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On the Hard Lefschetz property of stringy Hodge numbers

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On the Hard Lefschetz property of stringy Hodge numbers Jan Schepers ?Abstract For projective varieties with a certain class of ‘mild’isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities,we show that the stringy Hodge numbers satisfy the Hard Lefschetz property (i.e.h p,q st ≤h p +1,q +1st for p +q ≤d ?2,where d is the dimension of the variety).This result ?ts nicely with a 6-dimensional counterexample of Mustat ??a and Payne for the Hard Lefschetz property for stringy Hodge numbers in general.We also give such an example,ours is a hypersurface singularity.1Introduction 1.1.Stringy Hodge numbers of projective varieties with Gorenstein canoni-cal singularities were introduced by Batyrev in [Ba].They are de?ned if the stringy E -function ,which is in general a rational function of two variables u and v ,is in fact a polynomial.The idea is that they should be the Hodge numbers of a conjectural ‘string cohomology’.Several constructions of such string cohomology spaces were made by Borisov and Mavlyutov in [BM],and

they also made a connection to the orbifold cohomology of Chen and Ruan from [CR].Moreover,Yasuda showed that the stringy Hodge numbers are the Hodge numbers of the orbifold cohomology for varieties with Gorenstein quotient singularities (see [Ya,Remark 1.4(2)]).

1.2.In this paper we want to study under which conditions the Hard Lef-schetz property holds for stringy Hodge numbers.By the Hard Lefschetz

property we mean the inequalities that are imposed if there would be an analogue of the Hard Lefschetz Theorem for the conjectural string cohomol-ogy,i.e.

h p,q st≤h p+1,q+1

for p+q≤d?2,

where d is the dimension of the variety.Fernandez gave in[Fe]a crite-rion for the Hard Lefschetz Theorem to hold for orbifold cohomology.He also shows that this criterion fails in general for generic Calabi-Yau hyper-surfaces in the weighted projective space P(1,1,1,3,3)([Fe,Example4.4]). However,it follows for instance from the main theorem below that this can-not be seen from the Hodge or Betti numbers of the orbifold cohomology

(h p,q

orb =h p,q st≤h p+1,q+1

orb

=h p+1,q+1

for p+q≤1).In fact,the orbifold Hodge

numbers where in this case already studied by Poddar(see[Po],Section5 and especially Corollary2).

1.3.Mustat??a and Payne?rst gave an example of a6-dimensional projec-tive toric variety with stringy Betti number b6,st=h3,3st strictly smaller than b4,st=h2,2st(see[MP,Example1.1]).This example was used to disprove a conjecture of Hibi on the unimodality of the so-called h-vector orδ-vector of a re?exive polytope.This vector actually gives the stringy Betti numbers of the toric variety de?ned by the fan over the faces of the polytope([BD, Theorem7.2]and[MP,Theorem1.2]).For many more examples of re?exive polytopes with non-unimodal h-vector we refer to[Pa].

1.4.The main result of this paper is the following.The used notions are explained in Section

Main theorem.Let Y be either

?a projective variety of dimension d=3with Gorenstein canonical sin-gularities,or

?a projective variety of dimension d≥4with at most isolated Gorenstein singularities that admit a log resolution with all discrepancy coe?cients of exceptional components>?d?4

the stringy E-function of Y is a polynomial,then h p,q st(Y)≤h p+1,q+1

st (Y)for

p+q≤d?2.

For the proof of this theorem we refer to Section3.In Section4we compare the above theorem with the example of Mustat??a and Payne.We also discuss an explicit example of a6-dimensional projective variety with an isolated canonical hypersurface singularity that does not satisfy the Hard Lefschetz property.

Acknowledgements.I want to thank Wim Veys for helpful discussions and for encouraging me to communicate these results.I also want to thank the referee of an earlier paper for bringing references[Fe]and[MP]to my attention.

2Stringy Hodge numbers

2.1.Let X be an arbitrary complex algebraic set of dimension d.It is well known that the cohomology with compact support H?c(X)carries a natural mixed Hodge structure(we always use cohomology with coe?cients in the complex numbers).The data of this mixed Hodge structure are put in the Hodge-Deligne polynomial

H(X;u,v):=

p,q=0 2d i=0(?1)i h p,q(H i c(X)) u p v q,

where h p,q denotes the dimension of the(p,q)-component H p,q(H i c(X)).The Hodge-Deligne polynomial is a generalized Euler characteristic:if Y is a Zariski-closed subset of X,then H(X;u,v)=H(X\Y;u,v)+H(Y;u,v) and for a product of algebraic sets X and X′we have H(X×X′;u,v)= H(X;u,v)H(X′;u,v).

2.2.Now let Y be a normal irreducible variety.Assume that Y is Q-Gorenstein;i.e.rK Y is a Cartier divisor for some r∈Z>0.If K Y is already Cartier then Y is called Gorenstein.For example,all hypersurfaces and more generally all complete intersections are Gorenstein.Choose a log resolution f:X→Y of Y.This is a proper birational map f from a nonsingular variety X such that the exceptional locus is a divisor D with only smooth

components D i ,i ∈I,that have normal crossings.We can write rK X ?f ?(rK Y )= i ∈I b i D i ,with all b i ∈Z .Using Q -coe?cients this becomes K X ?f ?(K Y )= a i D i with a i =b i /r .The rational number a i is called the discrepancy coe?cient of D i .We call Y log terminal,canonical or terminal if all a i >?1,≥0or >0respectively (this does not depend on the chosen log resolution).

2.3.De?nition ([Ba,

De?nition

3.1]).Let Y be log terminal.Choose a log resolution f :X →Y with irreducible exceptional components D i ,i ∈I .Denote the discrepancy coe?cient of D i by a i .For a subset J of I we use the notations D J := j ∈J D j and D ?J :=D J \ i ∈I \J

D i .

This gives a strati?cation of X as

J ?I D ?J .The stringy E -function of Y is

de?ned by the formula

E st (Y ;u,v ):= J ?I H (D ?J ;u,v )

j ∈J uv ?1(uv )a j +1?1

2.5.Assume now that Y is in addition projective of dimension d.Batyrev proves the following relation in[Ba,Theorem

3.7]:

E st(Y;u,v)=(uv)d E st(Y;u?1,v?1).(1) If Y is also Gorenstein canonical and if E st(Y;u,v)is a polynomial p,q a p,q u p v q then Batyrev de?nes the stringy Hodge numbers of Y as h p,q st(Y):=(?1)p+q a p,q. Remark the following:

1.If Y is smooth then the stringy Hodge numbers are equal to the usual

Hodge numbers of Y.If Y has a crepant desingularization X,then the stringy Hodge numbers of Y are equal to the Hodge numbers of X.

2.h p,q st(Y)=h q,p st(Y).

3.From(1)above it follows that h p,q st(Y)can only be nonzero for0≤

p,q≤d and that h p,q st(Y)=h d?p,d?q

(Y).

4.From Remark2.4(3)we have h0,0st(Y)=1.

Batyrev also made the following very intriguing conjecture.

Conjecture([Ba,Conjecture3.10]).Stringy Hodge numbers are nonnega-tive.

2.6.Example.Canonical surface singularities are classi?ed and are pre-cisely the A-D-E singularities.It is well known that they admit a crepant resolution and thus the conjecture above is trivially true for them by Remark 2.5(1).Of course,the Hard Lefschetz property is also satis?ed in this case.

In[SV1]the conjecture was proved for the same class of varieties that is treated by the main theorem of this paper,and thus also for threefolds in full generality.

3Proof of the main theorem

3.1.Let us for convenience repeat the statement of the theorem.

Theorem.Let Y be either

?a projective variety of dimension d=3with Gorenstein canonical sin-gularities,or

?a projective variety of dimension d≥4with at most isolated Gorenstein singularities that admit a log resolution with all discrepancy coe?cients of exceptional components>?d?4

{(d ?1)/2,(d +1)/2}.The term S i,j has to be introduced for con-tributions of the exceptional components with the lowest allowed dis-crepancy coe?cients.If one does not put the above lower bound on the discrepancy coe?cients,it becomes more often nonzero,much more complicated and can even be negative (see the example in [SV2]).

If we denote ker H d ?i,d ?j (H 2d ?i ?j (X ))→H d ?i,d ?j (H 2d ?i ?j (D )) by K 2d ?i ?j i,j then it su?ces to prove that dim K 2d ?i ?j i,j ≤dim K 2d ?i ?j ?2i +1,j +1

for i +j ≤d ?2.Denote ker(H k (X )→H k (D ))by K k .

We will use the following construction of de Cataldo and Migliorini.Embed Y in a projective space P r and take a generic hyperplane section Y s of Y .So Y s is nonsingular and does not contain any of the singular points of Y .Let X s :=f ?1(Y s )and denote by ηs the fundamental class of X s in H 1,1(H 2(X )).If one dualizes the surjective map H k (X )→H k (D )for k ≥d and uses Poincar′e duality on X ,one obtains an injection H k (D )→H 2d ?k (X ).Now the spaces

H 0(X ),H 1(X ),H 2(X )

H d +1(D ),H d (X ),

K d +1,...,K 2d ?2,H 2d ?1(X ),H 2d (X )

satisfy the Hard Lefschetz Theorem with respect to the cup product with ηs .This result is discussed by de Cataldo and Migliorini in Section 2.3and the beginning of Section 2.4from [dCM2]for dimensions 3and 4,but it is not hard to see that their argument works in any dimension.It also follows from their earlier work [dCM1],see Section 2.4there.We note that the choice of the space H d (X )in the middle is somewhat arbitrary,it can be replaced by

any subspace containing the image of H d ?2(X )

should be a complex as well,where H d(X)→H d?2(X)corresponds to inter-secting with X s.And this is clear.

Summarizing,we obtain that the maps∪ηs:K d+k→K d+k+2are surjective for0≤k≤d?2.Since∪ηs is a morphism of Hodge structures of type(1,1),

we also get that∪ηs:K2d?i?j?2

i+1,j+1→K2d?i?j

i,j

is surjective for i+j≤d?2.

Now let Y be a projective threefold with arbitrary Gorenstein singularities. By the main theorem of[Re]we can?nd a projective variety Z with terminal singularities and a projective birational crepant morphism g:Z→Y.So E st(Z)=E st(Y)by Remark2.4(2).The point is that terminal threefold singularities are automatically isolated(see for instance[Ma,Corollary4-6-6])and thus we can apply the above reasoning for Z(the used results of de Cataldo and Migliorini remain valid for Z,as well as the description of the numbers(?1)i+j b i,j from[SV1]for i+j≤3,now with S i,j always zero). 4Examples

4.1.Example.We?rst compare Example1.1of[MP]with the main

theorem.Let f be the vector(1

3,1

)in R6and N the lattice Z6+Z·f.

Denote the standard basis vectors of R6by e1,...,e6.Mustat??a and Payne consider the polytope P with vertices

{e1,...,e6,e1?f,...,e6?f}.

It is a re?exive polytope and the projective toric variety Y de?ned by the fanΣover the faces of the polytope has stringy E-function

(uv)6+6(uv)5+8(uv)4+6(uv)3+8(uv)2+6uv+1.

The fanΣhas eight cones of maximal dimension,namely

?a coneσgenerated by e1,...,e6,

?τgenerated by e1?f,...,e6?f,

?and six non-simplicial conesρi generated by all vectors to vertices ex-cept e i and e i?f.

As sketched in[MP,Example3.1]one can make a triangulation of the bound-ary of P by taking the convex hulls of{e1,..., e j,...,e k,e k?f,...,e6?f} and{e1,...,e j,e j?f,...,

e k?f,...,e6}for1≤j

f codimension1where introduced.Thus E st(Z)=E st(Y).Since the singular locus of Z is given by the union of the orbits of the torus action correspondin

g to cones in the fan that cannot be generated by a part of the basis of the lattice,we see that Z has exactly two isolated singular points coming from the conesσandτ.The resolution of singularities of these points is particularly easy:we subdivide the fan by adding the rays generated by f and by?f.This does introduce two ir-reducible exceptional components of codimension1and from the theory of toric varieties it is well known that there discrepancy coe?cients are1.So the example of the variety Z shows that the lower bound on the discrepancies in the main theorem is crucial.Even for isolated singularities wit

h a very easy resolution,the theorem cannot be extended.

4.2.We want to conclude this paper by discussing another6-dimensional example that was obtained independently of the one of Mustat??a and Payne. The resolution of singularities is much more complicated,but it has the ad-vantage of being a hypersurface singularity.We will need the formula for the Hodge-Deligne polynomial of a Fermat hypersurface and for a quasihomoge-neous a?ne hypersurface with an isolated singularity at the origin.

Denote the Fermat hypersurface of dimension d and degree l by Y(d)

l .So Y(d)

is given by

{x l0+···+x l d+1=0}?P d+1

Dais shows in[Da,Lemma3.3]that the Hodge-Deligne polynomial of Y(d)

l is

given by

H(Y(d)

l ;u,v):=

p=0u p v p+(?1)d G(d+1,p+1|l?1,p)v d?p ,

where

G(κ,λ|ν,ξ):=

j=0(?1)j κ+1j ν(λ?j)+ξκ

for(κ,λ,ν,ξ)∈Z4≥0andκ≥λ(if m>n,the binomial coe?cient n m must be interpreted as0).

Let f∈C[x1,...,x r+1]be a quasihomogeneous polynomial of degree d with respect to the weights w1,...,w r+1and assume that the origin is an isolated singularity of Y:=f?1(0).According to[Da,Section2]the Hodge-Deligne polynomial of Y equals

(uv)r+(?1)r?1(uv?1)r?1

p=0dim C M(f)(p+1)d?(w1+···+w r+1)u p v r?1?p,

where M(f)(p+1)d?(w

1+···+w r+1)denotes the piece of degree(p+1)d?(w1+

···+w r+1)of the Milnor algebra

M(f):=

C[x1,...,x r+1]

?x1

,...,?f

(1?t w1)···(1?t w r+1)

4.3.Example.It is not so easy to give the variety Y whose stringy E-function we want to compute.Let us start from

Y′:={x51z3+x52z3+x83+x84+x85+x86+x87=0}?P7,

where we consider z=0as the hyperplane at in?nity.At in?nity there is a singular P1and the origin of the a?ne chart z=0is singular as well.Our

variety Y will consist of a resolution of the singular P1and will thus have one isolated hypersurface singularity.Let us?rst describe the resolution process at in?nity.Thereby we want to compute the Hodge-Deligne polynomial of the nonsingular part Y ns of Y,since the stringy E-function of Y can be written as H(Y ns;u,v)+contribution of the singular point.So we keep track of the contributions in every step.We blow up in the singular line.The exceptional locus after this?rst step consists of?ve disjoint components D∞1,...,D∞5(all isomorphic to P5)and one other component D∞6,also isomorphic to P5and singular for the strict transform of Y′.The intersection of D∞6and another D∞i is isomorphic to P4.Since we are only interested in the nonsingular part and since we will blow up in D∞6in the following step,the contribution of the?rst step to the Hodge-Deligne polynomial of Y ns will be5(uv)5(the nonsingular exceptional part are?ve disjoint A5’s).So in the second step we blow up with D∞6as center.Again there appear?ve new disjoint components E∞1,...,E∞5and one component E∞6intersecting the others transversally and singular for the strict transform of Y′.Here the E∞i are P1-bundles over P4 and E∞6is isomorphic to P5.The intersection of E∞6with an E∞i is isomorphic to P4and thus the contribution of this step to the Hodge-Deligne polynomial of Y ns is5(uv)5+5(uv)4+5(uv)3+5(uv)2+5uv.In the next step we blow up with E∞6as center.There are two new exceptional components.The?rst, F∞,is a P1-bundle over P4that contains?ve disjoint singular components for the strict transform of Y′,all isomorphic to P4.The second component G∞is isomorphic to a P1×P1-bundle over the Fermat hypersurface Y(3)8(notation as in4.2).The intersection F∞∩G∞is isomorphic to a P1-bundle over Y(3)8. All of this means that the contribution of this step to the Hodge-Deligne polynomial of Y ns equals

(uv?4)((uv)4+(uv)3+(uv)2+uv+1)+((uv)2+uv)H(Y(3)8;u,v).

In the?nal step we blow up in the?ve remaining singular components.This gives?ve new disjoint exceptional components H∞1,...,H∞5,whose Hodge-Deligne polynomial equal

(uv)5+2(uv)4+2(uv)3+2(uv)2+2uv+1+uvH(Y(3)8;u,v).

So the total contribution of the exceptional locus above the singular P1of Y′to the Hodge-Deligne polynomial of Y ns is

16(uv)5+12(uv)4+12(uv)3+12(uv)2+12uv+1+((uv)2+6uv)H(Y(3)8;u,v).

To compute the Hodge-Deligne polynomial of Y ns we must add the Hodge-Deligne polynomial of the nonsingular part of Y′.At in?nity this is the

minus the singular P1,with contribution double projective cone over Y(3)

(uv)2H(Y(3)8;u,v),

and the contribution of the nonsingular part of Y′in the a?ne chart z=0 can be computed by the method of Dais from4.2;it equals

(uv)6?1?(uv?1)(140u4v+140uv4+4060u2v3+4060u3v2).

The formula for H(Y(3)8;u,v)is

(uv)3+(uv)2+uv+1?35u3?35v3?1015u2v?1015uv2,

so?nally the Hodge-Deligne polynomial of Y ns equals

E:=(uv)6+18(uv)5+20(uv)4+20(uv)3+20(uv)2+18uv?210u5v2?210u2v5?6090u4v3?6090u3v4?70u4v?70uv4?2030u3v2

?2030u2v3.

Next we compute the contribution of the singular point given by the origin of

{x51+x52+x83+x84+x85+x86+x87=0}?A7

to the stringy E-function of Y by a log resolution.We remark that one can also use the combinatorial procedure of[SV2,Section4].We?rst blow up in the singular point itself.This gives?ve exceptional components D1,...,D5, all isomorphic to P5and intersecting in the new singular locus(isomorphic to P4).Blowing up in this intersection gives two new exceptional components, but one of them(isomorphic to a P1-bundle over P4)is singular for the strict transform of Y.The other one,called E,is isomorphic to a P2-bundle over Y(3)8.Its intersection with the D i is covered by the new singular locus. Blowing up in this singular locus creates six new exceptional components. Five of them behave similarly.They are isomorphic to a P1-bundle over P4, they are disjoint and each of them has an intersection with one D i isomorphic to P4(they are now the only components that intersect the D i).We call these components F1,...,F5and choose the numbering compatible with those of the D i.The sixth exceptional component is called G and is also isomorphic

to a P 1-bundle over P 4.The new singular locus is E ∩G and it is isomorphic to a P 1-bundle over Y (3)8.The intersection of an F i and G is isomorphic to

P 4and has a Y (3)8in common with the singular locus and the intersection

of E and an F i is only 3-dimensional and is covered by the singular locus.In the ?nal step we blow up in the remaining singular locus.There is one new exceptional component,called H .Its Hodge-Deligne polynomial equals ((uv )2+7uv +1)H (Y (3)8;u,v )and it splits o?E from the other components.The intersections of H with the F i ,with G and with E are all isomorphic to a P 1-bundle over Y (3)8.This ?nal blow up also adds uvH (Y (3)8;u,v )to the Hodge-Deligne polynomial of the F i .And there are threefold intersections F i ∩G ∩H isomorphic to Y (3)8.The result is a normal crossing divisor with the following intersection diagram:

u u u u

u u u u u u u u D 1D 2

D 3

D 4D 5G H

E d d d d d e e e e e e e e e ?????????d d d d d d d d d r r r r r r r r r ¨¨¨¨¨¨¨¨¨

F 1

F 2F 3F 4F 5The discrepancy coe?cients are 1for the D i and 0for all the other compo-nents.Thus one can compute that

F :=8(uv )5+15(uv )4+10(uv )3+15(uv )2+8uv +1?70u 5v 2?70u 2v 5

?2030u 4v 3?2030u 3v 4?210u 4v ?210uv 4?6090u 3v 2?6090u 2v 3.is the contribution of the singular point to the stringy E -function.Then the stringy E -function of Y is just E +F which equals

E st (Y ;u,v )=(uv )6+26(uv )5+35(uv )4+30(uv )3+35(uv )2+26uv +1

?280u 5v 2?280u 2v 5?8120u 4v 3?8120u 3v 4?280u 4v

?280uv 4?8120u 3v 2?8120u 2v 3,

and thus h2,2st(Y)>h3,3st(Y).

References

[Ba]V.V.Batyrev,Stringy Hodge numbers of varieties with Gorenstein canonical sin-gularities,Proc.Taniguchi Symposium1997,In‘Integrable Systems and Algebraic Geometry,Kobe/Kyoto1997’,World Sci.Publ.(1999),1-32.

[BD]V.V.Batyrev and D.Dais,Strong McKay correspondence,string-theoretic Hodge numbers and mirror symmetry,Topology35(1996),901-929.

[BG]W.Bruns and J.Gubeladze,Polytopes,rings and K-theory,preliminary version at http://www.math.uos.de/staff/phpages/brunsw/preprints.htm.

[BM]L.A.Borisov and A.R.Mavlyutov,String cohomology of Calabi-Yau hypersurfaces via mirror symmetry,Adv.Math.180(2003),355-390.

[CR]W.Chen and Y.Ruan,A new cohomology theory of orbifold,Comm.Math.Phys.

248(2004),1-31.

[Da] D.Dais,On the string-theoretic Euler number of a class of absolutely isolated sin-gularities,Manuscripta Math.105(2001),143-174.

[dCM1]M.A.de Cataldo and L.Migliorini,The Hodge theory of algebraic maps,Ann.

Sci.′Ecole Norm.Sup.,4e s′e rie,t.38(2005),693-750.

[dCM2]M.A.de Cataldo and L.Migliorini,Intersection forms,topology of maps and motivic decomposition for resolutions of threefolds,math.AG/0504554.

[Fe]J.Fernandez,Hodge structures for orbifold cohomology,Proc.Amer.Math.Soc.

134(2006),2511-2520.

[Ma]K.Matsuki,Introduction to the Mori program,Universitext,Springer-Verlag(2002). [MP]M.Mustat??a and S.Payne,Ehrhart polynomials and stringy Betti numbers,Math.

Ann.333(2005),787-795.

[Pa]S.Payne,Ehrhart series and lattice triangulations,math.CO/0702052,to appear in Discrete Comput.Geom.

[Po]M.Poddar,Orbifold Hodge numbers of Calabi-Yau hypersurfaces,Paci?c J.Math.

208(2003),151-167.

[Re]M.Reid,Canonical3-folds,Journ′e es de g′e om′e trie alg′e brique d’Angers1979,Si-jtho?&Noordho?(1980),273-310.

[SV1]J.Schepers and W.Veys,Stringy Hodge numbers for a class of isolated singularities and for threefolds,Int.Math.Res.Not.,Vol.2007,article ID rnm016,14pages. [SV2]J.Schepers and W.Veys,Stringy E-functions of hypersurfaces and of Brieskorn singularities,arXiv:0706.0798v1[math.AG].

[Ya]T.Yasuda,Twisted jets,motivic measures and orbifold cohomology,Compos.Math.

240(2004),396-422.

Jan Schepers

Universiteit Leiden

Mathematisch Instituut

Niels Bohrweg1

2333CA Leiden

The Netherlands

E-mail:jschepers@math.leidenuniv.nl

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英语造句

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学生造句--Unit 1

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介词后的动词要用—ing形式。介词加代词时，代词要用宾格。例如：give up her（him）这种形式是正确的，而give up she（he）这种形式是错误的。 7.冠词：冠词是指修饰名词，表名词泛指或特指。冠词有a an the 。 8.叹词：叹词表示一种语气。例如：OH. Ya 等 9.连词：连词是指连接两个并列的成分，这两个并列的成分可以是两个词也可以是两个句子。例如：and but or so 。 10.数词：数词是指表示数量关系词，一般分为基数词和序数词第二章节：英语句子成分主语：动作的发出者，一般放在动词前或句首。由名词. 代词. 数词. 不定时. 动名词. 或从句充当。谓语：指主语发出来的动作，只能由动词充当，一般紧跟在主语后面。宾语：指动作的承受着，一般由代词. 名词. 数词. 不定时. 动名词. 或从句充当. 介词后面的成分也叫介词宾语。定语：只对名词起限定修饰的成分，一般由形容

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M A: Has the case been closed yet? B: No, the magistrate still needs to decide the outcome. magistrate n.地方行政官，地方法官，治安官 A: I am unable to read the small print in the book. B: It seems you need to magnify it. magnify vt.１．放大，扩大；２．夸大，夸张 A: That was a terrible storm. B: Indeed, but it is too early to determine the magnitude of the damage. magnitude n.１．重要性，重大；２．巨大，广大 A: A young fair maiden like you shouldn’t be single. B: That is because I am a young fair independent maiden. maiden n.少女，年轻姑娘，未婚女子 a.首次的，初次的 A: You look majestic sitting on that high chair. B: Yes, I am pretending to be the king! majestic a.雄伟的，壮丽的，庄严的，高贵的 A: Please cook me dinner now. B: Yes, your majesty, I’m at your service. majesty n.１．［Ｍ－］陛下（对帝王，王后的尊称）；２．雄伟，壮丽，庄严 A: Doctor, I traveled to Africa and I think I caught malaria. B: Did you take any medicine as a precaution? malaria n.疟疾 A: I hate you! B: Why are you so full of malice? malice n.恶意，怨恨 A: I’m afraid that the test results have come back and your lump is malignant. B: That means it’s serious, doesn’t it, doctor? malignant a.１．恶性的，致命的；２．恶意的，恶毒的 A: I’m going shopping in the mall this afternoon, want to join me? B: No, thanks, I have plans already. mall n.（由许多商店组成的）购物中心 A: That child looks very unhealthy. B: Yes, he does not have enough to eat. He is suffering from malnutrition.

base on的例句

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英语造句大全

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(完整版)主谓造句

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初中英语造句

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The Kite Runner-美句摘抄及造句

《The Kite Runner》追风筝的人--------------------------------美句摘抄 1.I can still see Hassan up on that tree, sunlight flickering through the leaves on his almost perfectly round face, a face like a Chinese doll chiseled from hardwood: his flat, broad nose and slanting, narrow eyes like bamboo leaves, eyes that looked, depending on the light, gold, green even sapphire 翻译：我依然能记得哈桑坐在树上的样子，阳光穿过叶子，照着他那浑圆的脸庞。他的脸很像木头刻成的中国娃娃，鼻子大而扁平，双眼眯斜如同竹叶，在不同光线下会显现出金色、绿色，甚至是宝石蓝。 E.g.: A shadow of disquiet flickering over his face. 2.Never told that the mirror, like shooting walnuts at the neighbor's dog, was always my idea. 翻译：从来不提镜子、用胡桃射狗其实都是我的鬼主意。E.g.:His secret died with him, for he never told anyone. 3.We would sit across from each other on a pair of high

翻译加造句

一、翻译 1. The idea of consciously seeking out a special title was new to me., but not without appeal. 让我自己挑选自己最喜欢的书籍这个有意思的想法真的对我具有吸引力。 2.I was plunged into the aching tragedy of the Holocaust, the extraordinary clash of good, represented by the one decent man, and evil. 我陷入到大屠杀悲剧的痛苦之中，一个体面的人所代表的善与恶的猛烈冲击之中。 3.I was astonished by the the great power a novel could contain. I lacked the vocabulary to translate my feelings into words. 我被这部小说所包含的巨大能量感到震惊。我无法用语言来表达我的感情（心情）。 4，make sth. long to short长话短说 5.I learned that summer that reading was not the innocent(简单的) pastime(消遣) I have assumed it to be., not a breezy, instantly forgettable escape in the hammock(吊床)，( though I’ ve enjoyed many of those too ). I discovered that a book, if it arrives at the right moment, in the proper season, will change the course of all that follows. 那年夏天，我懂得了读书不是我认为的简单的娱乐消遣，也不只是躺在吊床上，一阵风吹过就忘记的消遣。我发现如果在适宜的时间、合适的季节读一本书的话，他将能改变一个人以后的人生道路。二、词组造句 1. on purpose 特意，故意 This is especially true here, and it was ~. (这一点在这里尤其准确，并且他是故意的) 2.think up 虚构，编造，想出 She has thought up a good idea. 她想出了一个好的主意。 His story was thought up. 他的故事是编出来的。 3. in the meantime 与此同时助记：in advance 事前in the meantime 与此同时in place 适当地... In the meantime, what can you do? 在这期间您能做什么呢？ In the meantime, we may not know how it works, but we know that it works. 在此期间，我们不知道它是如何工作的，但我们知道，它的确在发挥作用。 4.as though 好像，仿佛 It sounds as though you enjoyed Great wall. 这听起来好像你喜欢长城。 5. plunge into 使陷入 He plunged the room into darkness by switching off the light. 他把灯一关，房

改写句子练习2标准答案

The effective sentences:(improve the sentences!) 1.She hopes to spend this holiday either in Shanghai or in Suzhou. 2.Showing/to show sincerity and to keep/keeping promises are the basic requirements of a real friend. 3.I want to know the space of this house and when it was built. I want to know how big this house is and when it was built. I want to know the space of this house and the building time of the house. 4.In the past ten years,Mr.Smith has been a waiter,a tour guide,and taught English. In the past ten years,Mr.Smith has been a waiter,a tour guide,and an English teacher. 5.They are sweeping the floor wearing masks. They are sweeping the floor by wearing masks. wearing masks,They are sweeping the floor. 6.the drivers are told to drive carefully on the radio. the drivers are told on the radio to drive carefully 7.I almost spent two hours on this exercises. I spent almost two hours on this exercises. 8.Checking carefully,a serious mistake was found in the design. Checking carefully,I found a serious mistake in the design.

用以下短语造句

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英语造句

English sentence 1、(1)、able adj. 能句子：We are able to live under the sea in the future. (2)、ability n. 能力句子：Most school care for children of different abilities. (3)、enable v. 使。。。能句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的句子：We must have the accurate calculation. (2)、accurately adv. 精确地句子：His calculation is accurately. 3、(1)、act v. 扮演句子：He act the interesting character.（2）、actor n. 演员句子：He was a famous actor. (3)、actress n. 女演员句子：She was a famous actress. (4)、active adj. 积极的句子：He is an active boy. 4、add v. 加句子：He adds a little sugar in the milk. 5、advantage n. 优势句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n. 句子：He is in America. 10、appear 出现v. He appears in this place. 11. artist 艺术家n. He is an artist. 12. attract 吸引 He attracts the dog. 13. Australia 澳大利亚 He is in Australia. 14.base 基地 She is in the base now. 15.basket 篮子 His basket is nice. 16.beautiful 美丽的 She is very beautiful. 17.begin 开始 He begins writing. 18.black 黑色的 He is black. 19.bright 明亮的 His eyes are bright. 20.good 好的 He is good at basketball. 21.British 英国人 He is British. 22.building 建造物 The building is highest in this city 23.busy 忙的 He is busy now. 24.calculate 计算 He calculates this test well. 25.Canada 加拿大 He borns in Canada. 26.care 照顾 He cared she yesterday. 27.certain 无疑的 They are certain to succeed. 28.change 改变 He changes the system. 29.chemical 化学药品