be the cyclic group of order n and let ξn ∈ µn

INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY

YI ZHANG

Contents

1. Main result 1

2. Preliminary on commutative Algebra 3

3. Proof of Theorem 1.1 8

References 13

1. Main result

For any integer n ≥ 1, let G_n =< σ > be the cyclic group of order n and let ξ_n ∈ µ_n := {x ∈ C|xⁿ = 1} be a primitive n-th root of unity. Then G_n is isomorphic to the multiplicative group µn byσ 7→ξn for any integer n≥1.

For any non-singular complex varietyY with an invertible sheafLand an effective divisor D =

r

P

j=1

ν_jD_j satisfying Lⁿ = O_Y(D) for a positive integer n, we obtain a cyclic cover π:Z →Y by taking the n-th root out of Das follows : Let

F :=O_Y ⊕L⁻¹⊕L⁻²⊕ · · · ⊕L⁻⁽ⁿ⁻¹⁾

and let s be a section of Lⁿ defining the divisor D. The F becomes an O_Y-algebra generated byL⁻¹ due to the following laws :

• L^−a⊗L^−b =L^−(a+b);

1The author is supported in part by the NSFC Grant NSFC Grant(#11271070) and LNMS of Fudan University

Date: May 2012.

1

• for any positive integer m expressed as m =ln+k,0≤ k ≤n−1, one fixes an O_X-homomorphism L^−m −−→L^−kh7→hs^l.

Then we get a finite and flat morphism p : Spec(F) → Y of degree n ramified at D.The composite morphism π :=p◦Nor of the morphismp and the normalization Nor : Z → Spec(F) is called the cyclic cover over Y obtained by taking the n-th root out of D.

There is a well-known result as follows.

Theorem 1.1 (see [EV],[K85]). Let X be a non-singular complex variety, let D=

r

P

j=1

ν_jD_j be an effective divisor and let L be an invertible sheaf with L^N0 = O_X(D) for a positive integer N₀.

Let π :Z →X be the cyclic cover obtained by taking theN₀-th root out of D and let G be the cyclic group < σ > of orderN0. We have :

1.

(1.1.1) π∗O_Z =

N0−1

M

k=0

L^(k)−1 for L^(k):=L^k⊗ O_X(−[kD N₀]) where [^kD_N

0]denotes the integral part of the Q-divisor ^kD_N

0. So that the cover π is a finite and flat morphism.

2. Z is normal, the singularities of Z are lying over the singularities ofDred. 3. The cyclic group G acts on Z. One can choose a primitive N₀-th root of unit

ξ such that the sheaf L^(k) in 1) is the sheaf of eigenvectors for σ in π_∗O_Z with eigenvalue ξ^k.

4. X is irreducible if L^(k) 6=O_X for k = 1,· · · , N₀−1.In particular this holds true if the common factor of the integers N₀, ν₁,· · · , ν_r has gcd(N₀, ν₁,· · · , ν_r) = 1.

5. Assume that D=

r

P

j=1

ν_jD_j is an effective normal crossing divisor. Define D^(j):={jD

N0

}_red= X

{^iνj_N

0}6=0

D_j 0≤i≤N₀−1,

where {^kD_N

0}:= ^kD_N

0 −[^kD_N

0] is the fractional part of ^kD_N

0. For any p≥0, there is (1.1.2) π∗Ωc^p_Z ∼=

N0−1

M

i=0

Ω^p_X(logD⁽ⁱ⁾)⊗ L⁻⁽ⁱ⁾

where Ωc^p_Z is the reflexive hull of Ω^p_Z. In particular, Ω^p_X = (π∗Ωc^p_Z)^G.

6. Assume that X is a proper scheme and that D =

r

P

j=1

D_j is an effective normal crossing reduced divisor. There is

(1.1.3) (π_∗T_Z)^G =T_X(−logD).

2. Preliminary on commutative Algebra

As begining, we investigate the local structure of cyclic cover.

Let R be both a finitely generated C-algebra and an integral normal ring, let K be the fraction field of R.

Lemma 2.1. Let n ≥1 be an integer. Letf be an arbitrary nonzero element in R.

We have :

a) Any irreducible factor of Tⁿ−f in K[T] with coefficient 1 of the first term is of form Tⁿ⁰ −f⁰ ∈R[T] satisfying n⁰|n and f = (f⁰)^n/n0.

b) Let d|n be the maximal positive integer such that T^d−f has a root K, say g.

Then T^n/d−g is irreducible in K[T] and g ∈R.

c) Assume that f =uf₁^ν1· · ·f_m^νm is an element in R where f_i’s are prime elements inR andν_i’s are positive integers. Letdbe an integer. ThenT^d−f has a root in K if and only if d|gcd(ν1,· · · , νm) and there is a unit u0 ∈R satisfying u^d₀ =u.

Proof. a) Letφ(T) =T^m+am−1T^m−1+am−2T^m−2+· · ·+a₀ ∈K[T] be an irreducible factor of Tⁿ−f. Since R is C-algebra, φ(T) is separable and it has m different roots α₁,· · · , α_m in an algebraic closure K.Because every root of φ(T) is a root of Tⁿ−f, allαi’s are integral overR, so that the coefficients ofφ(T) are integral over R. Since R is integrally closed, we have φ(T)∈R[T].

Let y=α1 be a root of g(T). We have

Tⁿ−f = (T −y)(T −ξy)· · ·(T −ξⁿ⁻¹y) = (T −y₀)(T −y₁)· · ·(T −yn−1) where ξ is a primitive n-th root of unity. Then φ(T) is the minimal polynomial min(K, y) ofy inK[T],we write as

φ(T) =

m

Y

j=1

(T −y_ij) with y_i1 =y.

Thus a₀ = y^mξ^l for some integer l,then f⁰ := y^m ∈R. Hence φ(T)|T^m−y^m in K[T], we must have φ(T) =T^m−f⁰.

For any σ∈µ_n={x∈C|xⁿ= 1}, we defineφ^σ(T) :=

m

Q

j=1

(T −σy_ij). We have that each

φ^σ(T) =

m

Y

j=1

(T −σy_ij) = T^m+σam−1T^m−1+σ²am−2T^m−2+· · ·+σ^ma₀ is also an irreducible factor of Tⁿ −f in K[T] of degree m, so that φ^σ(T) ∈ R[T] is the minimal polynomial min(K, σy) of σy in K[T], and we have either φ(T) = φ^σ(T) or that φ(T) has no common factor with φ^σ(T) in K[T]. We list all different elements in {φ^σ(T)|σ ∈ µ_n} as φ₁(T) = φ(T), φ₂(T),· · · , φ_k(T).

Hence, we have

Tⁿ−f =φ₁(T)φ₂(T)· · ·φ_k(T),

so that n=mk. Therefore, we must have φ(T) = T^m−f⁰ with (f⁰)^n/m =f.

b) It is a consequence of (a).

c) If d|gcd(ν₁,· · · , ν_m) and u^d₀ =u for some unit u₀ ∈R, thenT^d−f ∈R[T] has a root u0f₁^ν1/d· · ·fm^νm/d ∈R.

Suppose that there is g ∈K withg^d =f. The T −g is an irreducible factor of T^d−f inK[T], thusg ∈R by (a). We writeg₀ :=g. Then we have

g₀^d=uf₁^ν1· · ·f_n^νmb^d₀ ∈(f₁).

We then haveg₀ =f₁g₁,and getf₁^dg₁^d=uf₁^ν1· · ·f_n^νm.Sincef₁ is a prime element, it is impossible that d > ν₁.So that ν₁−d≥0 and we have g₁^d=uf₁^ν1−d· · ·f_n^νm. If ν₁−d ≥1,we have g^d₁ =uf₁^ν1−d· · ·f_n^νm ∈ (f₁). We deal with g₁ similarly like g₀, and so on. We finally get thatν₁ =dk for some integer k. By symmetry, we have d|ν_i for otherν_i’s. Then d|gcd(n, ν₁,· · · , ν_m) and ( ^g

f₁^ν1/d···fn^νm/d

)^d=u.Again by (a), we have u₀ := ^g

f₁^ν1/d···fn^νm/d

is a unit in R.

Corollary 2.2. Let Tⁿ−f be a polynomial in R[T]. We have :

a) The Tⁿ−f is irreducible in K[T] if and only if it is irreducible in R[T].

b) If Tⁿ−f is irreducible in R[T] then it is a prime element in R[T].

For a polynomial Tⁿ−f(n ≥ 1) with nonzero f ∈ R, let d|n be the maximal positive integer satisfyingT^d−f has a rootg ∈K,we haveTⁿ−f =Qd

i−1(Tⁿ⁰−ζ_dⁱg) where ζ_d is a primitive d-th root of unit and each Tⁿ⁰ −ζ_dⁱg is a prime element in R[T] for n⁰ := n/d; the quotient ring K[T]/(Tⁿ−f) can be regarded as a finite K-linear space

K[T]/(Tⁿ−f) =

n−1

M

i=0

Ktⁱ

where t := T mod (Tⁿ−f) can be regarded as a root of Tⁿ−f in K, that t is also integral over R and the ring R[T]/(Tⁿ−f) is integral if and only if Tⁿ−f is irreducible, the group G_d =< σ > has a natural action on K[T]/(Tⁿ −f) by σ :T 7→ζ_nT and σ|_K = id_K, the µ_n-invariant part of K[T]/(Tⁿ−f) is K.

Lemma 2.3. Let f be an nonzero element in the domain R. Let n be a positive integer and d|n the maximal positive integer such that T^d−f has a root in K and n⁰ :=n/d. For any ζ ∈µ_d:={x∈C|xⁿ = 1}, define a polynomial

P_ζ = 1

Q

ζ⁰∈µ_d\ζ

(ζg−ζ⁰g)

Tⁿ−f Tⁿ⁰ −ζg where g ∈K is a root of T^d−f.

1. In the quotient ring _(T^K[Tn−f)^] ,theseP_ζ’s are idempotent,i.e. P_ζ·P_ζ⁰ = 0ζ 6=ζ⁰, P_ζ² = P_ζ with P

ζ∈µ_d

P_ζ = 1. Moreover, the cyclic group G_n =< σ > acts transitively on the set {P_ζ}ζ∈µ_d with (P_ζ)^σ =P_ξ−n0

n ζ∀ζ ∈µ_d. 2. Let K_ζ be the field ^K[T]

(Tⁿ⁰−ζg) for each ζ ∈µ_d. Any ξ ∈µ_n defines an isomorphism Kζ

∼=

−−→K_ξn⁰_ζ of fields by sending T toξT. Moreover, there is a naturalGn-action on the algebra Q

ζ∈µ_d

K_ζ.

3. There is a G_n-equivariant isomorphism of K-algebras

(2.3.1) P : Y

ζ∈µ_d

K[T] (Tⁿ⁰ −ζg)

∼=

−−→ K[T]

(Tⁿ−f),(α_ζ)_ζ 7→X

ζ

α_ζP_ζ.

Corollary 2.4. Let Tⁿ−f be a polynomial in R[T] with degree n ≥ 1. Let A = R[T]/(Tⁿ −f) be a ring and S := {a ∈ A|r is not zero divisor} a multiplicative system in A. Then S⁻¹A=K[T]/(Tⁿ−f).

Proof. Let h(T) ∈ R[T] such that it’s image h is not a zero-divisor in A. Consider the K-isomorphism 2.3.1, we get that P⁻¹(h) = (α_ζ)_ζ with α_ζ :=h(T) mod (T − ζg) in K[T]∀ζ ∈ µ_d. Thus that h is not a zero-divisor in R if and only if for each ζ ∈ µ_d there is a η_ζ(T) ∈ K[T] such that η_ζ(T)h(T) = 1 mod (Tⁿ⁰ −ζg) in K[T] We get ηh= 1 in K[T]/(Tⁿ−f), whereη is the projection image of the polynomial η(T) := P

ζ∈µ_dη_ζP_ζ. Thus, we have that any non zero divisor in A is an unit in

K[T]

(Tⁿ−f).Hence,

S⁻¹A=S⁻¹( K[T]

(Tⁿ−f)) = K[T] (Tⁿ−f).

Consider the ring homomorphismR →K[T]/(Tⁿ−f) for a polynomialTⁿ−f ∈ R[T] of degreen ≥1.Let ¯R be the integral closure of R in K[T]/(Tⁿ−f).

Lemma 2.5. Let f be an arbitrary nonzero element in R. If a polynomialTⁿ−f ∈ R[T] with degree n ≥ 1 is irreducible in R[T] then R¯ is the normalization of the domain _(T^R[Tn−f^]) in the field extension _(T^K[Tn−f)^] of K.

Therefore, by Corollary 2.4 the ¯R is also the normalization of _(T^R[Tn−f)^] even though Tⁿ−f is not irreducible sinceR is a normal integral ring.

Lemma 2.6. Let g be a nontrivial prime element in R. Then the principal prime ideal (g) is of height one.

Proof. We claim that any nonzero prime idea p of R has an irreducible element : Suppose the claim is not true. Let a0 be any nonzero element in p. Then a = bc such that both b and c are not unit. Then one of b, c is in p, say a1. The a1 is also irre4ducible and (a₀)((a₁)⊂p.We do same steps for a₁ as dealing with a₀ and so on. Thus we have an strictly increasing sequence of ideals

(a0)((a1)((a2)(· · ·(p, which contradicts that R is a Noetherian ring.

Let p be any nontrivial prime ideal of R such that 0 ( p ⊂ (g). Let a ∈ p be an irreducible element. We have a = gc due to a ∈ (g), but c is a unit since a is

irreducible. Hence p= (g).

For any height one prime idea p of R, the local ring R_p is a DVR, thus we can define div(x) for anyx∈K.

Theorem 2.7. Assume a finitely generated C-algebra R is integral and normal. Let K be the fraction field of R. Let f =uf₁^ν1· · ·f_l^νl(∀ν_i ≥1) be an element in R such that f_i’s are different prime elements in R and u is a unit in R. Let n ≥ 1 be an integer and d|n the maximal positive integer such that T^d−f has a root in K, say g, and denote by n⁰ :=n/d.

1. With respect to the injective ring homomorphismR→ _(T^K[Tn−f)^] ,the integral closure of R is

(2.7.1) R¯=

n−1

M

i=0

tⁱf^−[

ν1i n ]

1 · · ·f_l^−[νlin]R

where t:=T mod (Tⁿ−f).Moreover, the cyclic groupZn=< σ >has a natural action on R¯ by defining σ:t 7→ξ_nt, σ|_R = id_R, and the G_n-invariant part of R¯ is

R¯^Zn =R.

2. Let R¯_ζ be normalization of an integral ring R_ζ := ^R[T]

(Tⁿ⁰−ζg) in ^K[T]

(Tⁿ⁰−ζg) for each ζ ∈µ_d. Then

(2.7.2) R¯ζ =

n⁰−1

M

i=0

tⁱ_ζf^−[

ν1i n ]

1 · · ·f^−[

νli n]

l R

where t_ζ := T mod (Tⁿ−ζg). Moreover,R → R¯_ζ is Galois, with Galois group Zn⁰.

3. The group Zn has a natural action on the product R-algebra Q

ζ∈µd

R¯_ζ. The group permutes the factors of the decomposition, and there is a G_n-equivariant isomor- phism of R-algebras

Y

ζ∈µ_d

R¯_ζ −−→R,¯ (α_ζ)_ζ 7→X

ζ

α_ζP_ζ.

Proof. For anyx∈K, it is easy to verify the following conditions are equivalent : i. xⁿfⁱ ∈R,

ii. ndiv(x) +

l

P

j=1

iνjdiv(fj)≥0(we use div(u) = 0).

iii. div(xf^[

ν1i n ] 1 · · ·f^[

νli n] l ) +

l

P

j=1

(^iν_n^j −[^ν_n¹ⁱ])[f_j]≥0(we use that (f_i)’s are prime ideal of height one and so that div(f_j) is the Weil divisor [f_j] := Zero(f_j)),

iv. div(xf^[

ν1i n ]

1 · · ·f_l^[νlin])≥0(we use that div(xf^[

ν1i n ]

1 · · ·f_l^[νlin]) has integral coefficients and f_i’s are prime elements),

v. xf^[

ν1i n ] 1 · · ·f^[

νli n] l ∈R.

1. Thus there is

{x∈K;xⁿfⁱ ∈R}=f^−[

ν1i n ]

1 · · ·f^−[

νli n]

l R.

We now show

R¯=

n−1

M

i=0

{x∈K;xⁿfⁱ ∈R}tⁱ.

Letα =

n−1

P

i=0

x_itⁱ (x_i ∈K) be an element in _(T^K[Tn−f)^] .We claim that α∈R¯ if and only if x_itⁱ ∈R¯ for every i. Suppose α ∈ R.¯ We have α^ζ =Pn−1

i=0 ζⁱx_itⁱ ∈R, for¯ every ζ ∈µ_n.Let µ_n ={ζ₁, ζ₂, . . . , ζ_n}. The Vandermonde determinant

det









ζ₁⁰ ζ₁¹ · · · ζ₁ⁿ⁻¹ ζ₂⁰ ζ₁¹ · · · ζ₂ⁿ⁻¹

... ... . .. ... ζ_n⁰ ζ_n¹ · · · ζ_nⁿ⁻¹









= Y

1≤i<j≤n

(ζ_j−ζ_i)

is a unit in R, so that each x_itⁱ is a combination of the α^ζ’s with coefficients in R by Cramer’s rule. Therefore x_itⁱ ∈R.¯ The converse is obvious.

Let x ∈ K and 0 ≤ i < n. We claim that xtⁱ ∈ R¯ if and only if xⁿfⁱ ∈ R.

Indeed, we have that xtⁱ ∈R¯ if and only if y= (xtⁱ)ⁿ ∈R.¯ Due to y= (xtⁱ)ⁿ=xⁿ(tⁿ)ⁱ =xⁿfⁱ ∈K,

We have that y ∈R¯ if and only if y ∈K∩R¯ =R since R is integrally closed in K.

2. By the statement (1), the integral closure of R in ^K[T]

(Tⁿ⁰−ζg) is the integrally closed domain ¯R_ζ =

n⁰−1

L

j=0

{x ∈ K;xⁿ⁰(ζg)^j ∈ R}t^j_ζ. Since R is integrally closed in K, xⁿ⁰(ζg)^j ∈R if and only if (xⁿ⁰(ζg)^j)^d∈R. That is xⁿf^j ∈R. Therefore

R¯_ζ =

n⁰−1

M

j=0

{x∈K;xⁿf^j ∈R}t^j_ζ. 3. In the ring _(T^K[Tn−f^]), we have that f(P

ζα_ζP_ζ) = P

ζf(α_ζ)P_ζ for any polynomial f ∈R[T].Hence thatP

ζα_ζP_ζ is integral over R if and only if each α_ζ ∈ ^K[T]

(Tⁿ⁰−ζg)

is integral over _(T^R[T_n0 ^]

−ζg). The product decomposition of Lemma 2.3 induces an isomorphism of R-algebras Q

ζ∈µ_dR¯_ζ −−→^∼= R.¯

3. Proof of Theorem 1.1 Now we investigate the local structure of cyclic cover.

Lemma 3.1. Let U := SpecR be an irreducible complex normal affine variety of dimension m and f = uf₁^ν1· · ·f_e^νe(∀ν_i ≥ 0) an element in R such that f_i’s are prime elements in R and u is a unit in R. Let D=

e

P

i=1

ν_iD_i is zero locus of f, i.e., D_i = Zero(f_i)∀i.

Denote by U[√ⁿ

f] := {(p, y)∈U ×A¹ | yⁿ=f(p)} for an integer n ≥2.

a) Let p₀ be a smooth closed point of U.Then U[√ⁿ

f] is singular at a point (p₀,∗) if and only if p₀ ∈Sing(D). Moreover, U[√ⁿ

f] is smooth if and only ifU is smooth and D is reduced with only one smooth component.

b) Assume that U is smooth. If D=ν₁D₁ and f =uf₁^ν1(ν₁ ≥ 1) then the normal- ization U[√ⁿ

f] of U[√ⁿ

f] has only singularities over Sing(D₁).

Proof. The proof of (a) is directly follows from Jocobi’s criterion : Let (Z₁,· · ·Z_m) be an regular parameter near an open neighborhood V of p₀.Let

F(Z₀, Z₁,· · ·Z_m) =Z₀ⁿ−f(Z₁,· · · , Z_m).

The point q:= (p₀, y₀) in U[√ⁿ

f] is singular if and only if the F satisfies F(q) = 0

∂F

∂Z0

(q) = ∂F

∂Z1

(q) = · · ·= ∂F

∂Zm

(q) = 0.

Thus we prove the (a).

Now R = A(U) is an regular algebra. The problem (b) is now reduced to show that the U[√ⁿ

f] is smooth if theD₁ is a smooth divisor. In case of ν = 0,1 the (b) is obvious by (a.) With lost of generality, we assume that gcd(n, ν) = 1. Let R be the ring corresponding to the U. By Theorem 2.7, We get U[√ⁿ

f] = Spec( ¯R) such that

R¯ =

n−1

M

i=0

tⁱf^−[

ν1i n ]

1 R

where t:=T mod (Tⁿ−uf₁^ν1).

Since gcd(n, ν) = 1, we have an ring-isomorphism _n^Z

Z

×ν1

−−→ _n^Z

Z, and so we can get a∈ {0,· · ·n−1} with aν₁ = 1 +ln for somel ∈Z.Denote by g =t^af^−[

aν1 n ]

1 .We get that

gⁿ=u^af₁ and t=u^−lg^ν1. Thus

n−1

M

i=0

gⁱR ∼= R[W] (Wⁿ−u^af₁)

is an R-subalgebra of ¯R. Moreover, we have a tower of inclusions ofR-algebras R −−^⊂→ R[T]

(Tⁿ−uf^ν1)

−−−−−−→⊂ T7→u^−lg^ν1

n−1

M

i=0

gⁱR−−→^⊂ R.¯

such that the integral domains _(Tn^R[T−uf^]ν1),

n−1

L

i=0

gⁱR and ¯R all have same quotient field.

Since the algebra _(W^R[W]n−u^af1) is regular by (a) and the algebra ¯R is normal, we have to get

R¯∼= R[W] (Wⁿ−u^af₁). Therefore the U[√ⁿ

f] is smooth.

We only need to prove the statements 5) and 6) of Theorem 1.1.

Letd = dim_CX and let ξ be a primitive root of x^N0 −1. Since π is ´etale outside of D, there is

π∗Ωc^p_Z|_X−D ∼= (Ω^p_X ⊗π∗O_Z)|X−D

and the isomorphism 1.1.2 holds over X −D. On the other hand, the π_∗Ωc^p_Z is a reflexive sheaf on X by the following lemma 3.2, thus it is sufficient to check the assertion 1.1.2 only in codimension one on X.

Lemma 3.2 ([H80]). LetF be a coherent sheaf on a normal integral scheme X.The followin two conditions are equivalent:

i. F is reflexive;

ii. F is torsion-free, and is normal,i.e.,for every open set U ⊂X and every closed subsetB ⊂U of codimension2, the restriction mapF(U)→ F(U−Y)bijective.

Without lost of generality, we may assume that the D_red is smooth, moreover there is a local regular parameter (x₁,· · · , x_d) ofX such that

X = Spec(C[x₁,· · · , x_d]), D = (x^a₁).

Then X⁰ = Spec(C[x₁,· · · , x_d, t]/(T^N0 − ux^a₁)) where u is a unit in the domain C[x₁,· · · , x_d] and Z is normalization of X⁰.

Let R :=C[x₁, x₂,· · · , x_d] a normal integral ring. Let A:= _(TN^R[T0−ux^{] a}₁) and B the normalization of A. As Ω¹_B is generated as a B-module, by the image d :B →Ω¹_B. We know as that

∧^pΩ¹_B ={b₀db₁∧db₂· · · ∧db_p | b_i ∈B i= 1,· · · , N₀−1}.

Let t := T mod (T^N0 −ux^a₁). By Proposition 2.7 we have that B =

N0−1

L

i=0

q_iR as an R-module and that the groupG =< σ > has a natural action on ∧^pΩ¹_B, where q_i :=tⁱx^−[

ai N0]

1 0≤i≤N₀−1. Since q_iq_j =q_i :=t^i+jx^−[

a(i+j) N0 ]

1 x^[

a(i+j) N0 ]−[_N^ai

0]−[_N^aj

0]

1 ∀i, j ≥0,

we have that q_iq_j ∈ q_i+jR if i+j < N₀ and that q_iq_j ∈ qi+j−nR if i+j ≥ N₀. Let ξ be the primitive N₀-th root of unit as we choose in the statement 2). We then have an decomposition of ∧^pΩ¹_B =

N0−1

L

i=1

Q_i such that each Q_i is the R-submodule of eigenvectors forσ in∧^pΩ¹_B with eigenvalueξⁱ. We now find out theseQ_i’s. We have that

dt

t = a

N₀ dx₁

x₁ dq_i = {ai

N₀}q_idx₁ x₁

On the other hand, each b ∈ B can be written uniquely as b =

N0−1

P

i=1

qiri with each r_i ∈R and so db=

N0−1

P

i=1

(q_idr_i+r_idq_i). For each 0≤i≤N₀−1, the elements tⁱx^−[

ia N0]

1 dy_e1 ∧ · · · ∧dy_ep, e₁ < e₂ <· · ·< e_p < d fore₁ 6= 1, with





 tⁱx^−[

ia N0] 1 dx1

x1 ∧dxe2 ∧ · · · ∧dyep, 1< e2 <· · ·< ep < d if {_N^ai

0} 6= 0 tⁱx^−[

ia N0]

1 dx₁ ∧dx_e2 ∧ · · · ∧dx_ep, 1< e₂ <· · ·< e_p < d otherwise

becomes anR-basic of the moduleQ_i.In particular,G-invariant submodule of∧^pΩ¹_B has Q0 =∧^pΩ¹_R. Therefore, we get the formula 1.1.2 and Ω^p_X = (π∗Ωc^p_Z)^G.

Now we are going to prove the statement 6). For the normal varieties X and Z, by definition, the tangent sheaf

T_Z :=HomO_Z(Ω¹_Z,O_Z)

is a reflexive sheaf which is composed of derivations fromO_ZtoO_Z,andT_X(−logD⁽¹⁾) is the subsheaf ofTX of consisting of derivations which send the ideal sheafI ofD⁽¹⁾ into itself. Since X is smooth, we get(see [D72]) that

T_X(−logD⁽¹⁾) = Hom_OX(Ω¹_X(logD⁽¹⁾),O_X).

SupposeX is proper and Dis reduced. ThenD=D⁽ⁱ⁾ for all 1≤i≤N0−1 and by 1) Z is also a proper scheme. Thus the dualizing sheafωZ =π^!ωX has a natural O_Z-module(see Proposition 5.68. [KM]). Because π is an affine morphism, we can define an equivalent functor ∼ of categories between the category of quasi-coherent π_∗O_Z-modules and the category of quasi-coherent O_Z-module(see Ex. II. 5.17(e) [Hart]). Since π∗O_Z is locally free, we have

π^!ωX :=HomOX(π∗OZ, ωX)^∼ = (HomOX(π∗OZ,OX)⊗OX ωX)^∼ =π^!OX ⊗OZπ^∗ωX. Thus the relative dualizing sheaf ω_Z/X =π^!O_X.

We claim that

ω_Z/X =π^∗L^(N0−1) =π^∗(L^N0−1).

We can take a local regular parameter (x₁,· · · , x_d) ofX such that X = Spec(C[x₁,· · ·, x_d]), D = (x₁· · ·x_k).

Thus we get that

X⁰ = Spec(C[x₁,· · · , x_d, t]/(T^N0 −x₁· · ·x_k))

and Z is the normalization of X⁰. Let

R :=C[x₁, x₂,· · · , x_d] and A:= R[T]

(T^N0 −x1· · ·xk), and let B be the normalization of A. By Proposition 2.7 we have that

B =

N0−1

M

i=0

p_iR

as an R-module where {p_i :=tⁱ}i=0,···N₀−1 is an R-basis of B. Let η_i ∈B^∗ := Hom_R(B, R)

be dual ofq_i,i.e.,η_i(p_i) = 1, η_i(p_j) = 0∀j 6=i.TheB-module structure of Hom_R(B, R) is clear : b·φ :=φ_b where φ_b ∈Hom_R(B, R)is given by φ_b(b₁) :=φ(bb₁)∀b₁ ∈B.We observe that

p_ip_j =q_i+j if i+j < N₀ and p_ip_j ∈pi+j−nR if i+j ≥N₀. So that

p_i·η_N0−1 =η_N0−1−i

for alli= 0,· · ·N₀−1.As anB-module, the Hom_R(B, R) must be a module of rank one generated by η_N0−1.Hence π^!O_X =π^∗(L^(N0−1)).

Let Sing(D) be singularities ofD and X0 :=X\Sing(D). We know codimension of Sing(D) in X is more than two and π⁻¹(X0) is smooth open sub-scheme of Z.

Using the duality theory for finite morphism(cf. Theorem 5.67 [KM]), onX₀ we get π∗T_Z = π∗HomO_Z(Ω¹_Z,O_Z)

= π∗HomO_Z(ω_Z/X⊗O_ZΩ¹_Z, ω_Z/X)

= π∗HomO_Z(π^∗(L^(N0−1))⊗O_Z Ω¹_Z, π^!O_X)

= HomO_X(L^(N0−1)⊗O_X π∗Ω¹_Z,O_X)

= HomO_X(L^(N0−1)⊗O_X (

N0−1

M

i=0

Ω¹_X(logD⁽ⁱ⁾)⊗ L⁻⁽ⁱ⁾),O_X)

= (L^−(N0−1)⊗_OX T_X)⊕

N0−1

M

i=1

(L^(i−N0+1)⊗_OX T_X(−logD)).

Since both π∗T_Z and (L^−(N0−1) ⊗O_X T_X)⊕

N0−1

L

i=1

(L^(i−N0+1) ⊗O_X T_X(−logD)) are reflexive sheaves on X, there is

π∗T_Z =L^−(N0−1)⊗O_X T_X)⊕

N0−1

M

i=1

(L^(i−N0+1)⊗O_X T_X(−logD)) on X.

Hence (π∗T_Z)^G=T_X(−logD) on X.

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E-mail address: zhangyi math@fudan.edu.cn