D ) be the ramification ideal.
By a formula of Ragni Piene : CD Rπ = JD OD̃
For a free D it follows :
RD = OD̃ =⇒ CD = JD =⇒ Rπ = OD̃ =⇒ ΩD̃/D = 0
Then D̃ and D have smooth components D̃i
Michel Granger University of Angers, France
'
/ Di .
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Outline of the proof :
Let ϕ : Y → X be a morphism. Then
ΩY /X = 0 ⇔ ϕ is an immersion.
In codimension 1, D is free, D̃ is smooth.
Let Rπ := F 0 (ΩD̃/D ) be the ramification ideal.
By a formula of Ragni Piene : CD Rπ = JD OD̃
For a free D it follows :
RD = OD̃ =⇒ CD = JD =⇒ Rπ = OD̃ =⇒ ΩD̃/D = 0
Then D̃ and D have smooth components D̃i
'
/ Di .
Finally in codimension one the N.C. condition becomes
RD = ⊕i ODi , a case where the result is known by Saito.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Characterization of normal crossing divisors
A normal crossing divisor is free and JD is a radical ideal of OD .
A partial converse is :
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Characterization of normal crossing divisors
A normal crossing divisor is free and JD is a radical ideal of OD .
A partial converse is :
Theorem (E. Faber, G and M. Schulze.)
For a free divisor with smooth normalization, any of the conditions
The ideal Jh = (hx0 1 , · · · , hx0 n ) ⊂ OS is a radical ideal
Any of the equivalent conditions (A), (B), (C),
The Jacobian ideal JD is radical.
imply that D is a normal crossing divisor.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Characterization of normal crossing divisors
A normal crossing divisor is free and JD is a radical ideal of OD .
A partial converse is :
Theorem (E. Faber, G and M. Schulze.)
For a free divisor with smooth normalization, any of the conditions
The ideal Jh = (hx0 1 , · · · , hx0 n ) ⊂ OS is a radical ideal
Any of the equivalent conditions (A), (B), (C),
The Jacobian ideal JD is radical.
imply that D is a normal crossing divisor.
Question (E. Faber) In i) or iii), can one get rid of the smoothness
hypothesis?
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Going further.
When D does not satisfy (A), (B) or (C), How to determine
RD % OD̃
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Going further.
When D does not satisfy (A), (B) or (C), How to determine
RD % OD̃
Case of curves : Detailed answer in terms of the semigroup of
multivaluations (or values). Delphine Pol.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Going further.
When D does not satisfy (A), (B) or (C), How to determine
RD % OD̃
Case of curves : Detailed answer in terms of the semigroup of
multivaluations (or values). Delphine Pol.
There is a notion of multiresidues along a complete
intersection due to Aleksandrov and Tsikh.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Going further.
When D does not satisfy (A), (B) or (C), How to determine
RD % OD̃
Case of curves : Detailed answer in terms of the semigroup of
multivaluations (or values). Delphine Pol.
There is a notion of multiresidues along a complete
intersection due to Aleksandrov and Tsikh.
- Description of the dual residue module RC∨ . Results similar
to the above for curves, and other examples (D. Pol).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Going further.
When D does not satisfy (A), (B) or (C), How to determine
RD % OD̃
Case of curves : Detailed answer in terms of the semigroup of
multivaluations (or values). Delphine Pol.
There is a notion of multiresidues along a complete
intersection due to Aleksandrov and Tsikh.
- Description of the dual residue module RC∨ . Results similar
to the above for curves, and other examples (D. Pol).
- A notion of freeness (G, M. Schulze), and a cohomological
characterization of freeness, for multivector fields (G, Schulze)
and forms (D. Pol).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Going further.
When D does not satisfy (A), (B) or (C), How to determine
RD % OD̃
Case of curves : Detailed answer in terms of the semigroup of
multivaluations (or values). Delphine Pol.
There is a notion of multiresidues along a complete
intersection due to Aleksandrov and Tsikh.
- Description of the dual residue module RC∨ . Results similar
to the above for curves, and other examples (D. Pol).
- A notion of freeness (G, M. Schulze), and a cohomological
characterization of freeness, for multivector fields (G, Schulze)
and forms (D. Pol).
- Analogue of the normal crossing condition has been studied
by M. Schulze.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Semigroup of a curve.
S
Let (D, 0) = pi=1 Di ⊂ (S, 0) be a reduced curve, with
normalization :
OD ,→ OD̃ ' C{t1 } ⊕ · · · ⊕ C{tp }.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Semigroup of a curve.
S
Let (D, 0) = pi=1 Di ⊂ (S, 0) be a reduced curve, with
normalization :
OD ,→ OD̃ ' C{t1 } ⊕ · · · ⊕ C{tp }.
The value of g ∈ Q(OD ), is the p-uple of valuations w.r. to tj ’s
val(g) = (val1 (g), · · · , valp (g)) ∈ (Z ∪ {∞})p .
Let val(I) ⊂ Zp be the set of values on non zero divisors in I .
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Semigroup of a curve.
S
Let (D, 0) = pi=1 Di ⊂ (S, 0) be a reduced curve, with
normalization :
OD ,→ OD̃ ' C{t1 } ⊕ · · · ⊕ C{tp }.
The value of g ∈ Q(OD ), is the p-uple of valuations w.r. to tj ’s
val(g) = (val1 (g), · · · , valp (g)) ∈ (Z ∪ {∞})p .
Let val(I) ⊂ Zp be the set of values on non zero divisors in I .
Definition
The semigroup of D is Γ = val(OD ) ⊂ Np .
There is γ ∈ Np with val(CD ) = (γ1 , · · · , γp ) + Np .
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Semigroup of a curve.
S
Let (D, 0) = pi=1 Di ⊂ (S, 0) be a reduced curve, with
normalization :
OD ,→ OD̃ ' C{t1 } ⊕ · · · ⊕ C{tp }.
The value of g ∈ Q(OD ), is the p-uple of valuations w.r. to tj ’s
val(g) = (val1 (g), · · · , valp (g)) ∈ (Z ∪ {∞})p .
Let val(I) ⊂ Zp be the set of values on non zero divisors in I .
Definition
The semigroup of D is Γ = val(OD ) ⊂ Np .
There is γ ∈ Np with val(CD ) = (γ1 , · · · , γp ) + Np .
More generally each fractional ideal I ⊂ Q(OD ) has a conductor
ν ∈ Zp , val(I) ⊃ ν + Np
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
A symmetry result.
Theorem (Delgado)
The ring OD is Gorenstein iff the semi group Γ has the following
property:
∀v ∈ Zp , v ∈ Γ ⇐⇒ ∆(γ − v − (1, · · · , 1), OD ) = ∅
Here we set ∆(α, val(OD )) = val(OD ) ∩ α + ∆n−1 (Rn+ ),
∆n−1 (Rn+ ) is the n − 1-dimensional part of the upper quadrant
Rn+ ⊃ Nn+
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
A symmetry result.
Theorem (Delgado)
The ring OD is Gorenstein iff the semi group Γ has the following
property:
∀v ∈ Zp , v ∈ Γ ⇐⇒ ∆(γ − v − (1, · · · , 1), OD ) = ∅
Here we set ∆(α, val(OD )) = val(OD ) ∩ α + ∆n−1 (Rn+ ),
∆n−1 (Rn+ ) is the n − 1-dimensional part of the upper quadrant
Rn+ ⊃ Nn+
Let us restrict to plane curves D ⊂ (C2 , 0). We would like to
extend this result to a relation between val(JD ) and val(RD ).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Theorem (D. Pol)
Let D be a plane curve. Then the set of values of the module of
logarithmic residues is determined by JD :
v ∈ val(RD ) ⇐⇒ ∆(γ − v − (1, · · · , 1), JD ) = ∅
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Theorem (D. Pol)
Let D be a plane curve. Then the set of values of the module of
logarithmic residues is determined by JD :
v ∈ val(RD ) ⇐⇒ ∆(γ − v − (1, · · · , 1), JD ) = ∅
Remark: The same result holds in a more general context for any
Gorenstein reduced curve, and fractionnal ideals I , I ∨ .
In particular for a complete intersection curve C , we still have
RC , JC mutually dual. Here RC is as defined by Aleksandrov and
Tsikh.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The proof in the irreducible case is fairly direct.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The proof in the irreducible case is fairly direct.
In general we easily get the first part of the equivalence
val(RD ) ⊂ V := {v | ∆(γ − v − (1, · · · , 1), JD ) = ∅}
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The proof in the irreducible case is fairly direct.
In general we easily get the first part of the equivalence
val(RD ) ⊂ V := {v | ∆(γ − v − (1, · · · , 1), JD ) = ∅}
The converse (chasing all possible v ∈ V \ val(RD )) is much
harder.
D. Pol uses a combinatorial calculation of the dimensions of
RD /OD̃ or of JD /CD0 quotient of JD by its conductor. The
relationship with the set V ⊂ Zp is then the central point.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
The proof in the irreducible case is fairly direct.
In general we easily get the first part of the equivalence
val(RD ) ⊂ V := {v | ∆(γ − v − (1, · · · , 1), JD ) = ∅}
The converse (chasing all possible v ∈ V \ val(RD )) is much
harder.
D. Pol uses a combinatorial calculation of the dimensions of
RD /OD̃ or of JD /CD0 quotient of JD by its conductor. The
relationship with the set V ⊂ Zp is then the central point.
We give a picture for the case of f = (x 2 − y 3 )(x 4 − y 3 ), µ = 19,
δ = 10, τ = 17, γ = (8, 12), γJ = (12, 20).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
val2
20
15
γ
10
15
Figure : Semigroup of JD
Michel Granger University of Angers, France
val1
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
−5
val2
0
val1
−5
−10
Figure : Multi-values of RD
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
An example.
Notice that dim RD /OD = dim OD /JD = τ the Tjurina
number and dim RD /OD̃ = τ − δ.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
An example.
Notice that dim RD /OD = dim OD /JD = τ the Tjurina
number and dim RD /OD̃ = τ − δ.
Contrary to Γ, val(RD ) varies in the µ-constant stratum
S ⊂ Cµ of a semiuniversal unfolding F :
P
C2 × Cµ → C, F (x, y , s) = f (x, y ) + 1≤k≤µ t k mk (x, y )
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
An example.
Notice that dim RD /OD = dim OD /JD = τ the Tjurina
number and dim RD /OD̃ = τ − δ.
Contrary to Γ, val(RD ) varies in the µ-constant stratum
S ⊂ Cµ of a semiuniversal unfolding F :
P
C2 × Cµ → C, F (x, y , s) = f (x, y ) + 1≤k≤µ t k mk (x, y )
In fact τ -constant deformations are adapted to residue
calculations as flat deformations of f , fx0 , fy0 :
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
An example.
Notice that dim RD /OD = dim OD /JD = τ the Tjurina
number and dim RD /OD̃ = τ − δ.
Contrary to Γ, val(RD ) varies in the µ-constant stratum
S ⊂ Cµ of a semiuniversal unfolding F :
P
C2 × Cµ → C, F (x, y , s) = f (x, y ) + 1≤k≤µ t k mk (x, y )
In fact τ -constant deformations are adapted to residue
calculations as flat deformations of f , fx0 , fy0 :
Proposition
The partition of S by val(RD ) is a partition into analytic locally
closed subsets refining the τ -constant strata.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
An example.
Notice that dim RD /OD = dim OD /JD = τ the Tjurina
number and dim RD /OD̃ = τ − δ.
Contrary to Γ, val(RD ) varies in the µ-constant stratum
S ⊂ Cµ of a semiuniversal unfolding F :
P
C2 × Cµ → C, F (x, y , s) = f (x, y ) + 1≤k≤µ t k mk (x, y )
In fact τ -constant deformations are adapted to residue
calculations as flat deformations of f , fx0 , fy0 :
Proposition
The partition of S by val(RD ) is a partition into analytic locally
closed subsets refining the τ -constant strata.
Consider for example the irreducible branch x 5 − y 6 .
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
F (x, y , s1 , s2 , s3 ) = x 5 − y 6 + s1 x 2 y 4 + s2 x 3 y 3 + s3 x 3 y 4 ,
µ = 20, δ = 10 ,
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
F (x, y , s1 , s2 , s3 ) = x 5 − y 6 + s1 x 2 y 4 + s2 x 3 y 3 + s3 x 3 y 4 ,
µ = 20, δ = 10 ,
S1 = {0},
S2 = {(0, 0, s3 ), s3 6= 0}
S30 = {(s1 , s2 , s3 ), s1 6= 0} et S300 = {(0, s2 , s3 ), s2 6= 0}.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
F (x, y , s1 , s2 , s3 ) = x 5 − y 6 + s1 x 2 y 4 + s2 x 3 y 3 + s3 x 3 y 4 ,
µ = 20, δ = 10 ,
S1 = {0},
S2 = {(0, 0, s3 ), s3 6= 0}
S30 = {(s1 , s2 , s3 ), s1 6= 0} et S300 = {(0, s2 , s3 ), s2 6= 0}.
In the second colum is the value of
Strate τ − δ values < 0
S1
10
−1, −2, −3, −4,
S2
9
−1, −2, −3, −4,
S30
8
−1, −2, −3, −4,
00
S3
8
−1, −2, −3, −4,
Michel Granger University of Angers, France
dimC RDs /OD̃s = τ − 10
−7, −8, −9,
−7, −8, −9,
−7, −8, −9,
−7, −8, −9,
−13, −14,
−13, −14
−14
−13
−19
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Link with Kähler differentials
We consider the module Ω1C , and its set of values along a complete
intersection
P curve. 1
If ω = k ak dxk ∈ ΩC , and ti → ϕi (ti ) parametrizes the ith
branch,
!
∗
X
ϕi (ω)
0
vali (ω) = 1 + val
= 1 + val
(ak ◦ ϕi )(ti )xk (ti )
dti
k
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Link with Kähler differentials
We consider the module Ω1C , and its set of values along a complete
intersection
P curve. 1
If ω = k ak dxk ∈ ΩC , and ti → ϕi (ti ) parametrizes the ith
branch,
!
∗
X
ϕi (ω)
0
vali (ω) = 1 + val
= 1 + val
(ak ◦ ϕi )(ti )xk (ti )
dti
k
Proposition (G, D.Pol)
We have: val(JC ) = γ + val(Ω1C ) − (1, · · · , 1)
A direct calculation yields val(JC ) = val(Ω1C ) + λ
and λ = γ + (1, · · · , 1) by R. Piene’s formula.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Link with Kähler differentials
We consider the module Ω1C , and its set of values along a complete
intersection
P curve. 1
If ω = k ak dxk ∈ ΩC , and ti → ϕi (ti ) parametrizes the ith
branch,
!
∗
X
ϕi (ω)
0
vali (ω) = 1 + val
= 1 + val
(ak ◦ ϕi )(ti )xk (ti )
dti
k
Proposition (G, D.Pol)
We have: val(JC ) = γ + val(Ω1C ) − (1, · · · , 1)
A direct calculation yields val(JC ) = val(Ω1C ) + λ
and λ = γ + (1, · · · , 1) by R. Piene’s formula.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Corollary
For a plane curve v ∈ val(RD ) ⇐⇒ ∆(−v, val(Ω1D )) = ∅
In this way RD is related to the problem of moduli space for plane
branches with a given semi group Γ
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Corollary
For a plane curve v ∈ val(RD ) ⇐⇒ ∆(−v, val(Ω1D )) = ∅
In this way RD is related to the problem of moduli space for plane
branches with a given semi group Γ
Theorem (Hefez,Hernandez)
The set of analytic classes M of plane branches with given
topological type satisfies
S
M = Ω1 =Ω MΩ .
D
Each MΩ is separated and a quotient by a finite group of an affine
open space.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
A look at complete intersection case.
Let C ⊂ S = (Cn , 0) be a complete intersection f1 = · · · = fk = 0,
1
fq = P
D be the hypersurface f1 . . . fk = 0, and Ω
ΩqS .
j
S
b
f1 ···fj ···fk
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
A look at complete intersection case.
Let C ⊂ S = (Cn , 0) be a complete intersection f1 = · · · = fk = 0,
1
fq = P
D be the hypersurface f1 . . . fk = 0, and Ω
ΩqS .
j
S
b
f1 ···fj ···fk
Definition
1
(Aleksandrov, Tsikh) A form ω ∈ ΩqS (D) := f1 ···f
ΩqS is logarithmic
k
]
q
iff for all j, dfj ∧ ω ∈ Ωq−1
S . We write Ω (log C ).
Proposition
(Aleksandrov, Tsikh) A form ω is logarithmic iff there are ξ
]
holomorphic, and η ∈ Ωq−1
and g ∈ NZD(OC ) with
S
k
g ω = df1f∧···∧df
∧ξ+η
1 ···fk
We define the residue of ω as res ω = gξ |C ∈ Ωq−k ⊗ Q(OC ).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Definition (G, M. Schulze)
The complete intersection is called free iff JC (or OC /JC ) is
maximal Cohen Macaulay.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Definition (G, M. Schulze)
The complete intersection is called free iff JC (or OC /JC ) is
maximal Cohen Macaulay.
This property is equivalent to : Derk (− log C ) := ker(ΘkS −→ JC )
is of projective dimension k − 1.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Definition (G, M. Schulze)
The complete intersection is called free iff JC (or OC /JC ) is
maximal Cohen Macaulay.
This property is equivalent to : Derk (− log C ) := ker(ΘkS −→ JC )
is of projective dimension k − 1. The following result for forms is
more involved :
Theorem (Delphine Pol)
The complete intersection C is free iff Ωk (log C ) if of projective
dimension k − 1.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Definition (G, M. Schulze)
The complete intersection is called free iff JC (or OC /JC ) is
maximal Cohen Macaulay.
This property is equivalent to : Derk (− log C ) := ker(ΘkS −→ JC )
is of projective dimension k − 1. The following result for forms is
more involved :
Theorem (Delphine Pol)
The complete intersection C is free iff Ωk (log C ) if of projective
dimension k − 1.
We set RC = res(Ωk (log C )) ⊂ Q(OC ). Its does not depend on
the choice of fi ’s. By direct calculation or because RC = the set of
regular 0-forms (Aleksandrov).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
We end with a few facts about RC , noticed or proved in G,
M.Schulze, or in D. Pol;
We have JC∨ = RC ⊃ OC̃ , and for a free C also RC∨ = JC .
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
We end with a few facts about RC , noticed or proved in G,
M.Schulze, or in D. Pol;
We have JC∨ = RC ⊃ OC̃ , and for a free C also RC∨ = JC .
For a complete intersection curve this yields a result of
symmetry, completely similar to the planar case.
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
We end with a few facts about RC , noticed or proved in G,
M.Schulze, or in D. Pol;
We have JC∨ = RC ⊃ OC̃ , and for a free C also RC∨ = JC .
For a complete intersection curve this yields a result of
symmetry, completely similar to the planar case.
We can see that Ωq (log D) ⊂ Ωq (log C ). In particular
resC (Ωk (log D)) ⊂ RC
but in general this inclusion is strict (Example of D. Pol).
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.
Basic definitions.
Logarithmic residues and duality.
Normal crossing conditions
Residues along plane curves
The complete intersection case.
Thank you for your attention
Michel Granger University of Angers, France
Logarithmic differential forms and questions of residues.