For any non-singular complex variety Y with an invertible sheaf L and an effective divisor D =

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1 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY YI ZHANG Contents 1. Main resut 1 2. Preiminary on commutative Agebra 3 3. Proof of Theorem References Main resut For any integer n 1, et G n =< σ > be the cycic group of order n and et ξ n µ n := {x C x n = 1} be a primitive n-th root of unity. Then G n is isomorphic to the mutipicative group µ n by σ ξ n for any integer n 1. For any non-singuar compex variety Y with an invertibe sheaf L and an effective divisor D = r ν j D j satisfying L n = O Y (D) for a positive integer n, we obtain a cycic cover π : Z Y by taking the n-th root out of D as foows : Let F := O Y L 1 L 2 L (n 1) and et s be a section of L n defining the divisor D. The F becomes an O Y -agebra generated by L 1 due to the foowing aws : L a L b = L (a+b) ; 1 The author is supported in part by the NSFC Grant NSFC Grant(# ) and LNMS of Fudan University Date: May

2 2 YI ZHANG for any positive integer m expressed as m = n + k, 0 k n 1, one fixes an O X -homomorphism L m L k h hs. Then we get a finite and fat morphism p : Spec(F) Y of degree n ramified at D. The composite morphism π := p Nor of the morphism p and the normaization Nor : Z Spec(F) is caed the cycic cover over Y obtained by taking the n-th root out of D. There is a we-known resut as foows. Theorem 1.1 (see [EV],[K85]). Let X be a non-singuar compex variety, et D = r ν j D j be an effective divisor and et L be an invertibe sheaf with L = O X (D) for a positive integer. Let π : Z X be the cycic cover obtained by taking the -th root out of D and et G be the cycic group < σ > of order. We have : 1. (1.1.1) π O Z = N 0 1 k=0 L (k) 1 for L (k) := L k O X ( [ kd ]) where [ kd ] denotes the integra part of the Q-divisor kd. So that the cover π is a finite and fat morphism. 2. Z is norma, the singuarities of Z are ying over the singuarities of D red. 3. The cycic group G acts on Z. One can choose a primitive -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 eigenvaue ξ k. 4. X is irreducibe if L (k) O X for k = 1,, 1. In particuar this hods true if the common factor of the integers, ν 1,, ν r has gcd(, ν 1,, ν r ) = Assume that D = r ν j D j is an effective norma crossing divisor. Define D (j) := { jd } red = { iν j } =0 D j 0 i 1, where { kd } := kd [ kd ] is the fractiona part of kd. For any p 0, there is (1.1.2) π Ωp N 0 1 Z = Ω p X (og D(i) ) L (i) where Ω p Z is the refexive hu of Ωp Z. In particuar, Ωp X = (π Ω p Z )G.

3 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY 3 6. Assume that X is a proper scheme and that D = r D j is an effective norma crossing reduced divisor. There is (1.1.3) (π T Z ) G = T X ( ogd). 2. Preiminary on commutative Agebra As begining, we investigate the oca structure of cycic cover. Let R be both a finitey generated C-agebra and an integra norma ring, et K be the fraction fied of R. Lemma 2.1. Let n 1 be an integer. Let f be an arbitrary nonzero eement in R. We have : a) Any irreducibe factor of T n f in K[T ] with coefficient 1 of the first term is of form T n f R[T ] satisfying n n and f = (f ) n/n. b) Let d n be the maxima positive integer such that T d f has a root K, say g. Then T n/d g is irreducibe in K[T ] and g R. c) Assume that f = uf ν 1 1 fm νm is an eement in R where f i s are prime eements in R and ν i s are positive integers. Let d be an integer. Then T d f has a root in K if and ony if d gcd(ν 1,, ν m ) and there is a unit u 0 R satisfying u d 0 = u. Proof. a) Let φ(t ) = T m +a m 1 T m 1 +a m 2 T m 2 + +a 0 K[T ] be an irreducibe factor of T n f. Since R is C-agebra, φ(t ) is separabe and it has m different roots α 1,, α m in an agebraic cosure K. Because every root of φ(t ) is a root of T n f, a α i s are integra over R, so that the coefficients of φ(t ) are integra over R. Since R is integray cosed, we have φ(t ) R[T ]. Let y = α 1 be a root of g(t ). We have T n f = (T y)(t ξy) (T ξ n 1 y) = (T y 0 )(T y 1 ) (T y n 1 ) where ξ is a primitive n-th root of unity. Then φ(t ) is the minima poynomia min(k, y) of y in K[T ], we write as m φ(t ) = (T y ij ) with y i1 = y. Thus a 0 = y m ξ for some integer, then f := y m R. Hence φ(t ) T m y m in K[T ], we must have φ(t ) = T m f.

4 4 YI ZHANG For any σ µ n = {x C x n = 1}, we define φ σ (T ) := m (T σy ij ). We have that each φ σ (T ) = m (T σy ij ) = T m + σa m 1 T m 1 + σ 2 a m 2 T m σ m a 0 is aso an irreducibe factor of T n f in K[T ] of degree m, so that φ σ (T ) R[T ] is the minima poynomia 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 ist a different eements in {φ σ (T ) σ µ n } as φ 1 (T ) = φ(t ), φ 2 (T ),, φ k (T ). Hence, we have T n f = φ 1 (T )φ 2 (T ) φ k (T ), so that n = mk. Therefore, we must have φ(t ) = T m f with (f ) n/m = f. b) It is a consequence of (a). c) If d gcd(ν 1,, ν m ) and u d 0 = u for some unit u 0 R, then T d f R[T ] has a root u 0 f ν 1/d 1 fm νm/d R. Suppose that there is g K with g d = f. The T g is an irreducibe factor of T d f in K[T ], thus g R by (a). We write g 0 := g. Then we have g d 0 = uf ν 1 1 f νm n b d 0 (f 1 ). We then have g 0 = f 1 g 1, and get f1 d g1 d = uf ν 1 1 fn νm. Since f 1 is a prime eement, it is impossibe that d > ν 1. So that ν 1 d 0 and we have g1 d = uf ν 1 d 1 fn νm. If ν 1 d 1, we have g1 d = uf ν 1 d 1 fn νm (f 1 ). We dea with g 1 simiary ike g 0, and so on. We finay get that ν 1 = dk for some integer k. By symmetry, we g have d ν i for other ν i s. Then d gcd(n, ν 1,, ν m ) and ( ) d = u. Again by (a), we have u 0 := g f ν 1 /d 1 fn νm/d is a unit in R. Coroary 2.2. Let T n f be a poynomia in R[T ]. We have : f ν 1 /d 1 fn νm/d a) The T n f is irreducibe in K[T ] if and ony if it is irreducibe in R[T ]. b) If T n f is irreducibe in R[T ] then it is a prime eement in R[T ]. For a poynomia T n f(n 1) with nonzero f R, et d n be the maxima positive integer satisfying T d f has a root g K, we have T n f = d i 1 (T n ζd ig) where ζ d is a primitive d-th root of unit and each T n ζd i g is a prime eement in R[T ] for n := n/d; the quotient ring K[T ]/(T n f) can be regarded as a finite K-inear space n 1 K[T ]/(T n f) = Kt i

5 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY 5 where t := T mod (T n f) can be regarded as a root of T n f in K, that t is aso integra over R and the ring R[T ]/(T n f) is integra if and ony if T n f is irreducibe, the group G d =< σ > has a natura action on K[T ]/(T n f) by σ : T ζ n T and σ K = id K, the µ n -invariant part of K[T ]/(T n f) is K. Lemma 2.3. Let f be an nonzero eement in the domain R. Let n be a positive integer and d n the maxima positive integer such that T d f has a root in K and n := n/d. For any ζ µ d := {x C x n = 1}, define a poynomia P ζ = ζ µ d \ζ 1 T n f (ζg ζ g) T n ζg where g K is a root of T d f. 1. In the quotient ring K[T ], these P (T n f) ζ s are idempotent,i.e. P ζ P ζ = 0 ζ ζ, Pζ 2 = P ζ with P ζ = 1. Moreover, the cycic group G n =< σ > acts transitivey on ζ µ d the set {P ζ } ζ µd with (P ζ ) σ = P ξ n ζ µ n ζ d. K[T ] 2. Let K ζ be the fied for each ζ µ (T n ζg) d. Any ξ µ n defines an isomorphism = K ζ K ξ n of fieds by sending T to ξt. Moreover, there is a natura G ζ n-action on the agebra K ζ. ζ µ d 3. There is a G n -equivariant isomorphism of K-agebras (2.3.1) P : K[T ] = K[T ] (T n ζg) (T n f), (α ζ) ζ α ζ P ζ. ζ µ d ζ Coroary 2.4. Let T n f be a poynomia in R[T ] with degree n 1. Let A = R[T ]/(T n f) be a ring and S := {a A r is not zero divisor} a mutipicative system in A. Then S 1 A = K[T ]/(T n 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 ony if for each ζ µ d there is a η ζ (T ) K[T ] such that η ζ (T )h(t ) = 1 mod (T n ζg) in K[T ] We get ηh = 1 in K[T ]/(T n f), where η is the projection image of the poynomia η(t ) := ζ µ d η ζ P ζ. Thus, we have that any non zero divisor in A is an unit in K[T ]. Hence, (T n f) S 1 A = S 1 K[T ] ( (T n f) ) = K[T ] (T n f). Consider the ring homomorphism R K[T ]/(T n f) for a poynomia T n f R[T ] of degree n 1. Let R be the integra cosure of R in K[T ]/(T n f).

6 6 YI ZHANG Lemma 2.5. Let f be an arbitrary nonzero eement in R. If a poynomia T n f R[T ] with degree n 1 is irreducibe in R[T ] then R is the normaization of the domain R[T ] K[T ] in the fied extension of K. (T n f) (T n f) Therefore, by Coroary 2.4 the R is aso the normaization of T n f is not irreducibe since R is a norma integra ring. R[T ] (T n f) even though Lemma 2.6. Let g be a nontrivia prime eement in R. Then the principa prime idea (g) is of height one. Proof. We caim that any nonzero prime idea p of R has an irreducibe eement : Suppose the caim is not true. Let a 0 be any nonzero eement in p. Then a = bc such that both b and c are not unit. Then one of b, c is in p, say a 1. The a 1 is aso irre4ducibe and (a 0 ) (a 1 ) p. We do same steps for a 1 as deaing with a 0 and so on. Thus we have an stricty increasing sequence of ideas (a 0 ) (a 1 ) (a 2 ) p, which contradicts that R is a Noetherian ring. Let p be any nontrivia prime idea of R such that 0 p (g). Let a p be an irreducibe eement. We have a = gc due to a (g), but c is a unit since a is irreducibe. Hence p = (g). For any height one prime idea p of R, the oca ring R p is a DVR, thus we can define div(x) for any x K. Theorem 2.7. Assume a finitey generated C-agebra R is integra and norma. Let K be the fraction fied of R. Let f = uf ν 1 1 f ν ( ν i 1) be an eement in R such that f i s are different prime eements in R and u is a unit in R. Let n 1 be an integer and d n the maxima positive integer such that T d f has a root in K, say g, and denote by n := n/d. 1. With respect to the injective ring homomorphism R K[T ], the integra cosure (T n f) of R is n 1 (2.7.1) R = t i f [ ν 1 i 1 f νi [ where t := T mod (T n f). Moreover, the cycic group Z n =< σ > has a natura action on R by defining σ : t ξ n t, σ R = id R, and the G n -invariant part of R is R Zn = R. 2. Let R ζ be normaization of an integra ring R ζ := R[T ] in K[T ] (T n ζg) (T n ζg) ζ µ d. Then n 1 (2.7.2) Rζ = t i ζf [ ν 1 i 1 f R νi [ R for each

7 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY 7 where t ζ := T mod (T n ζg). Moreover,R R ζ is Gaois, with Gaois group Z n. 3. The group Z n has a natura action on the product R-agebra Rζ. The group permutes the factors of the decomposition, and there is a G n -equivariant isomorphism of R-agebras Rζ R, (α ζ ) ζ α ζ P ζ. ζ µ d Proof. For any x K, it is easy to verify the foowing conditions are equivaent : i. x n f i R, ii. ndiv(x) + iν j div(f j ) 0(we use div(u) = 0). iii. div(xf [ ν 1 i 1 f [ νi ) + ( iν j n ζ ζ µ d [ ν 1i )[f j] 0(we use that (f i ) s are prime idea of height one and so that div(f j ) is the Wei divisor [f j ] := Zero(f j )), iv. div(xf [ ν 1 i 1 f [ νi ) 0(we use that div(xf [ ν1i 1 f [ νi ) has integra coefficients and f i s are prime eements), v. xf [ ν 1 i 1 f [ νi R. 1. Thus there is We now show Let α = n 1 {x K; x n f i R} = f [ ν 1 i [ νi 1 f R. n 1 R = {x K; x n f i R}t i. x i t i (x i K) be an eement in K[T ] (T n f). We caim that α R if and ony if x i t i R for every i. Suppose α R. We have α ζ = n 1 ζi x i t i R, for every ζ µ n. Let µ n = {ζ 1, ζ 2,..., ζ n }. The Vandermonde determinant ζ1 0 ζ1 1 ζ n 1 1 ζ det 2 0 ζ1 1 ζ2 n = (ζ j ζ i ) ζn 0 ζn 1 ζn n 1 1 i<j n is a unit in R, so that each x i t i is a combination of the α ζ s with coefficients in R by Cramer s rue. Therefore x i t i R. The converse is obvious.

8 8 YI ZHANG Let x K and 0 i < n. We caim that xt i R if and ony if x n f i R. Indeed, we have that xt i R if and ony if y = (xt i ) n R. Due to y = (xt i ) n = x n (t n ) i = x n f i K, We have that y R if and ony if y K R = R since R is integray cosed in K. 2. By the statement (1), the integra cosure of R in is the integray cosed domain R ζ = n 1 K[T ] (T n ζg) {x K; x n (ζg) j R}t j ζ. Since R is integray cosed in K, j=0 x n (ζg) j R if and ony if (x n (ζg) j ) d R. That is x n f j R. Therefore n 1 R ζ = {x K; x n f j R}t j ζ. j=0 3. In the ring K[T ] (T n f), we have that f( ζ α ζp ζ ) = ζ f(α ζ)p ζ for any poynomia f R[T ]. Hence that ζ α ζp ζ is integra over R if and ony if each α ζ K[T ] is integra over R[T ] (T n ζg) isomorphism of R-agebras ζ µ d Rζ = R. (T n ζg). The product decomposition of Lemma 2.3 induces an 3. Proof of Theorem 1.1 Now we investigate the oca structure of cycic cover. Lemma 3.1. Let U := SpecR be an irreducibe compex norma affine variety of dimension m and f = uf ν 1 1 fe νe ( ν i 0) an eement in R such that f i s are prime eements in R and u is a unit in R. Let D = e ν i D i is zero ocus of f, i.e., D i = Zero(f i ) i. Denote by U[ n f] := {(p, y) U A 1 y n = f(p)} for an integer n 2. a) Let p 0 be a smooth cosed point of U. Then U[ n f] is singuar at a point (p 0, ) if and ony if p 0 Sing(D). Moreover, U[ n f] is smooth if and ony if U is smooth and D is reduced with ony one smooth component. b) Assume that U is smooth. If D = ν 1 D 1 and f = uf ν 1 1 (ν 1 1) then the normaization U[ n f] of U[ n f] has ony singuarities over Sing(D 1 ). Proof. The proof of (a) is directy foows from Jocobi s criterion : Let (Z 1, Z m ) be an reguar parameter near an open neighborhood V of p 0. Let F (Z 0, Z 1, Z m ) = Z n 0 f(z 1,, Z m ).

9 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY 9 The point q := (p 0, y 0 ) in U[ n f] is singuar if and ony if the F satisfies Thus we prove the (a). F (q) = 0 F (q) = F (q) = = F (q) Z 0 Z 1 Z m = 0. Now R = A(U) is an reguar agebra. The probem (b) is now reduced to show that the U[ n f] is smooth if the D 1 is a smooth divisor. In case of ν = 0, 1 the (b) is obvious by (a.) With ost of generaity, 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 n 1 R = t i f [ ν 1 i 1 R where t := T mod (T n uf ν 1 1 ). Since gcd(n, ν) = 1, we have an ring-isomorphism Z ν 1 nz Z, and so we can get nz a {0, n 1} with aν 1 = 1 + n for some Z. Denote by g = t a f [ aν 1 1. We get that g n = u a f 1 and t = u g ν 1. Thus n 1 g i R = R[W ] (W n u a f 1 ) is an R-subagebra of R. Moreover, we have a tower of incusions of R-agebras such that the integra domains Since the agebra to get Therefore the U[ n f] is smooth. R R[T ] n 1 g i R (T n uf ν 1 ) T u g ν 1 R[T ], n 1 (T n uf ν 1 ) R. g i R and R a have same quotient fied. R[W ] (W n u a f 1 ) is reguar by (a) and the agebra R is norma, we have R = R[W ] (W n u a f 1 ). We ony need to prove the statements 5) and 6) of Theorem 1.1.

10 10 YI ZHANG Let d = dim C X and et ξ be a primitive root of x 1. Since π is étae outside of D, there is π Ωp Z X D = (Ω p X π O Z ) X D and the isomorphism hods over X D. On the other hand, the π Ωp Z is a refexive sheaf on X by the foowing emma 3.2, thus it is sufficient to check the assertion ony in codimension one on X. Lemma 3.2 ([H80]). Let F be a coherent sheaf on a norma integra scheme X. The foowin two conditions are equivaent: i. F is refexive; ii. F is torsion-free, and is norma,i.e.,for every open set U X and every cosed subset B U of codimension 2, the restriction map F(U) F(U Y ) bijective. Without ost of generaity, we may assume that the D red is smooth, moreover there is a oca reguar parameter (x 1,, x d ) of X such that X = Spec(C[x 1,, x d ]), D = (x a 1). Then X = Spec(C[x 1,, x d, t]/(t ux a 1)) where u is a unit in the domain C[x 1,, x d ] and Z is normaization of X. R[T ] Let R := C[x 1, x 2,, x d ] a norma integra ring. Let A := and B the (T ux a 1 ) normaization of A. As Ω 1 B is generated as a B-modue, by the image d : B Ω1 B. We know as that Let t := T p Ω 1 B = {b 0 db 1 db 2 db p b i B i = 1,, 1}. mod (T ux a 1). By Proposition 2.7 we have that B = N 0 1 q i R as an R-modue and that the group G =< σ > has a natura action on p Ω 1 B, where ai [ ] q i := t i x1 0 i 1. Since a(i+j) q i q j = q i := t i+j [ ] N x 0 1 x [ a(i+j) ] [ ai ] [ aj ] 1 i, j 0, we have that q i q j q i+j R if i + j < and that q i q j q i+j n R if i + j. Let ξ be the primitive -th root of unit as we choose in the statement 2). We then have an decomposition of p Ω 1 B = N 0 1 Q i such that each Q i is the R-submodue of eigenvectors for σ in p Ω 1 B with eigenvaue ξi. We now find out these Q i s. We have that dt t = a dx 1 x 1 dq i = { ai }q i dx 1 x 1

11 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY 11 On the other hand, each b B can be written uniquey as b = N 0 1 q i r i with each r i R and so db = N 0 1 (q i dr i + r i dq i ). For each 0 i 1, the eements t i ia [ ] N x 0 1 dy e1 dy ep, e 1 < e 2 < < e p < d for e 1 1, with t i ia [ x ] 1 ia [ ] dx 1 x 1 dx e2 dy ep, 1 < e 2 < < e p < d if { ai } 0 t i x1 dx 1 dx e2 dx ep, 1 < e 2 < < e p < d otherwise becomes an R-basic of the modue Q i. In particuar, G-invariant submodue of p Ω 1 B has Q 0 = p Ω 1 R. Therefore, we get the formua and Ωp X = (π Ω p Z )G. Now we are going to prove the statement 6). For the norma varieties X and Z, by definition, the tangent sheaf T Z := Hom OZ (Ω 1 Z, O Z ) is a refexive sheaf which is composed of derivations from O Z to O Z, and T X ( ogd (1) ) is the subsheaf of T X of consisting of derivations which send the idea sheaf I of D (1) into itsef. Since X is smooth, we get(see [D72]) that T X ( ogd (1) ) = Hom OX (Ω 1 X(ogD (1) ), O X ). Suppose X is proper and D is reduced. Then D = D (i) for a 1 i 1 and by 1) Z is aso a proper scheme. Thus the duaizing sheaf ω Z = π! ω X has a natura O Z -modue(see Proposition [KM]). Because π is an affine morphism, we can define an equivaent functor of categories between the category of quasi-coherent π O Z -modues and the category of quasi-coherent O Z -modue(see Ex. II. 5.17(e) [Hart]). Since π O Z is ocay free, we have π! ω X := Hom OX (π O Z, ω X ) = (Hom OX (π O Z, O X ) OX ω X ) = π! O X OZ π ω X. Thus the reative duaizing sheaf ω Z/X = π! O X. We caim that ω Z/X = π L ( 1) = π (L 1 ). We can take a oca reguar parameter (x 1,, x d ) of X such that Thus we get that X = Spec(C[x 1,, x d ]), D = (x 1 x k ). X = Spec(C[x 1,, x d, t]/(t x 1 x k ))

12 12 YI ZHANG and Z is the normaization of X. Let R := C[x 1, x 2,, x d ] and A := R[T ] (T x1 x k ), and et B be the normaization of A. By Proposition 2.7 we have that B = N 0 1 p i R as an R-modue where {p i := t i }, N0 1 is an R-basis of B. Let η i B := Hom R (B, R) be dua of q i, i.e., η i (p i ) = 1, η i (p j ) = 0 j i. The B-modue structure of Hom R (B, R) is cear : b φ := φ b where φ b Hom R (B, R)is given by φ b (b 1 ) := φ(bb 1 ) b 1 B. We observe that p i p j = q i+j if i + j < and p i p j p i+j n R if i + j. So that p i η N0 1 = η N0 1 i for a i = 0, 1. As an B-modue, the Hom R (B, R) must be a modue of rank one generated by η N0 1. Hence π! O X = π (L (N0 1) ). Let Sing(D) be singuarities of D and X 0 := X \ Sing(D). We know codimension of Sing(D) in X is more than two and π 1 (X 0 ) is smooth open sub-scheme of Z. Using the duaity theory for finite morphism(cf. Theorem 5.67 [KM]), on X 0 we get π T Z = π Hom OZ (Ω 1 Z, O Z ) = π Hom OZ (ω Z/X OZ Ω 1 Z, ω Z/X ) = π Hom OZ (π (L ( 1) ) OZ Ω 1 Z, π! O X ) = Hom OX (L ( 1) OX π Ω 1 Z, O X ) N 0 1 = Hom OX (L (N0 1) OX ( Ω 1 X(og D (i) ) L (i) ), O X ) N 0 1 = (L (N0 1) OX T X ) (L (i N0+1) OX T X ( og D)). Since both π T Z and (L ( 1) OX refexive sheaves on X, there is T X ) N 0 1 (L (i N0+1) OX T X ( og D)) are N 0 1 π T Z = L (N0 1) OX T X ) (L (i N0+1) OX T X ( og D)) on X. Hence (π T Z ) G = T X ( ogd) on X.

13 INTRODUCTION TO CYCLIC COVERS IN ALGEBRAIC GEOMETRY 13 References [B78] Bogomoov, F.: Unstabe vector bundes and curves on surfaces. Proc.Intern.Congress of Math., Hesinki 1978, [D72] Deigne, P. :Théorie de Hodge II. I.H.É.S. Pub. Math. 40 (1972) 5 57 [EV] Esnaut, H.& Viehweg, E.: Lectures on vanishing theorems. DMV Seminar, 20. Birkhauser Verag, Base, [Hart] Hartshrone,R.: Agebraic Geometry. Graduate Texts in Math 52(1977) Springer, New York Heideberg Berin [H80] Hartshrone,R. Stabe Refexive Sheaves. Math. Ann. 254(1980) [K85] Kawamata, Y.: Minima modes and the Kodaira dimension of agebraic fibre spaces. Journ. Reine Angew. Math 363 (1985) 1-46 [KM] Koár, J. & Mori,S.: Birationa geometry of agebraic varieties, with the coaboration of C. H. Cemens and A. Corti. Cambridge University Press, 1998 [R80] Reid, M.: Canonica 3-fods. In: Géométrie Agébrique. Angers 1979, Sijthoff and Noordhoff Internationa Pubishers, Aphen aan den Rijn (1980) [V82] Viehweg, E.: Vanishing theorems. Journ. Reine Angew. Math. 335(1982) 1-8 [V83] Viehweg, E.: Weak positivity and the additivity of the Kodaira dimension II: the oca Torei map. Proc. Cassification of Agebraic and Anaytic Manifods, Katata Progress in Math. 39, Birkhauser 1983, [Vie] Viehweg, E.: Quasi-projective Modui for Poarized Manifods. Ergebnisse der Mathematik, 3. Foge 3 0 (1995), Springer Verag, Berin-Heideberg-New York Schoo of Mathematica Sciences, Fudan University, Shanghai , China E-mai address: zhangyi math@fudan.edu.cn