Construction Poincaré residue

1 construction

1.1 preliminary definition
1.2 definition of residue
1.3 algorithm computing class

construction
preliminary definition

given setup above, let




a

k


p


(
x
)


{\displaystyle a_{k}^{p}(x)}

space of meromorphic



p


{\displaystyle p}

-forms on





p


n




{\displaystyle \mathbb {p} ^{n}}

have poles of order upto



k


{\displaystyle k}

. notice standard differential



d


{\displaystyle d}

sends

d
:

a

k
−
1


p
−
1


(
x
)
→

a

k


p


(
x
)


{\displaystyle d:a_{k-1}^{p-1}(x)\to a_{k}^{p}(x)}

define

k



k


(
x
)
=




a

k


p


(
x
)


d

a

k
−
1


p
−
1


(
x
)





{\displaystyle {\mathcal {k}}_{k}(x)={\frac {a_{k}^{p}(x)}{da_{k-1}^{p-1}(x)}}}

as rational de-rham cohomology groups.

definition of residue

consider



(
n
−
1
)


{\displaystyle (n-1)}

-cycle



γ
∈

h

n
−
1


(
x
;

c

)


{\displaystyle \gamma \in h_{n-1}(x;\mathbb {c} )}

. if take tube



t
(
γ
)


{\displaystyle t(\gamma )}

around



γ


{\displaystyle \gamma }

(which locally isomorphic



γ
×

s

1




{\displaystyle \gamma \times s^{1}}

) lies within complement of



x


{\displaystyle x}

. since



n


{\displaystyle n}

-cycle, can integrate



ω


{\displaystyle \omega }

, number. if write as

∫

t
(
−
)


ω
:

h

n
−
1


(
x
;

c

)
→

c



{\displaystyle \int _{t(-)}\omega :h_{n-1}(x;\mathbb {c} )\to \mathbb {c} }

then linear transformation on homology classes. poincare duality implies cohomology class

res
⁡
(
ω
)
∈

h

n
−
1


(
x
;

c

)


{\displaystyle \operatorname {res} (\omega )\in h^{n-1}(x;\mathbb {c} )}

which call residue. notice if restrict case



n
=
1


{\displaystyle n=1}

, standard residue complex analysis (although extend our meromorphic



1


{\displaystyle 1}

-form of





p


1




{\displaystyle \mathbb {p} ^{1}}

.

algorithm computing class

there simple recursive method computing residues reduces classical case of



n
=
1


{\displaystyle n=1}

. recall residue of



1


{\displaystyle 1}

-form

res
⁡

(



d
z

z


+
a
)

=



d
z

z




{\displaystyle \operatorname {res} \left({\frac {dz}{z}}+a\right)={\frac {dz}{z}}}

if consider chart containing



x


{\displaystyle x}

vanishing locus of



w


{\displaystyle w}

, can write meromorphic



n


{\displaystyle n}

-form pole on



x


{\displaystyle x}

as

d
w


w

k




∧
ρ


{\displaystyle {\frac {dw}{w^{k}}}\wedge \rho }

then can write out as

1

(
k
−
1
)




(



d
ρ


w

k
−
1




+
d

(


ρ

w

k
−
1




)

)



{\displaystyle {\frac {1}{(k-1)}}\left({\frac {d\rho }{w^{k-1}}}+d\left({\frac {\rho }{w^{k-1}}}\right)\right)}

this shows 2 cohomology classes

[



d
w


w

k




∧
ρ
]

=

[



d
ρ


(
k
−
1
)

w

k
−
1





]



{\displaystyle \left[{\frac {dw}{w^{k}}}\wedge \rho \right]=\left[{\frac {d\rho }{(k-1)w^{k-1}}}\right]}

are equal. have reduced order of pole hence can use recursion pole of order



1


{\displaystyle 1}

, define residue of



ω


{\displaystyle \omega }

as

res
⁡

(
α
∧



d
w

w


+
β
)

=
α


|


x




{\displaystyle \operatorname {res} \left(\alpha \wedge {\frac {dw}{w}}+\beta \right)=\alpha |_{x}}