Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some examples of quasi-free divisors
Francisco-Jesús Castro-Jiménez (U. Seville)
Workshop on Free Divisors, Warwick, May 31st-June 4th; 2011
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Joint work with J.M. Ucha, F.J. Calderón and L. Narváez (U.
Seville) and D. Mond (U. Warwick).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free divisors
OX holomorphic functions on X = Cn .
DX linear differential operators on X .
Definition.- (J.M. Ucha, F.J.C.; 2004)
A germ of divisor (D, 0) in (Cn , 0) is called quasi-free if there
exists a free O-submodule L ⊂ Der(log D) of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
c) DL = DDer(log D).
In this case we say (D, L) is a quasi-free divisor (at 0 ∈ D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Conditions a), b) and c).
Condition c) is important.
L = f Der(O) satisfies conditions a) and b) (and doesn’t satisfy c)
in general).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Conditions a), b) and c).
Condition c) is important.
L = f Der(O) satisfies conditions a) and b) (and doesn’t satisfy c)
in general).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Conditions a), b) and c).
Condition c) is important.
L = f Der(O) satisfies conditions a) and b) (and doesn’t satisfy c)
in general).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
F implies QF
Free ⇒ Quasi-free
(taking L = Der(log D). Condition b) is implied by K. Saito’s
criterion for free divisors.)
Because of conditions a) and b), quasi-free notion is related to
Damon’s free∗ divisor structure notion. This relationship deserves
some explanation which I can not give.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
F implies QF
Free ⇒ Quasi-free
(taking L = Der(log D). Condition b) is implied by K. Saito’s
criterion for free divisors.)
Because of conditions a) and b), quasi-free notion is related to
Damon’s free∗ divisor structure notion. This relationship deserves
some explanation which I can not give.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
F implies QF
Free ⇒ Quasi-free
(taking L = Der(log D). Condition b) is implied by K. Saito’s
criterion for free divisors.)
Because of conditions a) and b), quasi-free notion is related to
Damon’s free∗ divisor structure notion. This relationship deserves
some explanation which I can not give.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
F implies QF
Free ⇒ Quasi-free
(taking L = Der(log D). Condition b) is implied by K. Saito’s
criterion for free divisors.)
Because of conditions a) and b), quasi-free notion is related to
Damon’s free∗ divisor structure notion. This relationship deserves
some explanation which I can not give.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
F implies QF
Free ⇒ Quasi-free
(taking L = Der(log D). Condition b) is implied by K. Saito’s
criterion for free divisors.)
Because of conditions a) and b), quasi-free notion is related to
Damon’s free∗ divisor structure notion. This relationship deserves
some explanation which I can not give.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (I)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 (it is not
free).
Der(log D) is generated by
η1 = 2y ∂y + z∂z
η3 = wy ∂z + z∂w
Francisco-Jesús Castro-Jiménez (U. Seville)
η2 = 2y ∂y − w ∂w
η4 = 2z∂y + w 2 ∂z
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (I)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 (it is not
free).
Der(log D) is generated by
η1 = 2y ∂y + z∂z
η3 = wy ∂z + z∂w
Francisco-Jesús Castro-Jiménez (U. Seville)
η2 = 2y ∂y − w ∂w
η4 = 2z∂y + w 2 ∂z
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (I)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 (it is not
free).
Der(log D) is generated by
η1 = 2y ∂y + z∂z
η3 = wy ∂z + z∂w
Francisco-Jesús Castro-Jiménez (U. Seville)
η2 = 2y ∂y − w ∂w
η4 = 2z∂y + w 2 ∂z
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (II)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L1 ⊂ Der(log D) generated by
δ1 = η1 + η2 , δ2 = zη4 − w 2 η2 δ3 = η3
(D, L1 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL1 DDer(log D).
L2 ⊂ Der(log D) generated by
δ4 = η4 , δ5 = η1 + η2 , δ6 = zη3 + yw η2
(D, L2 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL2 DDer(log D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (II)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L1 ⊂ Der(log D) generated by
δ1 = η1 + η2 , δ2 = zη4 − w 2 η2 δ3 = η3
(D, L1 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL1 DDer(log D).
L2 ⊂ Der(log D) generated by
δ4 = η4 , δ5 = η1 + η2 , δ6 = zη3 + yw η2
(D, L2 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL2 DDer(log D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (II)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L1 ⊂ Der(log D) generated by
δ1 = η1 + η2 , δ2 = zη4 − w 2 η2 δ3 = η3
(D, L1 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL1 DDer(log D).
L2 ⊂ Der(log D) generated by
δ4 = η4 , δ5 = η1 + η2 , δ6 = zη3 + yw η2
(D, L2 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL2 DDer(log D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (II)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L1 ⊂ Der(log D) generated by
δ1 = η1 + η2 , δ2 = zη4 − w 2 η2 δ3 = η3
(D, L1 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL1 DDer(log D).
L2 ⊂ Der(log D) generated by
δ4 = η4 , δ5 = η1 + η2 , δ6 = zη3 + yw η2
(D, L2 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL2 DDer(log D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (II)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L1 ⊂ Der(log D) generated by
δ1 = η1 + η2 , δ2 = zη4 − w 2 η2 δ3 = η3
(D, L1 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL1 DDer(log D).
L2 ⊂ Der(log D) generated by
δ4 = η4 , δ5 = η1 + η2 , δ6 = zη3 + yw η2
(D, L2 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL2 DDer(log D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (II)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L1 ⊂ Der(log D) generated by
δ1 = η1 + η2 , δ2 = zη4 − w 2 η2 δ3 = η3
(D, L1 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL1 DDer(log D).
L2 ⊂ Der(log D) generated by
δ4 = η4 , δ5 = η1 + η2 , δ6 = zη3 + yw η2
(D, L2 ) is not quasi-free (but it satisfies conditions a) and b)). We
have DL2 DDer(log D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (III)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L11 ⊂ Der(log D) generated by
δ1 = η1 − η2 , δ2 = zη4 − w 2 η2 , δ3 = η3
(D, L11 ) is quasi-free (in particular DL11 = DDer(log D)).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (III)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L11 ⊂ Der(log D) generated by
δ1 = η1 − η2 , δ2 = zη4 − w 2 η2 , δ3 = η3
(D, L11 ) is quasi-free (in particular DL11 = DDer(log D)).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (III)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L11 ⊂ Der(log D) generated by
δ1 = η1 − η2 , δ2 = zη4 − w 2 η2 , δ3 = η3
(D, L11 ) is quasi-free (in particular DL11 = DDer(log D)).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Damon’s examples (III)
The Whitney umbrella (D, 0) defined by h = yw 2 − z 2 .
Consider L11 ⊂ Der(log D) generated by
δ1 = η1 − η2 , δ2 = zη4 − w 2 η2 , δ3 = η3
(D, L11 ) is quasi-free (in particular DL11 = DDer(log D)).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Two examples of quasi-free arrangements
(D, 0) defined by x1 · · · xn (x1 + · · · + xn ) = 0 is quasi-free (and it is
not free, if n ≥ 3).
Orlik-Terao’s example:
Y
ai ∈{0,1};
P
(a1 x1 + a2 x2 + a3 x3 + a4 x4 ) = 0
i
ai 6=0
if quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Two examples of quasi-free arrangements
(D, 0) defined by x1 · · · xn (x1 + · · · + xn ) = 0 is quasi-free (and it is
not free, if n ≥ 3).
Orlik-Terao’s example:
Y
ai ∈{0,1};
P
(a1 x1 + a2 x2 + a3 x3 + a4 x4 ) = 0
i
ai 6=0
if quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Two examples of quasi-free arrangements
(D, 0) defined by x1 · · · xn (x1 + · · · + xn ) = 0 is quasi-free (and it is
not free, if n ≥ 3).
Orlik-Terao’s example:
Y
ai ∈{0,1};
P
(a1 x1 + a2 x2 + a3 x3 + a4 x4 ) = 0
i
ai 6=0
if quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
Representation of the quivers
2
(u, v ) .
1
& (w , z)
x
−→
1
The non-reduced equation of the corresponding discriminant is
h = x(uz − vw )2 .
(D, 0) defined by (h = 0) is quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
Representation of the quivers
2
(u, v ) .
1
& (w , z)
x
−→
1
The non-reduced equation of the corresponding discriminant is
h = x(uz − vw )2 .
(D, 0) defined by (h = 0) is quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
Representation of the quivers
2
(u, v ) .
1
& (w , z)
x
−→
1
The non-reduced equation of the corresponding discriminant is
h = x(uz − vw )2 .
(D, 0) defined by (h = 0) is quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
Representation of the quivers
2
(u, v ) .
1
& (w , z)
x
−→
1
The non-reduced equation of the corresponding discriminant is
h = x(uz − vw )2 .
(D, 0) defined by (h = 0) is quasi-free (and it is not free).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
h = x(uz − vw )2
L ⊂ Der(log D) the Lie algebra
action.

 
δ1
x 0
 δ2   0 u

 
 δ3  =  0 v

 
 δ4   0 0
δ5
0 0
associated with the infinitesimal
0 0
0 w
0 z
v 0
u 0
0
0
0
z
w






∂x
∂u
∂v
∂w
∂z






Let A be the coefficient matrix (Saito matrix); det(A) = h.
(D, L) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
h = x(uz − vw )2
L ⊂ Der(log D) the Lie algebra
action.

 
δ1
x 0
 δ2   0 u

 
 δ3  =  0 v

 
 δ4   0 0
δ5
0 0
associated with the infinitesimal
0 0
0 w
0 z
v 0
u 0
0
0
0
z
w






∂x
∂u
∂v
∂w
∂z






Let A be the coefficient matrix (Saito matrix); det(A) = h.
(D, L) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
h = x(uz − vw )2
L ⊂ Der(log D) the Lie algebra
action.

 
δ1
x 0
 δ2   0 u

 
 δ3  =  0 v

 
 δ4   0 0
δ5
0 0
associated with the infinitesimal
0 0
0 w
0 z
v 0
u 0
0
0
0
z
w






∂x
∂u
∂v
∂w
∂z






Let A be the coefficient matrix (Saito matrix); det(A) = h.
(D, L) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
h = x(uz − vw )2
L ⊂ Der(log D) the Lie algebra
action.

 
δ1
x 0
 δ2   0 u

 
 δ3  =  0 v

 
 δ4   0 0
δ5
0 0
associated with the infinitesimal
0 0
0 w
0 z
v 0
u 0
0
0
0
z
w






∂x
∂u
∂v
∂w
∂z






Let A be the coefficient matrix (Saito matrix); det(A) = h.
(D, L) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants
h = x(uz − vw )2
L ⊂ Der(log D) the Lie algebra
action.

 
δ1
x 0
 δ2   0 u

 
 δ3  =  0 v

 
 δ4   0 0
δ5
0 0
associated with the infinitesimal
0 0
0 w
0 z
v 0
u 0
0
0
0
z
w






∂x
∂u
∂v
∂w
∂z






Let A be the coefficient matrix (Saito matrix); det(A) = h.
(D, L) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free isolated singularity
f = uz − vw . (D, 0) ⊂ (C4 , 0) defined by f = 0.
(D, 0) is not free.
L generated by {δj }5j=2 .
 
u
δ2
 δ3   v
 

 δ4  =  0
δ5
0

0 w
0 z
v 0
u 0

0
∂u


0   ∂v
z   ∂w
∂z
w




The determinant of the coefficient matrix is f 2 . (D, L) is
quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free isolated singularity
f = uz − vw . (D, 0) ⊂ (C4 , 0) defined by f = 0.
(D, 0) is not free.
L generated by {δj }5j=2 .
 
u
δ2
 δ3   v
 

 δ4  =  0
δ5
0

0 w
0 z
v 0
u 0

0
∂u


0   ∂v
z   ∂w
∂z
w




The determinant of the coefficient matrix is f 2 . (D, L) is
quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free isolated singularity
f = uz − vw . (D, 0) ⊂ (C4 , 0) defined by f = 0.
(D, 0) is not free.
L generated by {δj }5j=2 .
 
u
δ2
 δ3   v
 

 δ4  =  0
δ5
0

0 w
0 z
v 0
u 0

0
∂u


0   ∂v
z   ∂w
w
∂z




The determinant of the coefficient matrix is f 2 . (D, L) is
quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free isolated singularity
f = uz − vw . (D, 0) ⊂ (C4 , 0) defined by f = 0.
(D, 0) is not free.
L generated by {δj }5j=2 .
 
u
δ2
 δ3   v
 

 δ4  =  0
δ5
0

0 w
0 z
v 0
u 0

0
∂u


0   ∂v
z   ∂w
w
∂z




The determinant of the coefficient matrix is f 2 . (D, L) is
quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free isolated singularity
f = uz − vw . (D, 0) ⊂ (C4 , 0) defined by f = 0.
(D, 0) is not free.
L generated by {δj }5j=2 .
 
u
δ2
 δ3   v
 

 δ4  =  0
δ5
0

0 w
0 z
v 0
u 0

0
∂u


0   ∂v
z   ∂w
w
∂z




The determinant of the coefficient matrix is f 2 . (D, L) is
quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants (II)
Representation of the quiver
1
↓
1→ 3 ←1
↓
1
The arrows are represented by
the array
x11
M = x21
x31
the columns (M1 , M2 , M3 , M4 ) of
x12 x13 x14
x22 x23 x24
x32 x33 x34
Discriminant
h = det(M1 , M2 , M3 )2 · hM1 , M4 i · hM2 , M4 i · hM3 , M4 i.
(D, 0) ⊂ (C12 , 0) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants (II)
Representation of the quiver
1
↓
1→ 3 ←1
↓
1
The arrows are represented by
the array
x11
M = x21
x31
the columns (M1 , M2 , M3 , M4 ) of
x12 x13 x14
x22 x23 x24
x32 x33 x34
Discriminant
h = det(M1 , M2 , M3 )2 · hM1 , M4 i · hM2 , M4 i · hM3 , M4 i.
(D, 0) ⊂ (C12 , 0) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants (II)
Representation of the quiver
1
↓
1→ 3 ←1
↓
1
The arrows are represented by
the array
x11
M = x21
x31
the columns (M1 , M2 , M3 , M4 ) of
x12 x13 x14
x22 x23 x24
x32 x33 x34
Discriminant
h = det(M1 , M2 , M3 )2 · hM1 , M4 i · hM2 , M4 i · hM3 , M4 i.
(D, 0) ⊂ (C12 , 0) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants (II)
Representation of the quiver
1
↓
1→ 3 ←1
↓
1
The arrows are represented by
the array
x11
M = x21
x31
the columns (M1 , M2 , M3 , M4 ) of
x12 x13 x14
x22 x23 x24
x32 x33 x34
Discriminant
h = det(M1 , M2 , M3 )2 · hM1 , M4 i · hM2 , M4 i · hM3 , M4 i.
(D, 0) ⊂ (C12 , 0) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Quasi-free (non-reduced) discriminants (II)
Representation of the quiver
1
↓
1→ 3 ←1
↓
1
The arrows are represented by
the array
x11
M = x21
x31
the columns (M1 , M2 , M3 , M4 ) of
x12 x13 x14
x22 x23 x24
x32 x33 x34
Discriminant
h = det(M1 , M2 , M3 )2 · hM1 , M4 i · hM2 , M4 i · hM3 , M4 i.
(D, 0) ⊂ (C12 , 0) is quasi-free.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic Comparison Problem (LCP)
LCP: To describe the class of divisors/hypersurfaces D ⊂ X = Cn
such that the inclusion
iD : (Ω• (log D), d) → (Ω• [∗D], d)
is a quasi-isomorphism (d exterior derivative).
We say that D satisfies the Logarithmic Comparison Property (or
we simply write LCP(D) holds) if iD is a quasi-isomorphism.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic Comparison Problem (LCP)
LCP: To describe the class of divisors/hypersurfaces D ⊂ X = Cn
such that the inclusion
iD : (Ω• (log D), d) → (Ω• [∗D], d)
is a quasi-isomorphism (d exterior derivative).
We say that D satisfies the Logarithmic Comparison Property (or
we simply write LCP(D) holds) if iD is a quasi-isomorphism.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic Comparison Problem (LCP)
LCP: To describe the class of divisors/hypersurfaces D ⊂ X = Cn
such that the inclusion
iD : (Ω• (log D), d) → (Ω• [∗D], d)
is a quasi-isomorphism (d exterior derivative).
We say that D satisfies the Logarithmic Comparison Property (or
we simply write LCP(D) holds) if iD is a quasi-isomorphism.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D-modules
D = DX (sheaf of) linear differential operators (LDO) with
holomorphic coefficients.
Any δ ∈ Der(log D) is a LDO of order 1.
F.J. Calderón (1997) associates a (left, coherent) D–module with
any divisor/hypersurface D ⊂ X :
Mlog D :=
D
DDer(log D)
Example: D ≡ (x 2 − y 3 = 0) ⊂ C2 .
Mlog D =
D
D(3x∂x +2y ∂y , 2x∂y +3y 2 ∂x )
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D-modules
D = DX (sheaf of) linear differential operators (LDO) with
holomorphic coefficients.
Any δ ∈ Der(log D) is a LDO of order 1.
F.J. Calderón (1997) associates a (left, coherent) D–module with
any divisor/hypersurface D ⊂ X :
Mlog D :=
D
DDer(log D)
Example: D ≡ (x 2 − y 3 = 0) ⊂ C2 .
Mlog D =
D
D(3x∂x +2y ∂y , 2x∂y +3y 2 ∂x )
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D-modules
D = DX (sheaf of) linear differential operators (LDO) with
holomorphic coefficients.
Any δ ∈ Der(log D) is a LDO of order 1.
F.J. Calderón (1997) associates a (left, coherent) D–module with
any divisor/hypersurface D ⊂ X :
Mlog D :=
D
DDer(log D)
Example: D ≡ (x 2 − y 3 = 0) ⊂ C2 .
Mlog D =
D
D(3x∂x +2y ∂y , 2x∂y +3y 2 ∂x )
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D-modules
D = DX (sheaf of) linear differential operators (LDO) with
holomorphic coefficients.
Any δ ∈ Der(log D) is a LDO of order 1.
F.J. Calderón (1997) associates a (left, coherent) D–module with
any divisor/hypersurface D ⊂ X :
Mlog D :=
D
DDer(log D)
Example: D ≡ (x 2 − y 3 = 0) ⊂ C2 .
Mlog D =
D
D(3x∂x +2y ∂y , 2x∂y +3y 2 ∂x )
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D-modules
D = DX (sheaf of) linear differential operators (LDO) with
holomorphic coefficients.
Any δ ∈ Der(log D) is a LDO of order 1.
F.J. Calderón (1997) associates a (left, coherent) D–module with
any divisor/hypersurface D ⊂ X :
Mlog D :=
D
DDer(log D)
Example: D ≡ (x 2 − y 3 = 0) ⊂ C2 .
Mlog D =
D
D(3x∂x +2y ∂y , 2x∂y +3y 2 ∂x )
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D-modules
D = DX (sheaf of) linear differential operators (LDO) with
holomorphic coefficients.
Any δ ∈ Der(log D) is a LDO of order 1.
F.J. Calderón (1997) associates a (left, coherent) D–module with
any divisor/hypersurface D ⊂ X :
Mlog D :=
D
DDer(log D)
Example: D ≡ (x 2 − y 3 = 0) ⊂ C2 .
Mlog D =
D
D(3x∂x +2y ∂y , 2x∂y +3y 2 ∂x )
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(D, 0) ⊂ (Cn , 0); with any L ⊂ Der(log D)
we associate ML :=
D
DL .
Assume L ⊂ Der(log D) is a free O–module of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(D, 0) ⊂ (Cn , 0); with any L ⊂ Der(log D)
we associate ML :=
D
DL .
Assume L ⊂ Der(log D) is a free O–module of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(D, 0) ⊂ (Cn , 0); with any L ⊂ Der(log D)
we associate ML :=
D
DL .
Assume L ⊂ Der(log D) is a free O–module of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(D, 0) ⊂ (Cn , 0); with any L ⊂ Der(log D)
we associate ML :=
D
DL .
Assume L ⊂ Der(log D) is a free O–module of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(D, 0) ⊂ (Cn , 0); with any L ⊂ Der(log D)
we associate ML :=
D
DL .
Assume L ⊂ Der(log D) is a free O–module of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(D, 0) ⊂ (Cn , 0); with any L ⊂ Der(log D)
we associate ML :=
D
DL .
Assume L ⊂ Der(log D) is a free O–module of rank n verifying:
a) L is a Lie subalgebra of Der(log D) (for [−, −]).
P
b) There exists a basis {δ1 , . . . , δn } of L such that if δi = j aij ∂j
then det((aij )) is an (non necessarily reduced) equation of (D, 0).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(F. Calderón, 1997) The Logarithmic Spencer complex
(Sp • (L), −• ) associated with L
is the following complex of vector spaces:
−p
· · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
p
X
−p (P ⊗ δ1 ∧ · · · ∧ δp ) =
(−1)i−1 Pδi ⊗ δ1 ∧ · · · δˇi · · · ∧ δp +
i=1
X
(−1)i+j P ⊗ [δi , δj ] ∧ δ1 ∧ · · · δˇi · · · δˇj · · · ∧ δp
i<j
and −1 (P ⊗ δ) = Pδ.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(F. Calderón, 1997) The Logarithmic Spencer complex
(Sp • (L), −• ) associated with L
is the following complex of vector spaces:
−p
· · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
p
X
−p (P ⊗ δ1 ∧ · · · ∧ δp ) =
(−1)i−1 Pδi ⊗ δ1 ∧ · · · δˇi · · · ∧ δp +
i=1
X
(−1)i+j P ⊗ [δi , δj ] ∧ δ1 ∧ · · · δˇi · · · δˇj · · · ∧ δp
i<j
and −1 (P ⊗ δ) = Pδ.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(F. Calderón, 1997) The Logarithmic Spencer complex
(Sp • (L), −• ) associated with L
is the following complex of vector spaces:
−p
· · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
−p (P ⊗ δ1 ∧ · · · ∧ δp ) =
p
X
(−1)i−1 Pδi ⊗ δ1 ∧ · · · δˇi · · · ∧ δp +
i=1
X
(−1)i+j P ⊗ [δi , δj ] ∧ δ1 ∧ · · · δˇi · · · δˇj · · · ∧ δp
i<j
and −1 (P ⊗ δ) = Pδ.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(F. Calderón, 1997) The Logarithmic Spencer complex
(Sp • (L), −• ) associated with L
is the following complex of vector spaces:
−p
· · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
p
X
−p (P ⊗ δ1 ∧ · · · ∧ δp ) =
(−1)i−1 Pδi ⊗ δ1 ∧ · · · δˇi · · · ∧ δp +
i=1
X
(−1)i+j P ⊗ [δi , δj ] ∧ δ1 ∧ · · · δˇi · · · δˇj · · · ∧ δp
i<j
and −1 (P ⊗ δ) = Pδ.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(F. Calderón, 1997) The Logarithmic Spencer complex
(Sp • (L), −• ) associated with L
is the following complex of vector spaces:
−p
· · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
p
X
−p (P ⊗ δ1 ∧ · · · ∧ δp ) =
(−1)i−1 Pδi ⊗ δ1 ∧ · · · δˇi · · · ∧ δp +
i=1
X
(−1)i+j P ⊗ [δi , δj ] ∧ δ1 ∧ · · · δˇi · · · δˇj · · · ∧ δp
i<j
and −1 (P ⊗ δ) = Pδ.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(F. Calderón, 1997) The Logarithmic Spencer complex
(Sp • (L), −• ) associated with L
is the following complex of vector spaces:
−p
· · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
p
X
−p (P ⊗ δ1 ∧ · · · ∧ δp ) =
(−1)i−1 Pδi ⊗ δ1 ∧ · · · δˇi · · · ∧ δp +
i=1
X
(−1)i+j P ⊗ [δi , δj ] ∧ δ1 ∧ · · · δˇi · · · δˇj · · · ∧ δp
i<j
and −1 (P ⊗ δ) = Pδ.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Definition
If (D, L) satisfies conditions a) and b) as before, we say that the
pair (D, L) is of Spencer type if
D
1) The logarithmic D–module ML = DL
is holonomic and
2) The augmented Logarithmic Spencer complex
0
Sp• (L) →
ML → 0
is exact (here 0 (P) = P + DL).
If D ⊂ Cn is free we simply say that D is a Spencer divisor if
(D, Der(log D)) is of Spencer type.
Theorem.- (F.J. Calderón, L, Narváez; 2004) If D ⊂ X is a free
divisor and LCT (D) holds then D is a Spencer divisor.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Definition
If (D, L) satisfies conditions a) and b) as before, we say that the
pair (D, L) is of Spencer type if
D
1) The logarithmic D–module ML = DL
is holonomic and
2) The augmented Logarithmic Spencer complex
0
Sp• (L) →
ML → 0
is exact (here 0 (P) = P + DL).
If D ⊂ Cn is free we simply say that D is a Spencer divisor if
(D, Der(log D)) is of Spencer type.
Theorem.- (F.J. Calderón, L, Narváez; 2004) If D ⊂ X is a free
divisor and LCT (D) holds then D is a Spencer divisor.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Definition
If (D, L) satisfies conditions a) and b) as before, we say that the
pair (D, L) is of Spencer type if
D
1) The logarithmic D–module ML = DL
is holonomic and
2) The augmented Logarithmic Spencer complex
0
Sp• (L) →
ML → 0
is exact (here 0 (P) = P + DL).
If D ⊂ Cn is free we simply say that D is a Spencer divisor if
(D, Der(log D)) is of Spencer type.
Theorem.- (F.J. Calderón, L, Narváez; 2004) If D ⊂ X is a free
divisor and LCT (D) holds then D is a Spencer divisor.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Holonomic
A (coherent) D–module M (on Cn ) is holonomic if either M = 0
or its characteristic variety Ch(M) ⊂ T ∗ Cn has dimension n.
Ch(Mlog D ) is defined by the set
{σ(P)(x, ξ) = 0 | P ∈ DDer(log D)}
In general it is not enough to consider
{σ(δ)(x, ξ) = 0 | δ ∈ Der(log D)}.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Holonomic
A (coherent) D–module M (on Cn ) is holonomic if either M = 0
or its characteristic variety Ch(M) ⊂ T ∗ Cn has dimension n.
Ch(Mlog D ) is defined by the set
{σ(P)(x, ξ) = 0 | P ∈ DDer(log D)}
In general it is not enough to consider
{σ(δ)(x, ξ) = 0 | δ ∈ Der(log D)}.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Holonomic
A (coherent) D–module M (on Cn ) is holonomic if either M = 0
or its characteristic variety Ch(M) ⊂ T ∗ Cn has dimension n.
Ch(Mlog D ) is defined by the set
{σ(P)(x, ξ) = 0 | P ∈ DDer(log D)}
In general it is not enough to consider
{σ(δ)(x, ξ) = 0 | δ ∈ Der(log D)}.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Holonomic
A (coherent) D–module M (on Cn ) is holonomic if either M = 0
or its characteristic variety Ch(M) ⊂ T ∗ Cn has dimension n.
Ch(Mlog D ) is defined by the set
{σ(P)(x, ξ) = 0 | P ∈ DDer(log D)}
In general it is not enough to consider
{σ(δ)(x, ξ) = 0 | δ ∈ Der(log D)}.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
All previous examples of quasi-free pairs (D, L) are of Spencer
type.
• D ≡ (xz + y )(x 4 + y 5 + xy 4 ) = 0 is free (and hence
quasi-free) but it is not of Spencer type (Calderón-Narváez).
In fact Mlog D is not holonomic. The dimension of its
characteristic variety is 4.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
All previous examples of quasi-free pairs (D, L) are of Spencer
type.
• D ≡ (xz + y )(x 4 + y 5 + xy 4 ) = 0 is free (and hence
quasi-free) but it is not of Spencer type (Calderón-Narváez).
In fact Mlog D is not holonomic. The dimension of its
characteristic variety is 4.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
All previous examples of quasi-free pairs (D, L) are of Spencer
type.
• D ≡ (xz + y )(x 4 + y 5 + xy 4 ) = 0 is free (and hence
quasi-free) but it is not of Spencer type (Calderón-Narváez).
In fact Mlog D is not holonomic. The dimension of its
characteristic variety is 4.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
All previous examples of quasi-free pairs (D, L) are of Spencer
type.
• D ≡ (xz + y )(x 4 + y 5 + xy 4 ) = 0 is free (and hence
quasi-free) but it is not of Spencer type (Calderón-Narváez).
In fact Mlog D is not holonomic. The dimension of its
characteristic variety is 4.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Assume f = 0 is a local equation of (D, 0) and δ is a logarithmic
derivation.
Then
1
δ(f )
=0
δ+
f
f
g
Der(log
f ) := {δ + δ(ff ) | δ ∈ Der(log D)}
AnnD f1 the annihilating ideal (in the ring D) of f1 .
(1)
g
DDer(log
f ) = AnnD
1
f
where the later is the left ideal in D generated by all the operators
P ∈ D annihilating f1 and with order less than or equal to 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Assume f = 0 is a local equation of (D, 0) and δ is a logarithmic
derivation.
Then
δ(f )
1
δ+
=0
f
f
g
Der(log
f ) := {δ + δ(ff ) | δ ∈ Der(log D)}
AnnD f1 the annihilating ideal (in the ring D) of f1 .
(1)
g
DDer(log
f ) = AnnD
1
f
where the later is the left ideal in D generated by all the operators
P ∈ D annihilating f1 and with order less than or equal to 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Assume f = 0 is a local equation of (D, 0) and δ is a logarithmic
derivation.
Then
δ(f )
1
δ+
=0
f
f
g
Der(log
f ) := {δ + δ(ff ) | δ ∈ Der(log D)}
AnnD f1 the annihilating ideal (in the ring D) of f1 .
(1)
g
DDer(log
f ) = AnnD
1
f
where the later is the left ideal in D generated by all the operators
P ∈ D annihilating f1 and with order less than or equal to 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Assume f = 0 is a local equation of (D, 0) and δ is a logarithmic
derivation.
Then
δ(f )
1
δ+
=0
f
f
g
Der(log
f ) := {δ + δ(ff ) | δ ∈ Der(log D)}
AnnD f1 the annihilating ideal (in the ring D) of f1 .
(1)
g
DDer(log
f ) = AnnD
1
f
where the later is the left ideal in D generated by all the operators
P ∈ D annihilating f1 and with order less than or equal to 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Assume f = 0 is a local equation of (D, 0) and δ is a logarithmic
derivation.
Then
δ(f )
1
δ+
=0
f
f
g
Der(log
f ) := {δ + δ(ff ) | δ ∈ Der(log D)}
AnnD f1 the annihilating ideal (in the ring D) of f1 .
(1)
g
DDer(log
f ) = AnnD
1
f
where the later is the left ideal in D generated by all the operators
P ∈ D annihilating f1 and with order less than or equal to 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
Assume f = 0 is a local equation of (D, 0) and δ is a logarithmic
derivation.
Then
δ(f )
1
δ+
=0
f
f
g
Der(log
f ) := {δ + δ(ff ) | δ ∈ Der(log D)}
AnnD f1 the annihilating ideal (in the ring D) of f1 .
(1)
g
DDer(log
f ) = AnnD
1
f
where the later is the left ideal in D generated by all the operators
P ∈ D annihilating f1 and with order less than or equal to 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(1)
g
So we have DDer(log
f ) = AnnD
1
f
⊂ AnnD
1
f
(J.M. Ucha; 1999) considers a new logarithmic D–module
flog D ) :=
(M
D
.
g
DDer(log
f)
flog D related?
How are Mlog D and M
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(1)
g
So we have DDer(log
f ) = AnnD
1
f
⊂ AnnD
1
f
(J.M. Ucha; 1999) considers a new logarithmic D–module
flog D ) :=
(M
D
.
g
DDer(log
f)
flog D related?
How are Mlog D and M
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(1)
g
So we have DDer(log
f ) = AnnD
1
f
⊂ AnnD
1
f
(J.M. Ucha; 1999) considers a new logarithmic D–module
flog D ) :=
(M
D
.
g
DDer(log
f)
flog D related?
How are Mlog D and M
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic D–modules
(1)
g
So we have DDer(log
f ) = AnnD
1
f
⊂ AnnD
1
f
(J.M. Ucha; 1999) considers a new logarithmic D–module
flog D ) :=
(M
D
.
g
DDer(log
f)
flog D related?
How are Mlog D and M
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic duality
Theorem
[J.M. Ucha, F.J.C.; 2002] If D ⊂ Cn is free + Spencer then
∗
flog D .
Mlog D ' M
∨
The dual (M)∗ = (RHomD (M, D)) is considered in the sense of
∨
D-module theory. In particular (−) denotes the left D–module structure
associated with a right one.
If M is holonomic then (M)∗ = ExtD (M, D)∨
Proof of the theorem (sketch).- Use Sp • (Der(log D)) (as a
locally free resolution of Mlog D ) to explicitly compute (Mlog D )∗ .
(The duality theorem has been generalized by F.J. Calderón and L.
Narváez).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic duality
Theorem
[J.M. Ucha, F.J.C.; 2002] If D ⊂ Cn is free + Spencer then
∗
flog D .
Mlog D ' M
∨
The dual (M)∗ = (RHomD (M, D)) is considered in the sense of
∨
D-module theory. In particular (−) denotes the left D–module structure
associated with a right one.
If M is holonomic then (M)∗ = ExtD (M, D)∨
Proof of the theorem (sketch).- Use Sp • (Der(log D)) (as a
locally free resolution of Mlog D ) to explicitly compute (Mlog D )∗ .
(The duality theorem has been generalized by F.J. Calderón and L.
Narváez).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic duality
Theorem
[J.M. Ucha, F.J.C.; 2002] If D ⊂ Cn is free + Spencer then
∗
flog D .
Mlog D ' M
∨
The dual (M)∗ = (RHomD (M, D)) is considered in the sense of
∨
D-module theory. In particular (−) denotes the left D–module structure
associated with a right one.
If M is holonomic then (M)∗ = ExtD (M, D)∨
Proof of the theorem (sketch).- Use Sp • (Der(log D)) (as a
locally free resolution of Mlog D ) to explicitly compute (Mlog D )∗ .
(The duality theorem has been generalized by F.J. Calderón and L.
Narváez).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic duality
Theorem
[J.M. Ucha, F.J.C.; 2002] If D ⊂ Cn is free + Spencer then
∗
flog D .
Mlog D ' M
∨
The dual (M)∗ = (RHomD (M, D)) is considered in the sense of
∨
D-module theory. In particular (−) denotes the left D–module structure
associated with a right one.
If M is holonomic then (M)∗ = ExtD (M, D)∨
Proof of the theorem (sketch).- Use Sp • (Der(log D)) (as a
locally free resolution of Mlog D ) to explicitly compute (Mlog D )∗ .
(The duality theorem has been generalized by F.J. Calderón and L.
Narváez).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Logarithmic duality
Theorem
[J.M. Ucha, F.J.C.; 2002] If D ⊂ Cn is free + Spencer then
∗
flog D .
Mlog D ' M
∨
The dual (M)∗ = (RHomD (M, D)) is considered in the sense of
∨
D-module theory. In particular (−) denotes the left D–module structure
associated with a right one.
If M is holonomic then (M)∗ = ExtD (M, D)∨
Proof of the theorem (sketch).- Use Sp • (Der(log D)) (as a
locally free resolution of Mlog D ) to explicitly compute (Mlog D )∗ .
(The duality theorem has been generalized by F.J. Calderón and L.
Narváez).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Theorem.- (J.M. Ucha, F.J.C.; 2004)
Assume (D, 0) is a (germ of) free Spencer divisor and let f = 0 be
a local equation. Then (D, 0) satisfies LCP if and only if AnnD ( f1 )
is generated by differential operators of order 1.
(1)
LCP(D) holds ⇔ AnnD
Francisco-Jesús Castro-Jiménez (U. Seville)
1
f
=AnnD
1
f
.
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Theorem.- (J.M. Ucha, F.J.C.; 2004)
Assume (D, 0) is a (germ of) free Spencer divisor and let f = 0 be
a local equation. Then (D, 0) satisfies LCP if and only if AnnD ( f1 )
is generated by differential operators of order 1.
(1)
LCP(D) holds ⇔ AnnD
Francisco-Jesús Castro-Jiménez (U. Seville)
1
f
=AnnD
1
f
.
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Theorem.- (J.M. Ucha, F.J.C.; 2004)
Assume (D, 0) is a (germ of) free Spencer divisor and let f = 0 be
a local equation. Then (D, 0) satisfies LCP if and only if AnnD ( f1 )
is generated by differential operators of order 1.
(1)
LCP(D) holds ⇔ AnnD
Francisco-Jesús Castro-Jiménez (U. Seville)
1
f
=AnnD
1
f
.
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Two main ingredients in its proof (for any free Spencer divisor
(D, 0)):
(a) (F.J. Calderón; 1997. J.M. Ucha, F.J.C.; 2002):
The solution complex Sol(Mlog D ) is quasi-isomorphic to
Ω• (log D).
(b) (J.M. Ucha, F.J.C.; 2002)
flog D .
(Mlog D )∗ ' M
Conjecture.- (T. Torrelli; 2004)
A germ (D, 0) satisfies LCP if and only if AnnD ( f1 ) is generated by
differential operators of order 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Two main ingredients in its proof (for any free Spencer divisor
(D, 0)):
(a) (F.J. Calderón; 1997. J.M. Ucha, F.J.C.; 2002):
The solution complex Sol(Mlog D ) is quasi-isomorphic to
Ω• (log D).
(b) (J.M. Ucha, F.J.C.; 2002)
flog D .
(Mlog D )∗ ' M
Conjecture.- (T. Torrelli; 2004)
A germ (D, 0) satisfies LCP if and only if AnnD ( f1 ) is generated by
differential operators of order 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Two main ingredients in its proof (for any free Spencer divisor
(D, 0)):
(a) (F.J. Calderón; 1997. J.M. Ucha, F.J.C.; 2002):
The solution complex Sol(Mlog D ) is quasi-isomorphic to
Ω• (log D).
(b) (J.M. Ucha, F.J.C.; 2002)
flog D .
(Mlog D )∗ ' M
Conjecture.- (T. Torrelli; 2004)
A germ (D, 0) satisfies LCP if and only if AnnD ( f1 ) is generated by
differential operators of order 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Two main ingredients in its proof (for any free Spencer divisor
(D, 0)):
(a) (F.J. Calderón; 1997. J.M. Ucha, F.J.C.; 2002):
The solution complex Sol(Mlog D ) is quasi-isomorphic to
Ω• (log D).
(b) (J.M. Ucha, F.J.C.; 2002)
flog D .
(Mlog D )∗ ' M
Conjecture.- (T. Torrelli; 2004)
A germ (D, 0) satisfies LCP if and only if AnnD ( f1 ) is generated by
differential operators of order 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
LCP and annihilating ideals
Two main ingredients in its proof (for any free Spencer divisor
(D, 0)):
(a) (F.J. Calderón; 1997. J.M. Ucha, F.J.C.; 2002):
The solution complex Sol(Mlog D ) is quasi-isomorphic to
Ω• (log D).
(b) (J.M. Ucha, F.J.C.; 2002)
flog D .
(Mlog D )∗ ' M
Conjecture.- (T. Torrelli; 2004)
A germ (D, 0) satisfies LCP if and only if AnnD ( f1 ) is generated by
differential operators of order 1.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
Can we extend previous theorem to quasi-free Spencer pairs
(D, L)?
∗
e
Theorem (Duality) [Ucha-F.J.C.J. 2004]: ML ' ML
Notations: Assume f = 0 is a reduced equation of (D, 0) and f α if
the determinant of the coefficient matrix of a basis of Q
L (here
f = f1 · · · fr is an irreducible decomposition and f α = i fi αi with
α = (α1 , . . . , αr ) ∈ Nr ).
Le := {δ +
e
ML
:=
δ(f α )
fα
| δ ∈ L}.
D
e.
DL
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
Can we extend previous theorem to quasi-free Spencer pairs
(D, L)?
∗
e
Theorem (Duality) [Ucha-F.J.C.J. 2004]: ML ' ML
Notations: Assume f = 0 is a reduced equation of (D, 0) and f α if
the determinant of the coefficient matrix of a basis of Q
L (here
f = f1 · · · fr is an irreducible decomposition and f α = i fi αi with
α = (α1 , . . . , αr ) ∈ Nr ).
Le := {δ +
e
ML
:=
δ(f α )
fα
| δ ∈ L}.
D
e.
DL
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
Can we extend previous theorem to quasi-free Spencer pairs
(D, L)?
∗
e
Theorem (Duality) [Ucha-F.J.C.J. 2004]: ML ' ML
Notations: Assume f = 0 is a reduced equation of (D, 0) and f α if
the determinant of the coefficient matrix of a basis of Q
L (here
f = f1 · · · fr is an irreducible decomposition and f α = i fi αi with
α = (α1 , . . . , αr ) ∈ Nr ).
Le := {δ +
e
ML
:=
δ(f α )
fα
| δ ∈ L}.
D
e.
DL
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
Can we extend previous theorem to quasi-free Spencer pairs
(D, L)?
∗
e
Theorem (Duality) [Ucha-F.J.C.J. 2004]: ML ' ML
Notations: Assume f = 0 is a reduced equation of (D, 0) and f α if
the determinant of the coefficient matrix of a basis of Q
L (here
f = f1 · · · fr is an irreducible decomposition and f α = i fi αi with
α = (α1 , . . . , αr ) ∈ Nr ).
Le := {δ +
e
ML
:=
δ(f α )
fα
| δ ∈ L}.
D
e.
DL
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
Can we extend previous theorem to quasi-free Spencer pairs
(D, L)?
∗
e
Theorem (Duality) [Ucha-F.J.C.J. 2004]: ML ' ML
Notations: Assume f = 0 is a reduced equation of (D, 0) and f α if
the determinant of the coefficient matrix of a basis of Q
L (here
f = f1 · · · fr is an irreducible decomposition and f α = i fi αi with
α = (α1 , . . . , αr ) ∈ Nr ).
Le := {δ +
e
ML
:=
δ(f α )
fα
| δ ∈ L}.
D
e.
DL
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
Can we extend previous theorem to quasi-free Spencer pairs
(D, L)?
∗
e
Theorem (Duality) [Ucha-F.J.C.J. 2004]: ML ' ML
Notations: Assume f = 0 is a reduced equation of (D, 0) and f α if
the determinant of the coefficient matrix of a basis of Q
L (here
f = f1 · · · fr is an irreducible decomposition and f α = i fi αi with
α = (α1 , . . . , αr ) ∈ Nr ).
Le := {δ +
e
ML
:=
δ(f α )
fα
| δ ∈ L}.
D
e.
DL
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
The proof of this duality theorem uses the locally free resolution
Sp• (L) of M L to explicitly compute its dual (M L )∗ and then
check the isomorphism
∗
e
ML ' ML
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
The proof of this duality theorem uses the locally free resolution
Sp• (L) of M L to explicitly compute its dual (M L )∗ and then
check the isomorphism
∗
e
ML ' ML
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
The proof of this duality theorem uses the locally free resolution
Sp• (L) of M L to explicitly compute its dual (M L )∗ and then
check the isomorphism
∗
e
ML ' ML
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some results and future work
−p
Recall Spp (L) = D ⊗O · · · −→ D ⊗ ∧p L −→ D ⊗ ∧p−1 L −→ · · ·
where for p > 1,
Theorem (Solution complex): The solution complex
Sol(ML ) = RHomD (ML , O) is isomorphic to a
complex of meromorphic differential forms Ω• (L) defined as
follows:
Ω0 (L) = O; Ω1 (L) := HomO (L, O)
Ωp (L) := ∧p Ω1 (L).
Ω• (L) with the exterior derivative is a complex of vector spaces.
To do: Compare Ω• (L) with Ω• (∗D).
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some references
• Calderón, F.J. Logarithmic differential operators and logarithmic de Rham
complexes relative to a free divisor. Ann. Sci. École Norm. Sup. (4) 32 (1999),
no. 5, 701–714.
• Calderón, F.J.; Mond, D.; Narváez, L.; Castro, F.J. Logarithmic cohomology
of the complement of a plane curve. Comment. Math. Helv. 77 (2002), no. 1,
24–38.
• Calderón, F. J.; Narváez, L. On the logarithmic comparison theorem for
integrable logarithmic connections. Proc. Lond. Math. Soc. (3) 98 (2009), no.
3, 585–606.
• Castro, F.J.; Narváez, L.; Mond, D. Cohomology of the complement of a free
divisor. Trans. Amer. Math. Soc. 348 (1996), no. 8, 3037–3049.
• Castro Jiménez F. J. and Ucha Enrı́quez J. M. Free Divisors and Duality for
D-Modules. Proc. Steklov Inst. of Math., volume 238, (2002), 97–105.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors
Free and quasi-free divisors.
J. Damon examples
QF arrangements, QF I.S., quivers ...
LCP
Logarithmic D–modules
Some references (continued)
• Castro-Jiménez F.J. and Ucha J.M. Testing the logarithmic comparison
theorem for free divisors, Experiment. Math. 13 (2004) 441–449.
• Castro-Jimenez, F.J.; Ucha, J.M. Quasi-free divisors and duality. C.R. Acad.
Sci. Paris, Ser. I 338 (2004) 461-466.
• Narváez, L. Linearity conditions on the Jacobian ideal and
logarithmic-meromorphic comparison for free divisors. Singularities I, 245–269,
Contemp. Math., 474, Amer. Math. Soc., Providence, RI, 2008.
Francisco-Jesús Castro-Jiménez (U. Seville)
Some examples of quasi-free divisors