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TORICGEOMETRYANDCALABI-YAUCOMPACTIFICATIONS Tori Geometry and Calabi-Yau Compa ti(cid:28) ations Maximilian KREUZER Institute for Theoreti al Physi s Vienna Universityof Te hnology, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria e-mail: Maximilian.Kreuzertuwien.a .at 7 0 0 These notes ontain a brief introdu tion to the onstru tion of 1. Basi s of Tori Geometry 2 tori Calabi(cid:21)Yau hypersurfa esand ompleteinterse tions witha fo usonissuesrelevantforstringduality al ulations.Thelasttwo n T =( ∗)n n n se tions an be read independently and report on re ent results The (algebrai ) -torus is a produ t ∗ = C\{0}of o- a and work in progress, in luding torsion in ohomology, lassi(cid:28) a- piesofthepun tured omplexplaneC C ,whi h J tionissuesandtopologi altransitions weregardasamultipli ativenabelianUg(r1o)unp.Itishen ea 2 omplexi(cid:28) ation of the real -torus . A tori vari- T ety is de(cid:28)ned as a (partial) ompa ti(cid:28) ation of in the 2 X v follonwingseTnse: It is a (normal) variety that ontains 7 an -torus as a dense open subset su h that the nat- T 0 ural a tion of the torus on itself extends to an a tion Introdu tion T X 3 of on the variety . 2 1 Tori geometry is a beautiful part of mathemati s that Thebeautyofthetori geometry omesfromthefa t 6 relates dis rete and algebrai geometry and provides thatthedataisen odedin ombinatorialtermsandthat 0 an elegant and intuitive onstru tion of many non- this stru ture an be used to derive simple formulas for h/ trivial examples of omplex manifolds [1(cid:21)6℄. Sparked sophisti ated topologi al and geometΣri al obje ts. More t by Batyrev's onstru tion of tori Calabi(cid:21)Yau hyper- pre isely, the data is given by a fan , whi h is a (cid:28)nite - p surfa es, whi h relates mirror symmetry to a ombi- olle tion of strongly onvex rational polyhedral ones e natorial duality of onvex polytopes [7℄, tori geome- (i.e. onesgeneratedby a (cid:28)nite number of latti e points h try also be ame a pivotal tool in string theory. It pro- andnot ontaininga ompleteline)su hthatallfa esof : v vides e(cid:30) ient tools for the onstru tion and analysis ones and all interse tions of any two ones also belong Xi of large lasses of models, and for omputing quan- to the fan. Σ tum ohomology and symple ti invariants [8(cid:21)11℄, (cid:28)- The spa e in whi h lives an be obtained as fol- r a bration stru tures [12(cid:21)15℄ for non-perturbative duali- l(otw,s..[.6,℄t: )If we parametrize theMtor=us{bχy : oTor→dinat∗e}s 1 n ties[16(cid:21)19℄,andlagrangiansuFbmanifoldsforopenstring T the hara ter group M ∼= n C andD-branephysi s[20(cid:21)22℄. -theory ompa ti(cid:28) ations omf ∈M an be identi(cid:28)ed with a latti eχm((t ,.Z..,,tw)h)er=e 1 n [23,24℄,whi harebasedonellipti Calabi(cid:21)Yau4-folds,is tm1...tm onrr≡esptomnds to the hara ter maybe the most promising approa h to realisti uni(cid:28)ed 1 n . Another natural latti e that omes T string models for parti le physi s [25℄, but also 3-folds with the torus an be identi(cid:28)edNwi∼=th{tλhe: g∗ro→upTo}f with torsion in ohomology have been used for su ess- algebrai one-parameter subgroups, C u ∈ N ful model building [26℄. wλuh(eτr)e=(τu1,. .o.r,rτesupno)n∈dsTto thτe∈gro∗up homomorphism In the present notes we des ribe some tools that are (χ◦λ)(τ)=χ(λ(τ))=τhχ,λfior C . The omposition provided by tori geometry and report on some re ent hχm,λui=m·u dNe(cid:28)nesaM a∼=noHnoi mal(Npa,iri)ng results.Se tion1. ontainsthebasi de(cid:28)nitionsand on- whi hmakes and Z a stru tionsoftori varieties,workingmainlywiththeho- dual paχirmof latmti ∈esM(or free abelian groups). The har- mogeneous oordinate ring. In se tion 2. we re all the a ters for an be regarded as holomorphi T stringtheory ontextinwhi hCalabi(cid:21)Yaugeometrybe- fun tionsonthetorus andhen e asrationalfun tions X N omes important for parti le physi s and des ribe the on the tori variety . WeNwil=l sNee⊗that∼=thenlatti e , tori onstru tion of hypersurfa es and omplete inter- orratheriσts∈reΣalextension R ZR R , iswhere se tions.Se tion3.explains(cid:28)brationsandtorsionin o- the ones of the fan live. ρ j = 1,...,r j homology in terms of the ombinatori s of polytopes. We (cid:28)rst onstru t the rays with , i.e. ρ ∈ Σ(1) j In se tion 4. we summarize re ent results and work in the none-dimensΣio(nn)al ones n of the fan where progress and on lude with a list of open problems. the -skeleton ontains the -dimensional ones of 1 M.KREUZER Σ z j . Re all that divXisors are formal linear ombinations in tZerm⊂s orf homogeneous oordinates ,(an∗)erx− nep×tiGonal of subvarieties of of ( omplex) odimension 1, i.e. of set C , and the identi(cid:28) ation group C as n−1 X dimension . It an bdeivs(χhomw)n that normality of X =( r−Z)/(( ∗)r−n×G), implies that the divisors χm =,0i.e. the hypersurfa es C C (4) de(cid:28)rned by the equations , are equal to sums a D D P1 jdivj(χfomr)someD(cid:28)nite sTet of irredu ible divisors j. where thΣe quotient an be shown to be (cid:16)geometri al(cid:17) if Like the j are -invariant and hen e unions the fan is simpli ial [27℄ ( f., howveve∈r, exnample 2 be- j oaPfj( amojmD) jparldeetee(cid:28)unnoeirqsbuilteisnseooafrtthmhaeatpttoshrmeusd→ae atoimjo(nmp.o)Tsi=thieohmn o,devi(cid:30)jvi χ.imeTnht=es l ρohjwo∈)o.siΣFno(g1r)daiw(cid:27)geeirve enantnlsa etottnio sftersgueN nte⊂rdaiR(cid:27)tonerrstehnattt oorRni tavioanfrtitehhteeievsrjabyayss D v ∈ N irredu ible divisors j hen e de(cid:28)ne ve tors j for primitive latti e ve tors.These are all abelian quotients j ≤ r with aj(m) = hm,vji. These ve tors are the of the variety for whi h N is the integral span of {vj}. ρ j primitive generators of the rays that onstitute the The last pie e of information that we need is the ex- Σ(1) Σ Z (zj) ∈ X \T = SDj 1-skeleton ofthefan .Ifwelo allywritetheequa- eptional set . The limit points D = {z = 0} z T j j j tion of the divisor as with a se tion thatareaddedto aredeterminedbythe onditionsun- of some lo al line bundle then we an write the torus der whi h homogeneous oordinates are allowed to van- Σ aosotrid=inQatezsjh,eiw,vijtihoanpspormoperidaetnes ehsouib eseotfonfoTrm⊆alXiz.ations, issuhb.sTethiosfitshweh eoroertdhineaitnefsorzmjaσitsi∈oanlΣloowfetdhetofanvanisehntseirms:uAl- taneously i(cid:27) there is a one ontaining all of the ρ j 1.1 Homogeneous oordinates orresponding rays . In geomeDtri al terms thisXmeans j {z } that the orresponding divisors interse t in . The j Z We now regard as global homogeneous oordinates (z :...:z ) ex eptional set hen e is the union of sets 1 r in genneralizatio∗n of the onstrnu+ 1tio\n{0o}f the Z ={(z :...:z )|z =0∀j ∈I} proje tive spa e P(z a:s..a.:Cz -)q≡uo(tλieznt:.o.f.C:λz ) vzia I 1 r j (5) 0 n 0 n j the identi(cid:28) ation . If all σ ∈Σ ρ ⊆σ j are non-zero then the oordinates forwhi hthereisno one su hthat forall j ∈I I (λq1z1 :...:λqrzr)∼(z1 :...:zr), λ∈ ∗ .Minimalindexsets withthispropertyare alled C (1) primitive olle tions. They orrespond to the maximal Z = Z S I T irredu ible omponents of . des ribe the same point of the torus with oordinates ti =Yzjhei,vji ∈ T =X \[Dj (2) 1.2 Torus orbits Pqjvj =0 vj In terms of homogeneous oordina(tes∗)trhe torus a tion iρfj ∈ Σ(1) of,thwehfearne Σ.aSrien tehethgeenveer attoorrssvojf∈thNe rabyes- aomnotuhnetsqutoottihenete(cid:27)(4e )tibvyepinadretpoefntdheentCs alian gtsionofinzdjua nedd n T X l{oqnjg}∈to aΣ(l1a)t∼=ti erof dimension the s aling exponents tXhus extends from to . The torus orbits into whi h Z Z in the identi(cid:28) ation (1) are restri ted de omposesarehen e hara terizedby theindexsets n by independent linear equations. Naively we therefore ofthevanishing oordinates.It anbeshown[3℄thatthe X O σ might expe (t rth\atZt)h/e t(or∗i )rv−anr,iety {a(nzjb)e}w=rittren\aZs torus orbitσs ∈ Σare in onOeσ-to∼=-o(ne∗) no−rrdeimspσonden e with a quotient C C where C the ones , where C is the inter- D ρ ⊂σ j j is the set of allow(ed∗)vr−alnues for the h{o(mzo)g}eneous oor- se tionofalldivisors for wOithall omplements j σ dinates and the C (cid:21)a tion on implements oftheremainingdivisors.Theorbit isanopensubva- q v =0 V P j j σ the identi(cid:28) ation (1) with . The identi(cid:28) a- riety of the orbit losure , whi h is the interse tion of D ρ ⊂σ j j tion g(zro)up→ o(trre=sponzdhse,i,vhjoiw)ever(,zto) ∈th(e k∗e)rrnel of thve nall−ddivimisoσrs for and whi h also has dimension map j i Q j for j C . If the j . N Uσ do not span the (cid:21)latti e, i.e. if the quotient Further important sets are the a(cid:30)ne open sets , X \D j G∼=N/(span {v1,...,vr}) whi hρjar6∈e tσhe interse tions of all omplementsX Z (3) with and whi h provide a overing of . The |G| > 1 relations between these sets an be summarized as niselais(cid:28)n(CGit∗e)ra/b(eCli∗a)nng∼=ro(uCp∗)orf−onXr×deGr and on,tatihnesnatdhiiss kreetre- Vσ = [ Oτ, Uσ = [ Oτ, X = [ Uσ, (6) fa tor . The tori variety an hen e be onstru ted τ⊇σ τ⊆σ σ∈Σ 2 TORICGEOMETRYANDCALABI-YAUCOMPACTIFICATIONS FF12 z@BssAB2AB@Aw@ss ss (λ2µz0:λz1:λz2:µw) ∈(C4−Z)/(C∗)2 Ja(cid:17)Ja(cid:17)aaaαaβ(cid:17)a(cid:10)(cid:17)a(cid:10) z2Ja(cid:17)Ja(cid:17)aza1aza0(cid:17)a(cid:10)(cid:17)a(cid:10)z3 Ja(cid:17)vJa(cid:17)ayaAaAaA(cid:17)aux(cid:10)a(cid:17)(cid:10) F0 s BAs(cid:0)@sz1 Z ={z0=w=0}∪{z1=z2=0} J (cid:3)(cid:3)(cid:10) J (cid:3)(cid:3)(cid:10) J (cid:3)(cid:3)(cid:10) s B(cid:0)s s J(cid:3)(cid:10) J(cid:10)(cid:3) J(cid:10)(cid:3) z 0 (2a) (2b) (2 ) W 2 211 Fig. 1: The Hirzebru h surfa e F as blowup of P . Fig. 2: Tori desingularizations of the onifold. V U v = (0,1), v + v = 0 σ σ 3 0 3 where and are disjoint unions and the last union and with linear relations , X U nv +v +v = 0 µ λ σ 0 1 2 anberestri tedtoa overingof bythe 'sformaxi- and s aling parameters and as σ mal ones .Inthetraditionalapproa h[1(cid:21)4℄atori va- shown in (cid:28)gure 1. X n = 2 riety is onstru ted by gluing its open a(cid:30)ne pat hes We onsider the ase . If we drop the vertex U v σ 3 along their interse tions the fan would onsist of 3 ones and we obtain the W 211 U ∩U =U ⊇T =U . weighted proje tive spa e P with s aling weights σ τ σ∩τ {0} (2,1,1) (7) .Thisspa eissingularbe ausethe onespanned (v ,v ) Uσ by 1 2 has volume 2. Indeed, if we drop the oor- Thepat hes are onstru tedintermsoftheirringsof w µ = 1 (1:0:0) dinate and set then is a (cid:28)xed point rχemgularfun tions, whi haregeneratedbythe hara ters of the C∗ identi(cid:28) ation for λ = −1, i.e. we have a Z2 that are nonsingular on the relevant pat h. Sin e 2 quotient singularity. The Hirzebru h surfa e F is a de- ti =Yzjhei,vji ⇒ χm =tm =Yzjhm,vji (8) singularization of th(vis,svur)fa e orresponding to a(vsu,bvdi)- 1 2 1 3 vision of the one into two basi ones m ∈ M (v ,v ) 3 2 the relevant exponent ve tors are the latti e and . The ex eptionalsetisa ordinglymodi(cid:28)ed Z = {z = w = 0}∪{z = z = 0} 0 1 2 points in the dual one to . We an now w 6= 0 µ = 1/w w σ∨ ={x∈M : hx,vi≥0 ∀ v ∈σ}. w on=si1der two ases: If (tzhe:nz :z :1)≡s( zale:sz :zto) R (9) . This yields all points 0 1 2 0 1 2 W (1:0:0) Aσ = [σ∨∩M] of P211 ex ept for its singular point , whi h Moreabstra tly,thσe∨s∩emMigroupalgebra C is ex luded due to the subdivision of the one (v1,v2) of the semigroup Uσ is, by de(cid:28)nition, thUeσring of by v3. If w = 0 we an s ale z0 6= 0 to z0 = 1 and (cid:28)nd orebgtualianredfuans ttihoensspoen trum, soSptehcamt(tAheσ)pooifnmtsaoxfimal idaenablse. (1:z1:z2:0) ∈ F2. We thus1observe that the singular Uσ point has been repla e by a P with homogeneous oor- The Zariski topology of an be onstru ted in terms (z1:z2) dinates . This pro ess of repla ing a point by a of the prime ideals and the gluing anbe workedout by proje tive spa e is alled blow-up. In the present exam- relating the hara ters in di(cid:27)erent pat hes ( f. example ple it desingularizes a weighted proje tive spa e. It an 2 below). We now state two important theorems [1(cid:21)4℄: be shown [3℄ that all singularities of tori varieties an Theorem1:Atori varietyis ompa tifandonlyifthe resolved by a sequen e of blow-ups that orrespond to fNan is omp|Σle|t=e, i.e.σif=thNe support of the fan overs the subdivisions of the fan. S latti e Σ R. Example 2: The onifold singularity We prove the only if: For an in omplete fan we on- u ∈ N \|Σ| A ordingtotheorem2these ondsour eofsingularities sider sopmλe= (λu1t1,...,λaunndtna)o∈neT-parameter familyχomf is nonsimpli ity of a one, whi h is only possible in at pointsχm(p )=λhm,uiχm(p ) . Evaluation opf least 3 dimensions. We hen e onsider a quadrati one yields λ 1 .BuUtthelimitpχomin(tp )λ→0 σ as displayed in (cid:28)gure 2b with generators annotbeλ →ont0ainedmin∈aσn∨ypat uh6∈σσbe ausehm,uiλ<d0i- v0 =(1,0,0), v1 =(0,1,0), v2 =(1,0,1), v3 =(0,1,−1) vergesas for and sothat . Pqivi =0 q =(1,1,−1,−1) and relation with so that Theorem 2: A tori variety is non-singular if and only (z :z :z :z )=(λz :λz , 1z : 1z ), σ ∈Σ 0 1 2 3 0 1 λ 2 λ 3 (10) ifall onesaresimpli ialandbasi ,i.e.ifall ones N X = (1,1,−1,−1) σ∨ are generated by a subset of a latti e basis of . or P .Thedual one hasgenerators m =(1,0,0) m =(0,1,0) m =(0,1,1) m =(1,0,−1) 0 1 2 3 To illustrate these theorems we work out two examples: , , , . A σ A ordingto (8) the oordinatering is generatedby HEixrazembrpul eh1s:uTrfvah ee=sHF(inr0z,a−erbe1r)Pu,1 vhbus=nudr(ef1las, 0oe)v,ervP1=th(−at1 ,ann) xu==χχmm02 ==zz0zz2,, yv ==χχmm13 ==zz1zz3,, (11) 0 1 2 1 2 0 3 be de(cid:28)ned by (12) 3 M.KREUZER c = 0 1 whi h are regular,xiynv=ariuavnt under tAhσe s= ali[nσg∨(∩10M) a]n∼=d on the other hand, never have . We will hen e obey the relation . Hen e C not only be interested in tori varieties themselves but [x,y,u,v]/hxy−uvi X=U σ C and an be identi(cid:28)ed with also in hypersurfa es or omplete interse tions thereof, xy = uv 4 the hypersurfa e in C , whi h has a (cid:16) onifold whi h under appropriate onditions are smooth Calabi(cid:21) singularity(cid:17) at the origin. As shown in (cid:28)gure 2 the one Yau spa es. Their de(cid:28)ning equations will be se tions of σ an be triangulated in two di(cid:27)erent ways. In the (cid:28)rst non-trivialline bundles. The relevantdataofthese bun- σ = hv ,v ,v i σ = hv ,v ,v i α 0 2 3 β 1 2 3 ase , and the dual dles is the transition fun tions between di(cid:27)erent pat h- σ∨ = (m ,m ,m ), σ∨ = (m ,m ,m ) ones are α α 0 3 β β 1 2 with es.Thisdatais loselyrelatedto the topologi aldataof m =−m =(−1,1,1) α β so that we obtain the algebras Cartier divisors, whi h by de(cid:28)nition are lo ally given in f = 0 f /f α α β A = (m ,m ,m )∋(z /z , x=z z , v =z z ), terms of rational equations with regular α α 0 3 1 0 0 2 0 3 C (13) and nonzero on the operlap of two pat hes. Sin e mul- A = (m ,m ,m )∋(z /z , u=z z , y =z z ). β β 2 1 0 1 1 2 1 3 C (14) tipli ation by a rational fun tion does not hange the Z ={z =z =0} line bundle we are interested in divisors lasses with re- 0 1 With the ex eptional set we observe x = y = u = v = 0 spe t to linear equivalen e, i.e. modulo additionof prin- div(f) that the singular point has been 1 ∋(z :z ) ipal divisors , whi h are the divisors of rational 0 1 f repla edby a P and the transition fun tions X fun tions . Cartier divisor lasses hen e determine the showthat {σα,σβ} anbeidenti(cid:28)edwiththetotalspa e Pic(X) O(−1)⊕O(−1)→ 1 Pi ard group of holomorphi line bundles. oftheranktwobundle P . Theread- Finite formal sums of irredu ible varieties of o- er may verify that the homogeneous oordinates work dimension one are alled Weil divisors (whi h may not straightforwardly (geometri al) in the simpli ial ases be Cartier, i.e. lo ally prin ipal, on singular varieties). (2a) and (2 ). In the non-simpli ial ase (2b) the quo- On a tori variety it an be shown that the Chow group A (X) tient∗(4)hast(oλbzet:aλkzen:in0th:e0)(cid:16)GIT(cid:21)(s0en:se0(cid:17):[217z℄be: a1uzse) n−1 of Weil divisors modulo linear equivalen e is all C orbits 0 1 and λ 2 λ 3 T Dj 0 ∈ Uσ generated by the -invariantdivir(rχedmu) ible dmiv∈isoMrs map to the tip of the onifold (the ategori- modulo the prin ipal divisors with , i.e. al quotient of geometri invariant theory (GIT) involves there is an exa t sequen e the dropping of (cid:16)bad(cid:17) orbits). 0→M → Σ(1) →A (X)→0 The two tori resolutions orrespond to bl1owups re- Z n−1 (16) pla ing the singular point bSy2two di(cid:27)erent P 's (whi h M ∋ m → (hm,v i) ∈ Σ(1) Σ(1) ∋ (a ) → topologi ally are 2-spheres ). We an hen e go from where j Z and Z j a D A (X) r−n one (cid:16)small resolution1(cid:17) to the other via the onifold by P j j,TheChowgroup n−1 hen ehasrank . blowing down the P to a point and blowing up that It ontains the Pi ard group as subgroup, whi h is tor- X point in a di(cid:27)erent way. This is alled a (cid:29)op transition. sion free if Σ is ompa t [3℄. D = a D There is also a non-tori possibility to resolve the A Weil divisor of the form P j j is Cartier, O(D)∈Pic(X) singularity by deforming the hypersurfa e equation to and hen e de(cid:28)nes aline bundle , if there xy−uv=ε m ∈ M σ ∈ Σ .Topologi allSy3thisamountstorepla ingthe existshman,vσi=−a for ea hρm∈axσimal one su h σ j j j Bsinygaullainrietayrb yhaan3g-espohfevraeriab.lTesh{isx ∓2ayn,ub±2evs}ee↔na{s4ilfwollwl}o2ww=se: ttihoants of Oχ(mDσ)−bmeτtweefXnortahlelpat he.sTUhσeatnrdanUsiτtiaorneftuhne n- an write the deformed onifold equation as Pl=1 l given by . If is smooth then all Weil divisors ε w =a +ib kD l l l .With itsrealandimaginarypartbe ome are Cartier. For a simpli ial fan is Cartier for some k Σ Xl≤4a2l =ε+Xl≤4b2l, Xl≤4albl =0. (15) lpinoseiatrivreeainltfeugne rtio.nFψoDr ConarNtieRrddei(cid:28)vniseodrsbythe -pie ewise ψ (v)=hm ,vi for v ∈σ For ε > 0 the four real variables a′k = ak/pε+Plb2l D σ (17) b a′b = 0 parametrize a 3-sphere and the k with Pl l l X D = T∗S3 is aalDled support fun Otio(Dn.)If is ompa t and parametrize the (cid:28)bers of the otangent bundle . P j j Cartier then is generated by global se - ψ D The topology hange between the small resolution and tions i(cid:27) the support fun tion D is onvex and is ψ hm ,v i>−a the deformation is alled onifold transition. D σ j j ample i(cid:27) is stri tly onvex, i.e. if for dimσ =n ρ 6⊂σ j and . For onvex support fun tions 1.3 Line bundles ∆ = {m∈M :hm,v i≥−a ∀ j ≤r} D j j The onifold is the standard example of a non- ompa t R (18) = {m∈M :hm,ui≥ψ (u) ∀ u∈N} D (lo al) Calabi(cid:21)Yau geometry. A ompa t tori varieties, R (19) 4 TORICGEOMETRYANDCALABI-YAUCOMPACTIFICATIONS ∆ ⊂ M D de(cid:28)nes a onvex latti e polytope R whose lat- belong to the same one ( f. the de(cid:28)nition of the ex- Z = Z S I ti e points providethe globalse tions of the line bundle eptional set in se tion 1.1). The ideal in O(D) D [D ,...,D ] R 1 r I orresponding to the divisor (the (cid:28)rst equali- Z generated by these is alled Stanley(cid:21) ∆ D J [D ,...,D ]/J D 1 r ty de(cid:28)nes also if is not Cartier). In parti ular, Reisner ideal , and Z is the Stanley(cid:21) ∆ =k∆ ∆ =∆ −m kD D D+div(χm) D and sothatthepoly- Reisner ring. M A (X) X k tope anbetranslatedinthe latti ewithout hanging The Chowgroups of avariety aregenerat- k X the divisor lass and the transition fun tions. ed by -dimensional irredu ible losed subvarietiesof D Intermsofthepolytope(18) isgeneratedbyglobal modulorationalequivalen ebydivisorsofrationalfun - ∆ {m } D k+1 D σ se tion∆s i(cid:27) n is the onvexhull of mand σ ∈isΣa(mn)- tionsonsubvaXrietiesofdimension .AFor(Xan)arbitrary D σ Σ k plei(cid:27) mis -6=dimmensionσal6=wτith∈vΣer(nti) es for tori variety it an be shown that is Vgener- σ τ σ and with for ∆. In the latter asΣe ated bσy∈thΣe(ne−quk)ivalen e lasses of orbit losures for D there is a bije tion between fa es of and ones in ones .Theinterse tionringofanon-singular Σ ∆ X D Σ or,morepri isely, isthe normal fan of : By de(cid:28)ni- ompa t tori variety is [1℄ σ Σ ∆ τ ∆ A (X )= [D ,...,D ]/ R , hm,v iD tion the ones of the normal fan∆−xof a polyxto∈peM ∗ Σ Z 1 r (cid:10) I Pj j j(cid:11) (22) arethedual onesofthe onesover where R τ ⊂ ∆ (for a de(cid:28)nition of the interse tion produ t see [3℄). is any point in the relative interior of a fa e . If 0∈M ∆ The interse tion ring an hen e be obtained from the is in the interior of , as an alwaysbe a hieved ∆ δx ∈ M Stanley(cid:21)Reisner ring by adding the linear relations by a rationaΣl∆translation of by Q, then the Pjhm,vjiDj ≃0,whereitissu(cid:30) ienttotakeformaset normal fan oin ides wi∆th◦ t⊆heNfan of ones over the M ofbasisve torsofthe -latti e.TheChowringalsode- fa es of the polar polytope R de(cid:28)ned by H2k(XΣ, ) = Ak(X, ) ∆◦ ={y ∈N : hx,yi≥−1 ∀x∈∆}. termines the homology groups Z Z . R (20) These results a tually generalize to the simpli ial proje tive ase with the ex eption that one needs to admit On smooth ompa t tori varieties it an be shown that T rational oe(cid:30) ients [2,3℄. In parti ular, for a maximal- σ v ,...,v everyχammple -invaOria(nDt)divisor is very ample. The se - dimensionalsimpli ial one spannedby j1 jn the tXioΣns oKf−su1 han(χm1 :.h..en: χempKro)videanKem=be|∆ddDin∩gMof| interse tion number of the orresponding divisors is intoP via where D ·...·D =1/Vol(σ) O(D) j1 jn (23) isthedimensionofthespa eofglobalse tionsof . X Σ Vol(σ) Σ A tori variety therefore is proje tive i(cid:27) is the where is the latti e-volume (i.e. the geometri al ∆⊂M 1/n! normal fan of a latti e polytope R [2,3℄. volume divided by the volume of a basi simplex). Summarizing,theequationsde(cid:28)ningCalabi(cid:21)Yauhy- Having dis ussed the y les we not turn to di(cid:27)eren- persurfa es or omplete interse tions will be se tions of tial forms. The anoni al bundle of a non-singular tori O(D) line bundles given by Laurent polynomials variety anbeobtainedby onsideringtherationalform dx dx f = X cmχm = X cm Yzjhm,vji (21) ω = x11 ∧...∧ xnn (24) m∈∆D∩M m∈∆D∩M j whi h by an appropriate hoi e of the orientation (i.e. m x i whose∆expo⊆neMnt ve tors span the onvexlatti e poly- the order of the lo al oordΩinnates in the a(cid:30)ne pat h- Utoσpes D R de(cid:28)nfeσd=infe/qχ.m(σ18). In an a(cid:30)ne pat h es) is a rationarl se tion of X. This implies fσ ∈thAeσlob ea lauses eti∆onD−mσ ⊂σ∨. is a regular fun tion ΩnX =OX(−jP=1Dj) (25) −D = − D P j 1.4 Interse tion ring and Chern lasses and for the anoni al divisor . The om- If a olle tion ρj1,...,ρjk of rays is not ontained in a putation of the total Chern lass requires an expression for the ( o)tangent bundle, for whi h there is an exa t single onethenthe orrespondinghomogeneous oordi- tnhaete sozrjrlesaproenndointgaldloivwiseodrstoDvjalnhisahvesinmou ltoamnemoounslyinatenrd- 0seq→uenΩ e1X → Ω1X(logD) r→es LjO(Dj) (26) sneo nt-iolinn.eaFrorreltahteioinnsteRrsIe= tiDonj1r·in..g.·wDejhke=n 0e,ewxhpeer etitthies mwhaepretaΩk1Xes(lωog=DP) tfujrndzsjo/uztjtr→oesb⊕eftjr|iDvjia[l3℄a.nAd t hael ruelsaitdioune su(cid:30) ient to take into a ount the primitive olle tions I = {j ...j } 1 k yields the total Chern lass of the tangent bundle as de(cid:28)ned by Batyrev, i.e. the minimal r c(T )= (1+D )= [V ] index sets su h that the orresponding rays do not all X Q1 j Pσ∈∆ σ (27) 5 M.KREUZER and the Todd lass thesuperpotential.Theimaginarypartsofthe omplex- t θ td(T )= r Dj =1+ 1c + 1 c2c +... i(cid:28)ed radii a thus orrespond to -angles. X Q1 1−exp(−Dj) 2 1 12 1 2 (28) r The (cid:28)rst Chern lass c1 =PDj is positive for ompa t formUnΩde=r tQhedszyjmopnle Ctri dreesd uen tdisontothaehhoolloommoorrpphhii n-- tori varieties,butiDtva+niDshe∼sf0orDthe+ oDnif∼old0be auDseo∼f formq(ai(cid:27)) =it0is invaraia≤ntru−nnder the group a tion, i.e. if thelinearrelations 0 2 , 1 3 and 1 Pj j forall .Theseequations anbein- D U(1) 3 . (Impli ations for volumes and numbers of latti e terpretedas gaugeanomaly an ellation onditions points an now be derived by applying the Hirzebru h(cid:21) in the linear sigma model [28℄. Sin e the existen e of a χ(X,E) = ch(E)Td(X) n X c = 0 R Σ 1 Riemann(cid:21)Ro h formula to holomorphi -from on is equivalent to we the ase of line bundles of Cartier divisors as des ribed, thus obtain a simple form of the Calabi(cid:21)Yau ondition q for example, in the last hapter of [3℄.) ( f. the vanishing of Pj j for the onifold). Instead of the linear ombinations (30) we an on- 1.5 Symple ti redu tion µj =|zj|2 siderallmomentmaps , whose(cid:29)owsarephase z There is another approa hto tori geometryin terms of rotations of the homogeneous oordinaGte∼=s Uj(.1A)nfter the symple ti insteadof omplexgeometry,whi hisimpor- sUy(m1)prle ti redu tionthee(cid:27)e tivepart ofthis tant be ause, in addition to the omplex stru ture, we -a tionyieldsthe ompa tpart ofthetorusa tion G ⊂T willalsoneedaK(cid:4)ahlermetri .Moreover,thesymple ti .Theimageofthe orrespondingmomentummap ∆(t ) approa h an be given a dire t physi al interpretation is a onvexpolytope, the Delzant polytope a , whose G in terms of supersymmetri gau∗ged linear sigma mod- orners orrespondto (cid:28)xedpoints of . Sin e theLieal- g G els [28℄. The idea is that the C (cid:21)quotient an be per- gebra of anbeidenti(cid:28)edwiththetherealextension N ∆(t ) g∗ ≡M formedintwosteps:We(cid:28)rstUd(i1v)ideoutthephaseparts, of the -latti e a is a polytope in R. whi h amount to ompa t quotients, and then (cid:21) n qj = 1 Example 3: For the proje tive spa e P all instead of a radial identi(cid:28) ation (cid:21) (cid:28)x the values of ap- ta = r2 propriate (cid:16)radial(cid:17) variables to ertain sizes ta that will and{|zw0i|t2h+...+|znw|2e=obrt2a}in/Ut(h1e)symple ti qtauo=tiern2t as . The value parameterize the K(cid:4)ahler metri s. of the moment map of the symple ti redu tion hen e InorderthatthequotientinheritsaK(cid:4)ahlerform(and parametrizesthe sizeofthe proje tivespa e.The image hfoernm eoan Csyrmple ti stru ture) from the natural K(cid:4)ahler orifettyheismtohmeesnimtpmleaxp{fotir≥G0⊂,rT2−onPthn1etire=sutl0tin≥g0t}or⊂i MvaR- ω =i dz ∧dz¯ =2 dx ∧dy = dr2∧dϕ , k P j j P j j P j j (29) whose fka es of odimensiton orrespond to the vanish- i z = x+iy = reiϕ ingU(o1f)k moment mapGs and hen e to (cid:28)xed points onf with we use the sGymple ti redu t- a subgroup of ∆. The fan of the tori varietynP ionformalism.Thisrequiresthattheµ :(cid:21)a rti→ongis∗hamil- is the normal fan of . We thus an onstru ∆t P∈ ans toniang∗, i.e. given by a momgent mGa=p U(1C)r−n to the a ( ompa t) torus (cid:28)bration over a polytope R dual of the Lie algebra of su h that whose (cid:28)bers degenerate to lower-dimensional tori over µ ∆ thehamiltonian(cid:29)owsde(cid:28)nedby generatethein(cid:28)nites- the fa es of . This (cid:28)bration stru ture has been used G imal -transformations. Then the symple ti redu tion by Strominger, Yau and Zaslow [20℄ for an interpreta- T theorem guarantees that the restri tion of the image of tion of mirror symmetry as -duality on a torus-(cid:28)bered t a = 1,...,n−r a the moment map to (cid:28)xed values for Calabi(cid:21)Yau manifold. indu es a symple ti stru ture on the quotient of the µ−1(t )/G t Example 4: As a non- ompa t example we onsider a a preimage for regular values of . the onifold whose K(cid:4)ahler metri is parametrized by t = |x |2 +|x |2 −|x |2 −|x |2 t = 0 In tori geometry we onsider the moment maps 0 1 2 3 . Obviously is a µ = q(a)|z |2 q(a)v =0 t=±ε2 →0 ε a Pj j j with Pj j j (30) singularvalue,1whilefor thesize ofoneof theblown-upP 'sshrinksto0.Regularvaluesofthemo- With ω−1 ∼ Pj ∂∂ϕj ∧ ∂∂rj2 the orresponding hamilto- mentmapsta leadtoasmoothsymple ti quotientand, ω−1(µ ) ∼ q(a) ∂ in parti ular, to a (proje tive) triangulation of the fan. nian (cid:29)ows a Pj j ∂ϕj generate the ompa t ∗ The orresondingsmoothK(cid:4)ahlermetri isparametrized r−n t subgroups of the C a tions of the homomorphi qu(oa)- bythe values a,whi h anbeinterpretedasizesof q r−n tients(4).Inthegraugedlinearsigmazmodel[28℄Uth(1e)r−jn H er(tXain)two- y les,ina ordwiththedimension of j 2 Σ arethe hargesof hiralsuper(cid:28)elds undera .Theregularvalues orrespondtoopen onesof µ D a gauge group and the moment maps are (cid:21)terms in the se ondaryfan, whi h parametrizes the K(cid:4)ahler mod- 6 TORICGEOMETRYANDCALABI-YAUCOMPACTIFICATIONS H21 uli spa es and whose hambers are separated by walls metri s, an be related to omplex stru ture defor- that orrespond to (cid:29)op transitions [29℄ between di(cid:27)er- mΩatioδnJsλvi∈a Ho2n1tra tion with the holomorphi 3-form, µνλ ρ¯ ent smooth phases (in the physi ist's language [28℄). At . Sin e ex hange of parti les and anti- U(1) thetransitiona y leshrinkstoapointthatisblownup parti les, as well as the orresponding sign of a Σ (t) ∆ a ordingtoadi(cid:27)erenttriangulation ontheother harge in the sigma model des ription of the Calabi(cid:21) side of the wall [29℄. Yau ompa ti(cid:28) ations, are mere onventions, physi ists ame up with the idea of mirror symmetry [36℄, whi h was used by Candelas et al. [37℄ to onstru t a mirror 2. Strings, geometry and re(cid:29)exive polytopes map between the K(cid:4)ahler and omplex stru ture moduli X spa es of a Calabi(cid:21)Yau manifold and its mirror dual At observable energy s alesstring theory leadsto an ef- X∗ , whose topologies are related by fe tive theory that orresponds to a 10-dimensional su- pergravity ompa ti(cid:28)edon a6-dimensionalmanifoldK. h11(X)=h21(X∗), h21(X)=h11(X∗). M ×K (32) 4 At small distan es spa e-time hen e looks like , M 4 where isour4-dimensionalMinkowskispa e,aslong The power series expansions of this map ould be in- as quantum (cid:29)u tuations of the metri are su(cid:30) iently terpreted as instanton orre tionsin the quantum theo- smallto allowforasemi lassi algeometri alinterpreta- ry, and thus lead to a predi tion of numbers of rational tion. urves [8,9℄. Forphenomenologi alreasonsweusuallyrequirethat supersymmetrysurvivesthe ompa ti(cid:28) ation,whi him- ∇η = 2.1 Tori hypersurfa es pliestheexisten eofa ovariantly onstantspinor 0 K on the internal manifold . In the simplest situation The beauty of the tori onstru tion of Calabi(cid:21)Yau B RR ba kground (cid:28)elds and the (cid:28)eld vanish. Cande- spa es is based on the fa t that it relates mirror sym- las, Horowitz, Strominger and E. Witten [30℄ showed metry to a ombinatorial duality of latti e polytopes, K that this implies that is a omplex K(cid:4)ahler manifold as was dis overed by Batyrev [7℄. He showed that the B with vanishing (cid:28)rst Chern lass. (The in lusion of Calabi(cid:21)Yau onditionforahypersurfa e,i.e.thevanish- (cid:28)eldswasalreadydis ussedinabeautifulpaperbyStro- ing of the (cid:28)rst Chern lass, requires as a ne essary and ∆ ⊆ M D minger [31℄, but RR (cid:29)uxes were largely for a long time su(cid:30) ient ondition that the polytope R of the O(D) untiltheirimportan eformodulistabilizationintypeII linebundle whosese tiond∆e∗(cid:28)n=es∆th◦eh⊆ypNersurfa e theories was re ognized [32℄. The investigation of their ∆is∗polar to the latti e polytope Dv R where j geometry lead to the beautiful new on ept of general- ρ ∈isΣth(1e) onvex hull of the generators of the rXays j Σ ized omplex stru tures [33(cid:21)35℄.) Expli itly, the K(cid:4)ahler of the fan of the ambient tori variety . ω Ω form andtheholomorphi 3-form oftheCalabi(cid:21)Yau A latti e polytope whose polar polytope (20) is again a η an be onstru ted in terms of as latti e polytope is alled re(cid:29)exive. Batyrev also derived ω =iη†γ γ η, Ω∼η†γ γ γ η, a ombinatorialformula for the Hodge numbers ij [i j] [i j k] (31) h11(X∆)=h2,1(X∆◦)=l(∆◦)−1−dim∆ (33) and the integraJbillit=y ωongdjitlion Nijk = 0 for the om- − X l∗(θ◦)+ X l∗(θ◦)l∗(θ) pNlexk s=truJ tlu∂reJ ki− J l∂ijJ k −wi∂thJtlhJekN+ije∂nhJuilsJtkensor codim(θ◦)=1 codim(θ◦)=2 ij i l j j l i i j l j i l is a θ θ◦ ∆ ∆◦ trivially satis(cid:28)ed for the torsion(cid:21)free metri - ompatible where and is a dual pair of fa es of and , re- η =0 l(θ) onne tion that stabilizes . spe tively. is the number of latti e points of a fa e c = 0 θ l∗(θ) 1 The ondition , whi h is equivalent to the ex- , and is the number of its interior latti e points. Ω ∆ isten eofaholomorphi 3-form ,hasbeen onje tured Mirror symmetry now amounts to the ex hange of ∆◦ by Calabi and proven by Yau to be also equivalent to and and the formula for the Hodge data makes the the existen e of a Ri hi-(cid:29)at K(cid:4)ahler metri , so that the topologi al duality (32) manifest. va uum Einstein equations are satis(cid:28)ed. The formula (33) has a simple interpretation: The h 11 Inthestandard onstru tionofanomalyfreeheterot- prin ipal ontributions to ome from the tori divi- E D ∆◦ 6 j i stringswithgaugegroup itturnsoutthat harged sors that orrespondto latti epoints in di(cid:27)erent dim(∆) parti les and anti-parti les show up in onjun tion with fromtheorigin.Thereare linearrelationsamong H11 elements of the Dolbeault ohomology groups and these divisors. The (cid:28)rst sum orresponds to the sub- H21 H11 , respe tively. While parametrizes the K(cid:4)ahler tra tion of interior points of fa ets. The orresponding 7 M.KREUZER s s s s s s s s s s s s s s s s s s s s s s @s s s s s s s s (cid:0)@ (cid:1)A (cid:0)@ s s s s s s s s s s s (cid:0)s s@s s @s s s s(cid:1)s s s (cid:0) @ (cid:1) A (cid:8) @ (cid:0) s s @s s s(cid:1)(cid:8)s s s (cid:0)s s s s@s (cid:1)s s As s s s s@(cid:0)s s s s s s s s s s s s s s s s s s s s s s @ (cid:0)@ @ (cid:1)@ (cid:0)@ (cid:1)@ s s@s (cid:0)s s@s s s s s s s@s (cid:0)s s s s s s@s @ (cid:1) @ (cid:1) (cid:0) (cid:0) s s s s (cid:0)s s s s s@s (cid:1)s (cid:0)s s s s s@s (cid:1)s s s s s s s s s s s s s s s s (cid:1)@ A (cid:0)@ @ s s s s s s s (cid:0)s s@s s s@s (cid:1) @ A @ (cid:1)s s s@s s s As s s s s@s s Fig. 3: All 16 re(cid:29)exive polygons in 2D: The (cid:28)rst 3 dual pairs are maximal/minimal and ontain all others as subpolygons, while the last 4 polygons are selfdual. divisors of the ambient spa e do not interse t a gener- useof omputersandwasa hievedin[38,39℄and[40,41℄, i Calabi(cid:21)Yau hypersurfa e. Lastly, the bilinear terms respe tively. The ode was later in luded into the soft- in the se ond sum an be understood as multipli ities ware pa kage PALP [42℄, whi h an be used for the re- of tori divisors and their presen e indi ates that only onstru tion of the data as well as for many other pur- a subspa e of the K(cid:4)ahler moduli is a essible to tori posesliketheanalysisof(cid:28)brationsandintegral ohomol- methods. ogy(seese tion3.),aswellasthe onstru tionofhigher- 6ph11+h12 ≤502 p dimensionalexamples.Allresults anbea essedonthe web page [43℄ and we just note that the numbers of re- p p (cid:29)exivepolytopes in 3d and 4d are4319and 473800776, p p p p p respe tively.Theresulting30108di(cid:27)erentHodgedataof p ppp pp p pp pp 3-folds amount to 15122 mirror pairs as shown in (cid:28)gure p ppp pppppppp pppppp 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∆∇ rtd S42eeoo.io.◦rs(cid:29)2nro=s red=neoexi v∆stitpaChveii1o∆freootone+nne1ldmsadrs,.t.ssa[ptt.u.4nih.t.l m4oeoe+,e℄tt∆m.hhpem∆Ierayonriii(rrplrnc∆yirbtoezttohnreleoev,srdiaprsu∇usseirgntmeysfiea mf)ant uthreme≥lieeor e lni−ioftanz∆∇osrmsvaδlyell◦otbom=ilowoi=Bvnnfe∇iaandhtttt1∇gyooww+rrr1eoeiii,q tav..hpul.. aoa.d.totln+h,uami∇deoar∇pnlrdipBlstireea,ycot(cid:28)oierrnnt(isvh3siinoa4nogvt)-f - r χ where is the odimension of the Calabi(cid:21)Yau and the f =0 O(∆ ) 100 500 900 i i de(cid:28)ningequations arese tions of . The de- 4 h ≤h M ∆ ⊂ M Fig. : Hh1y1p+erhsu12rfa o emsepsef rtorma f(o2r51,12151)1a2n.dT(h4e91m,1a1x)i.mal M oimnkpoowsistkioinsuomf th∆e = -∆la1tt+i e..p.o+lyt∆orpeis now duRailnttoo aa NEF (numeri ally e(cid:27)e tive) partition of the verti es of ∇⊂N The enumeration of all re(cid:29)exive polytopes has been adi(cid:27)erentre(cid:29)exivepolytope R su hthatthe on- ∇ 0 ∈ N i a hieved by Batyrev for the two-dimensional ase, as vex hulls of the respe tive verti es and only shown in (cid:28)gure 3, many years ago. In dimensions 3 and interse t at the origin [5,44℄. h pq 4, whi h are relevant for K3 surfa es and Calabi(cid:21)Yau The Hodge numbers of the orresponding om- 3-folds,respe tively, the enumerationrequired extensive plete interse tions have been omputed and shown to 8 TORICGEOMETRYANDCALABI-YAUCOMPACTIFICATIONS obey (32) in [45℄. They are summarized for arbitrary interse tions.We(cid:28)rstneedtodis usssmoothness ondi- n−r dimensiEon(t,t¯) oftheCalabi(cid:21)Yauin ageneratingpoly- tions and fo us on the hypersurfa e a∆se.⊆IfMwe onsiXder Σ nomial as thenormal fan ofare(cid:29)exivepolytope R then E(t,t¯)=X(−1)p+q hpq tpt¯q will generi ally have singularities whi h have to be re- (35) solved if they have positive dimension while point-like = X (−()tρt¯x)rtρyS(Cx,tt¯)S(Cy∗,tt¯)BI(t−1,t¯) hsiynpgeurlsaurritfaie se eaqnuabteionav.oTidheedrebsyoluatigoenne rain b heopi eerfoorfmthede I=[x,y] by the hoi e of a onvex (or oherent) triangulation of Σ ∆ n+r the fan [10,29,46℄ wh∆o◦se rays should onsist of all in terms of the ombinatorial data of the dimen- Γ({∆i}) raysoverlatti epoint∆s◦of (thisamountstoamNax\im∆a◦l sional Gor(een,stve)in one e spanned by ve trors of startriangulationof ; raysof latti epoints in i i c the form , where is a unit ve tor in R and 1 v ∈ ∆ x,y C would ontributeto andhen edestroytheCalabi(cid:21)Yau i x . In this formula label fa es of dimen- ρ Γ({∇ }) C∨ ondition if the orresponding divisors interse t the hy- x i x sion of and denotes the dual fa e of the Γ({∆ }) I = [x,y] persurfa e).ForK3-surfa essu hatriangulationalready i dual one of . The interval labels all C C leads to a smooth tori ambient spa e be ause re(cid:29)exivi- y x ones that are fa es ofB (t, t¯o)ntaining . The Batyrev(cid:21) ty implies that the fa ets are at distan e one and every I Borisov polynomials en ode ertain ombina- maximaltriangulationofapolygon onsistsofbasi sim- tSo(rCiaxl,td)a=ta(1o−ftt)hρxePfan ≥e0ltantltni( Cex[)4o5f℄.deTghreeeρpxo−ly1noamreiarels- pli es.ForCY3-foldsonlythe odimension-two onesof l (C ) n the triangulation are basi while maximal-dimensional n x latedtothenumbers oflatti epointsatdegree C ones may ontain pointlike singularities. This is still x in andhen etotheEhrhartpolynomialoftheGoren- C o.k. be ause pointlike singularities an be avoided by a x steinpolytopegenerating . (TheGorensteinpolytope ∆ generi hypersurfa e. Tori 4-fold hypersurfa es, on the C onsists of the degree 1 points of a Gorenstein one C r =1 ∆ =∆=∆ otherhand, mayhaveterminal singularitiesthat annot C 1 . In the hypersurfa e ase and .) be avoided, so that many 5-dimensional re(cid:29)exive poly- Without goinginto details let us emphasize that the topes annotbeusedforthe onstru tionofsmoothCY formula (35) ontains positive and negative ontribu- hypersurfa es. For omplete interse tions the situation tions whose interpretation in terms of individual on- D is analogous:3-folds are generi ally smooth be ause the j dim∆−3 tributions from tori divisors is, in ontrast to the odimension oftheCalabi(cid:21)Yauisalwayslarg- dim∆−4 hypersurfa e formula (33), unfortunately unknown. In erthanthe dimension of the singularlo us of X additionaltoe(cid:30) ien yproblemsin on rete al ulations Σ . ∆⊂ 4 (theformulaisimplementedinthenef-partofPALP[42℄, r > 2 If we now onsider a (cid:28)xed polytope R without but be omes quite slow for odimension ) this en- spe i(cid:28) ation of the latti e then re(cid:29)exivity, i.e. integral- ∆ ∆◦ N tailsimportanttheoreti alproblems,whi hwewill om- ity of the verti es of and of , imply that is a M ment on below. V sublatti e of the dual of the latti e generated by ∆ N N V the verti es of and that ontains the latti e ∆◦ 3. Fibrations and torsion in ohomology generated by the verti es of N ⊆N ⊆M◦. V V (36) Fibrationstru turesplayanimportantroleinstringthe- N ory, like e.g. in heteroti (cid:21)type II duality [10,16(cid:21)18℄, F- A re(cid:28)nement of the latti e amounts to a geometri al G theory [23(cid:21)25℄, but also for the onstru tion of ve tor quotientbyagroupa tion ,whi hwe alltori be ause bundlesinheteroti ompa ti(cid:28) ationswheretheyareof- ita tsdiagonallyonthehomogeneous oordinates.Su h ten ombined with non-trivial fundamental groups [26℄. are(cid:28)nementalwaysentailsadditionalquotientsingular- We now dis uss how these topologi al properties mani- ities in the ambient spa e and no ontributions to its fest themselves in ombinatorial properties of the poly- fundamental group [1℄. If, however, a Calabi(cid:21)Yau does topes that de(cid:28)ne tori Calabi(cid:21)Yau varieties. notinterse tthesingularlo usofthatquotientthenthe π 1 group a ts freely on that variety and ontributes to . This is the ase if the re(cid:28)nement of the latti e does not 3.1 Torsion in ohomology ∆◦ lead to additional latti e points of (more pre isely, We begin with a dis ussion of the fundamental group, latti epointsof odimensionone anbeignoredbe ause, whi h is trivial for every ompa t tori variety [1℄ but a ording to eq. (33), the orresponding divisors do not may be ome non-trivial for hypersurfa es and omplete interse t the hypersurfa e). 9 M.KREUZER Foragivenpair ofre(cid:29)exivepolytopes thereisonlya propriatetriangulations[12(cid:21)15,41℄.These (cid:28)brations de- N (cid:28)nitenumberoflatti es thatobey(36)andhen eon- s end from tori morphisms of the ambient spa e [3,4℄, φ:Σ→Σ N b lya(cid:28)nite numberofpossibletori freequotients.In[47℄ whi h orrespondto amap offansin and N φ : N → N b b we have shown that the fundamental group of a tori , respe tively, where is a latti e homo- σ ∈Σ CY hypersurfa e is isomorphi to the latti e quotient of morphism su h that for ea h one there is a one N N(3) σ ∈Σ σ N b b f the -latti e divided by its sublatti e generated that ontainsthe imageof . The latti e for Σ φ N by the latti e points on 3-fa es of . All fundamental the (cid:28)ber is the kernel of in . groups of tori hypersurfa es thus ome from tori quo- π 1 tients andHa2re abelian, so that is isomorphi to the ,(cid:12)lL torsionin .Wealsofounda ombinatorialformulafor ,(cid:12) Ll B , l (cid:12) L the BrauerHg3roup , whi h is the torsionN(t2h)e third o- (cid:8),(cid:8) (cid:12) L HlH homology , in terms of the sΣublatti e genBera×teBd H(cid:8)l,H (cid:12) L (cid:8)H(cid:8),l by latti e points on 2-faN /eNs o(2f) . Here, however, lH(cid:12) L(cid:8), must be a subgroup of . le %, le %, Various dualities imply that the omplete torsion in le,% π B 1 the ohomologygroupisdeterminedintermsof and K andwe onje tured,basedonsome -theoryarguments, ∆◦ ⊆N Fig. 5: CY (cid:28)bration from re(cid:29)exivese tions of R. that these groups are ex hanged with the duals of the other under mirror duality [47℄. This onje ture ould If we are interested in (cid:28)brations whose (cid:28)bers are beveri(cid:28)edfor all tori hypersurfa esby expli it al ula- Calabi(cid:21)Yauvarietiesoflowerdimensionthentherestri - 5 Σf Nf tion. Two well-known examples are the free Z quotient tion of the de(cid:28)ning equationsto the fan in needs 3 ofthe quinti2 ×and2the free Z quotientofthe CY hyper- to satisfy Ba∆ty◦re=v'∆s ◦r∩iteNrion. We hen e require that the surfa e in P P . For the omplete list of 473800776 interse tion f f is re(cid:29)exive,like the interse - re(cid:29)exive polytopes one (cid:28)nds 14 more examples of tori tionwiththehorizontalplanein(cid:28)gure5.The sear hfor 3 free quotients 9[42℄: The ellipt4i ally (cid:28)bered Z quotient tori (cid:28)brati∆on◦s hen e amounts to a sear h for re(cid:29)exive of the degree surfa e in P11133, whose group a tion se tions of with appropriate dimension (or, equiva- M on the homogeneous oordinates is given by the phases lently, for re(cid:29)exive proje tions in the -latti e, whi h (1,2,1,2,0)/3 , and 13 ellipti K3 (cid:28)brations where the was used in a sear h for K3 (cid:28)brations in [12℄). In order latti e quotient has index 2. The 16 non-trivial Brauer to guaranteethe existen eof the proje tionwe hoosea ∆◦ groups showed up, as expe ted, exa tly for the 16 poly- triangulationof f andthenextendittoatriangulation ∆◦ topesthatarepolartotheonesthatleadtoanon-trivial of (this may not always be possible if the odimen- fundamental group.For omplete interse tions it is pos- sionislargerthat1,aswaspointedoutandanalyzedby sibletohavebothanon-trivialfundamentalgroupanda Rohsiepe[15℄).Forea hsu h hoi ewe aninterpretthe ∆◦ non-trivial Brauer group at the same time and our on- homogeneous oordinatesthat orrespondto raysin f je ture was veri(cid:28)ed for a odimension-2 Calabi(cid:21)Yau for as oordinates of the (cid:28)ber and the others as parameters × 3 3 whi h both groups are Z Z [48℄. of the equations and hen e as moduli of the (cid:28)ber spa e. For hypersurfa es the geometry of the resulting (cid:28)- bration has been worked out in detail in [14℄. Even in 3.2 Fibrations the ase of omplete intersetions re(cid:29)exivity of the (cid:28)ber ∆◦ ForgeneralK3surfa esandCalabi(cid:21)Yau3-foldsthereex- polytope f ensures that the (cid:28)ber also is a omplete ∆◦ istsa riterionbyOguisofortheexisten eofellipti and interse tion Calabi(cid:21)Yau be ause a nef partition of ∆◦ K3(cid:28)brationsintermsofinterse tionnumbers[18,49℄.In automati ally indu es a nef partition of f [10℄. The r f t∆h◦e⊆torNi ontext the data of a given re(cid:29)exive polytope odimension r ofthe(cid:28)bergeneri ally oin rid>es1withthe R has to be supplemented by a triangulation of odimension Σ1of the (cid:28)bered spa e but for it may the fan, as dis ussed above,and the (cid:28)bration properties happen that f does not interse t one (or more) of the ∆ i like interse tions depend on the hosen triangulation. of the nef partition, in whi h ase the odimension Computation of all interse tion numbers for all tri- de reases [10℄. In [10℄ we performed an extensive sear h angulations is omputationally quite expensive, but for for K3-(cid:28)brations in omplete interse tions whi h, due tori Calabi(cid:21)Yau spa es there is, fortunately, a moredi- to modular properties, ould be used for all-genus al- re twaytosear hfor(cid:28)brationsthatmanifestthemselves ulationsoftopologi alstringamplitudes.Inthis sear h inthegeometryofthepolytopeandtosingleouttheap- wealsoen ounteredanexamplewherethe(cid:28)brationdoes 10