Derived Categories of Toric Varieties

Transcription

1 Michigan Math. J Deived Categoies of Toic Vaieties Yujio Kawamata 1. Intoduction The pupose of this papes to investigate the stuctue of the deived categoy of a toic vaiety. We shall pove the following esult. Theoem 1.1. Let X be a pojective toic vaiety with at most quotient singulaities, let B be an invaiant Q-diviso whose coefficients belong to the set { 1 ; } Z >0, and let X be the smooth Deligne Mumfod stack associated to the pai X, B as in [12]. Then the bounded deived categoy of coheent sheaves D b CohX has a complete exceptional collection consisting of sheaves. An object of a tiangulated categoy a T is called exceptional if { Hom p e, e C fo p = 0, = 0 fo p = 0. A sequence of exceptional objects {e 1,..., e m } is said to be an exceptional collection if Hom p e i, e j = 0 fo all p and i>j. The sequence is said to be stong if, in addition, Hom p e i, e j = 0 fo p = 0 and all i, j; it is called complete if T coincides with the smallest tiangulated subcategoy containing all the e i cf. [2]. It is usually had to detemine the explicit stuctue of a deived categoy of a vaiety. But it is known that some special vaieties, such as pojective spaces o Gassmann vaieties, have stong complete exceptional collections consisting of vecto bundles [1; 8; 9; 10]. Such sheaves ae useful fo futhenvestigation of the deived categoies see e.g. [6; 7; 14; 18]. We use the minimal model pogam fo toic vaieties as developed in [17] and coected in [15] in ode to pove the theoem. A special featue of this appoach is that, even if we stat with the smooth and nonbounday case B = 0, we ae foced to deal not only with singulaities but also with the case B = 0 because Moi fibe spaces have multiple fibes in geneal. Thus we ae inevitably led to conside the geneal situation concening Deligne Mumfod stacks even if we need esults fo smooth vaieties only. The stacky sheaves need caeful teatment because thee exist nontivial stabilize goups on the stacks cf. Remak 5.1. Received Apil 5, Revision eceived June 1,

2 518 Yujio Kawamata We stat with the Beilinson theoem fo the case of pojective spaces and then build up exceptional collections following the pocedue of the minimal model pogam. We use a coveing tick to poceed fom pojective spaces to log Fano vaieties Section 3. Then we poceed by induction on the dimension. Fist we conside a Moi fibe space in Section 4, whee the base space is assumed to have aleady a complete exceptional collection by the induction hypothesis. Though a Moi fibe space has singula fibes, the associated mophism of stacks is poved to be smooth Coollay 4.2, and we can define a complete exceptional collection on the total space by using twisted pull-backs. The behavio of deived categoies unde biational tansfomations such as divisoial contactions o flips was studied in [12]. We use this esult, togethe with esults of Section 4, in Sections 5 and 6. Indeed, the exceptional locus of a divisoial contaction o a flip has the stuctue of a Moi fibe space itself. The agument of the poof is a genealization of that in [16], which consideed the deived categoies of pojective space bundles and blow-ups of smooth vaieties with smooth centes. 2. Toic Minimal Model Pogam Let X be a pojective toic vaiety of dimension n that is quasi-smooth i.e., X has only quotient singulaities. We note that a toic vaiety is quasi-smooth if and only if it is Q-factoial. We conside a Q-diviso B on X whose pime components ae invaiant divisos with coefficients contained in the set { } 1 ; Z >0. Let X be the smooth Deligne Mumfod stack associated to the pai X, B with the natual mophism π X : X X as in [12]. The pai X, Bhas only log teminal singulaities. We wok on the log minimal model pogam fo X, B see [15; 17]. Let φ : X Y be a pimitive contaction mophism coesponding to an extemal ay with espect to K X + B. Then Y is also a pojective toic vaiety and φ is a toic mophism. If φ is a biational mophism, then the bounday diviso C on Y is defined to be the stict tansfom of B. Othewise, it will be defined late. Let N X be the lattice of 1-paamete subgoups of the tous acting on X, and let X be the fan in N X,R coesponding to X. Let w = v 3,..., v n+1 be a wall in X coesponding to an extemal ational cuve, whee the v i ae pimitive vectos in N X on the edges of w. Let v 1, v 2 N X be two pimitive vectos, each of which foms an n-dimensional cone in X when combined with w. Let D i be the pime divisos on X coesponding to the v i, and let D i be the coesponding pime divisos on X. Let 1 be the coefficients of the D i in B. Then the natual mophism π X : X X amifies along D i such that πx D i = D i. The contaction mophism is descibed by the equation whee the ae integes such that a 1 v 1 + +a n+1 v n+1 = 0, 2.1

3 Deived Categoies of Toic Vaieties 519 a 1,..., a n+1 = 1; > 0 fo 1 i α, = 0 fo α + 1 i β, < 0 fo β + 1 i n + 1; 2 α β n + 1. Note that we use slightly diffeent notation fom the liteatue in which the ae ational numbes. Since K X + B is negative fo φ,wehave > 0. The following lemma assets that the set of integes { } is well pepaed. Lemma 2.1. Let i 0 be an intege such that 1 i 0 α o β + 1 i 0 n + 1. Then the set of n + α β integes, whee 1 i α and β + 1 i n + 1 except one i = i 0, is copime fo any i 0. Poof. Let c be the lagest common diviso of these integes, and set = cā i. Then we have 0, c = 1. Let x, y be integes such that 0 x + cy = 1. Then 1 c v i 0 = 1 c v i 0 x v i = x ā i v i + yv i0 N X ; c i i =i 0 hence c = 1. One of the following cases occus. 1 Moi fibe space: β = n + 1; then dim Y = n + 1 α. 2 Divisoial contaction: β = n. 3 Small contaction: β<n. We teat these cases sepaately in the following sections. Accoding to the minimal model pogam, Theoem 1.1 follows fom the combination of Coollaies 4.4, 5.3, and Fano Case We stat with the case whee X, B is a log Q-Fano vaiety with ρ = 1. We have α = β = n + 1. In this case, thee ae no edges in X besides R 0 v i. Such a vaiety X is not necessaily a weighted pojective space, as noted in [15]. But it is coveed by a weighted pojective space via a finite mophism that is étale in codimension 1. Indeed, a weighted pojective space is chaacteized by the popety that the diviso class goup has no tosion cf. Lemma 3.1. Let N X be the sublattice of N X geneated by the v i. By equation 2.1, the toic vaiety X coesponding to the fan X in N X,R with the lattice N X is isomophic to the weighted pojective space Pa 1,..., a n+1. The natual mophism

4 520 Yujio Kawamata σ 1 : X X is étale in codimension 1. Let D i be the pime divisos on X coesponding to the v i, let X be the smooth Deligne Mumfod stack associated to the pai X, 1 i D i with the pojections πx : X X and σ 1 : X X, and let D i be the pime divisos on X such that πx D i = D i. Let be a positive intege such that s divisible by fo any i, and let N X be the sublattice of N X geneated by the vectos ṽ i = v i. We have ṽ i = 0, and the toic vaiety X coesponding to the fan X in N X,R with the lattice N X is isomophic to the pojective space P n. Let σ 2 : X X be the natual mophism, and set σ = σ 1 σ 2, σ 2 = π X σ 2, and σ = π X σ. Let D i be the pime divisos on X coesponding to the vectos ṽ i. Moeove, let N X be the sublattice of N X geneated by the v i. Obseve that the vectos v i ae not necessaily pimitive in this lattice. Lemma 3.1. and only if 1 A diviso i k id i is tosion in the diviso class goup of X if i = 0. 2 The goup of tosion diviso classes on X is dual to the quotient goup N X /N X. 3 The goup of tosion Weil diviso classes on X is dual to the quotient goup N X /N X. Poof. 1 A diviso i k id i is linealy equivalent to 0 if and only if thee exists an m M X = NX such that m, v i = k i, because the mophism π X : X X is biational. Thus i k id i is tosion if and only if thee exists m M X,R such that = 0. m, v i = k i. The latte condition is equivalent to the equality i 2 Fo m M X,R,wehavem, v i Z fo all v i if and only if m M X = NX. Theefoe, the goup of tosion diviso classes is isomophic to M X /M X. 3 is a paticula case of 2. Remak 3.2. If B = 0 i.e., if = 1 fo all i, then the diviso class goups of X and X ae isomophic. Example 3.3. Tosion diviso classes coespond to étale coveings of the stack. Fo example, let X = P n be the pojective space and let X be the smooth stack associated to the pai X, 1 n+1 H i, whee the Hi ae coodinate hypeplanes. Let H i be the pime divisos on X above the H i so that πx H i = H i fo the pojection π X : X X. Let X = P n be anothe pojective space, and let σ : X X be the Kumme coveing with Galois goup Z/ n obtained by taking the th oots of the coodinates. Then σ is étale, and we have

5 σ O X = Deived Categoies of Toic Vaieties l 1,...,l n =0 O X n l i H i + n l i H n+1. n O X l 1,...,l n =0 We note that the diect summands ae invetible sheaves on X coesponding to the tosion diviso classes. In the usual language, if we set σ = π X σ : X X then σ O 1 n X = O X l i H n+1, l 1,...,l n =0 because n n n π X O X l i H i + l i H n+1 = O X l i H n+1. Moe geneally, we have σ O X p 1 n = l i H i + p l i H n+1 and σ O X p = 1 l 1,...,l n =0 O X p + n l i H n+1. Fo example, the diect images of the sheaves O X p fo 0 p n, which geneate the deived categoy D b Coh Xsee [1], have diect summands of the fom O X q fo 0 q n cf. [11]. Lemma Let G 1 = N X /N X be the Galois goup of the coveing σ 1 : X X. Then we have the following decomposition into eigenspaces with espect to the G 1 -action: σ 1 O X d i D i = O X d i + k i D i, i k i whee the sequences of integes k = k i in the summation ae detemined by the equation k i = m, v i fo the epesentatives m of the goup of tosion Weil diviso classes M X /M X of X. 2 Let G 2 = N X /N X be the Galois goup of the coveing σ 2 : X X. Then thee is the following decomposition into eigenspaces with espect to the G 2 - action: σ 1 O X p = O X l i D i + l n+1 p n+1 D n+1 a n+1 1 l i, l 1 i n whee the sequences of integes l = l i in the summation un unde the conditions that 0 l i < and n+1 l i. 3 Let G = N X /N X be the Galois goup of the coveing σ : X X. Then we have the following decomposition into eigenspaces with espect to the G-action:

6 522 Yujio Kawamata σ O X p = O X k i a k 1 i n i D i + k n+1 p n+1 D n+1, a n+1 whee the sequences of integes k = k i satisfy the equation k i = 0. Poof. 1 is clea. 2 We have an exact sequence n+1 0 Z/ Z/ G 2 0, whee 1 in the fist tem is sent to in the second tem. Thus { G n+1 } 2 = l i Z/ ; l i = 0 mod. We have σ 1 D i = D i fo the pime diviso D i on X above D i. Since l i = l i, we obtain the fomula. We emak that O X 1 is well-defined because X has no tosion diviso classes. 3 Combining 1 and 2 yields σ O X p = O X k,l 1 i n k i + l i whee the k = k i satisfy D i + = 1 k n+1 n+1 + l n+1 p n+1 a n+1 l i D n+1, and whee the summation on l = l i is unde the estiction that 0 l i < and i l i. If we eplace k i + l i by k i, then we obtain ou assetion. Theoem An invetible sheaf O n+1 X k id i on X is an exceptional object fo any sequence of integes k = k i fo 1 i n If n+1 > n+1 Hom q > n+1 O X k i D i k i 1, then, O X k i D i = 0 fo all q, whee k = k i is anothe sequence of integes. 3 If n+1 = n+1 and n+1 k id i n+1 k i D i, then Hom q O X k i D i, O X k i D i = 0 fo all q.

7 Deived Categoies of Toic Vaieties If n+1 n+1, then Hom q O X k i D i, O X k i D i = 0 fo q = 0. 5 The set of invetible sheaves O X n+1 k id i fo 0 > i geneates the tiangulated categoy D b CohX. Poof. The canonical diviso of X is given by ω X = π X ω X O X 1D i i = O X i D i. An invetible sheaf O X i k a id i is ample if and only if i k i i > 0. Theefoe, assetions 1 4 follow immediately fom the vanishing theoem [13]. 5 follows fom a simila genealization of the Beilinson esolution theoem [1] as in [11, Sec. 5]. Indeed, the integal functo coesponding to an object e on X X given by e ={0 [σ O X n σ ñ X n]g [σ O X 1 σ 1 X 1]G [σ O X σ O X ]G 0} is isomophic to the identity functo, whee the goup G acts diagonally on the tenso poducts. Thus the deived categoy D b CohX is geneated by the diect summands of the sheaves σ O X p fo 0 p n given in Lemma Since i k i = 0, it follows that Then we calculate 0 n n k i + a k n+1 p n+1 n+1 a n+1 = p n+1. k i + a k n+1 p n+1 n+1 a n+1 n+1 i 1 1si n n+1 whee we set = s i fo some integes s i. + 1, Coollay 3.6. Let X, B be a Q-factoial pojective toic vaiety such that K X + B is ample, ρx = 1, and the coefficients of B belong to the set { 1 ; } Z >0. Let X be the smooth Deligne Mumfod stack associated to the pai

8 524 Yujio Kawamata X, B. Then the deived categoy D b CohX has a stong complete exceptional collection consisting of invetible sheaves. Poof. Thee is a finite numbe of isomophism classes of the set of invetible sheaves O X i k id i fo 0 i > i. 4. Moi Fibe Space We conside a toic Moi fibe space φ : X Y with espect to K X + B. This fibation is not necessaily locally tivial, because thee may be multiple fibes. Yet it becomes locally tivial afte taking coveings, as we now show. Lemma 4.1. Let Y 0 be an invaiant open affine subset of Y, and let X 0 = φ 1 Y 0. Then thee exist finite sujective toic mophisms τ X0 : X 0 X 0 and τ Y0 : Y 0 Y 0, with a toic sujective mophism φ 0 : X 0 Y 0, that satisfy the following conditions: 1 τ X0 is étale in codimension 1; 2 φ τ X0 = τ Y0 φ 0 ; 3 X 0 is isomophic to the diect poduct of Y 0 and a weighted pojective space, and φ 0 coesponds to the pojection. Poof. Let N Y be the lattice of 1-paamete subgoups of the tous fo Y, and let Y be the fan in N Y,R coesponding to Y. We take the wall w descibed in the fomula 2.1 such that the coesponding extemal ational cuve is contained in X 0. We have / α N Y = N X Rv i N X. Let h: N X N Y be the pojection. We wite hv i = s i v i fo pimitive vectos v i in N Y and positive integes s i, whee α + 1 i n + 1. Then these v i give the set of edges of an n + 1 α-dimensional cone σ 0 in Y coesponding to Y 0. Let E i be the pime divisos on Y coesponding to the vectos v i. Now X 0 coincides with the toic vaiety coesponding to the fan X h 1 σ 0 in N X,R. Let N X 0 be the sublattice of N X geneated by the v i fo 1 i n + 1, and let N Y 0 esp. N Y of N 0 Y be geneated by the v i esp. hv i fo α + 1 i n + 1. Let X 0 be the toic vaiety coesponding to the fan X h 1 σ 0 in N X 0,R, and let Y 0 esp. Y 0 be the one coesponding to the cone σ 0 in N Y 0,R esp. N Y Y ae étale in codi- Then the natual mophisms τ X0 : X 0 X and τ Y 0 : Y 0 mension 1, while τ Y 0 : Y 0 Y 0 is not in geneal. Since α v i = 0, 0,R. it follows that X 0 is isomophic to the poduct of Y 0 with a weighted pojective space Pa 1,..., a α.

9 Deived Categoies of Toic Vaieties 525 We define the bounday Q-diviso C on Y by assigning coefficients s i 1 s i to the ieducible components E i, whee the s i ae as defined in the poof of Lemma 4.1. We note that, even if we stat with the nonbounday case B = 0, the natually defined bounday diviso C on Y is nonzeo in geneal because thee may be multiple fibes fo φ. Let Y be the smooth Deligne Mumfod stack associated to the pai Y, C. Then Lemma 4.1 implies the following coollay. Coollay 4.2. The natual mophism ψ : X Y is smooth. Theoem The functo ψ : D b CohY D b CohX is fully faithful. Let D b CohY k denote the full subcategoy of D b CohX defined by α D b CohY k = ψ D b CohY O X k i D i fo a sequence of integes k = k i with 1 i α. 2 If α > α > α k i k i 1, then Hom q D b CohY k, D b CohY k = 0 fo all q, whee k = k i is anothe sequence of integes. 3 If α = α k i and O X α k i k i D i / ψ D b CohY, then Hom q D b CohY k, D b CohY k = 0 fo all q. 4 The set of subcategoies D b CohY k fo α α 0 > geneates the tiangulated categoy D b CohX. Poof. 1 By [4] o [5], it is sufficient to pove the following statement: If A and B ae skyscape sheaves on Y of length 1, then the natual homomophism Hom p A, B Hom p ψ A, ψ Bis bijective. This follows fom the facts that a X 0 is isomophic to the poduct of Y 0 with a weighted pojective space Pa 1,..., a α and b the natual homomophism of Galois goups N X /N X 0 N Y /N Y is sujective. 0 Fo 2 and 3, we use a spectal sequence α E p,q 2 = H p Y, HomA, B R q ψ O X k i D i α Hom p+q ψ A, ψ B O X k i D i fonvetible sheaves A, B on Y. The diect image sheaves vanish in ou case since the elative canonical diviso fo ψ is given by

10 526 Yujio Kawamata α ω X/Y = OX D i > and since an invetible sheaf O α X k id i is ψ-ample if and only if α 0 [13]. 4 In geneal, a full tiangulated subcategoy B of a tiangulated categoy A is said to be ight esp. left admissible if A is geneated by B and B esp. B and B, whee B esp. B denotes the ight esp. left othogonal complement of B in A [3]. The tiangulated subcategoy T of D b CohX geneated by the subcategoies D b CohY k is admissible by [3, 1.12, 2.6, 2.11]. Theefoe, it is sufficient to pove that the left othogonal T consists of zeo objects. Let A be an abitay skyscape sheaf of length 1 on X suppoted at a point P. Then, by Theoem 3.6, thee exists a skyscape sheaf B of length 1 on Y suppoted at Q = φp such that A is contained in the subcategoy geneated by the sheaves of the fom ψ B O α X k id i fo 0 α Theefoe, A is contained in T. Hence T = 0, because such A span D b CohX [4] o [5]. > α Coollay 4.4. Assume that D b CohY has a complete exceptional collection consisting of sheaves. Then so has D b CohX.. 5. Divisoial Contaction We conside a toic divisoial contaction φ : X Y. Hee K X + B is negative fo φ, and C = φ B is the stict tansfom. Let D be the exceptional diviso of the contaction. Then the estiction φ : D F = φdis a Moi fibe space, which was teated in the pevious section. Let Y be the stack associated to the pai Y, C. We note that thee is no mophism of stacks fom X to Y in geneal. But thee is still a fully faithful functo : D b CohY D b CohX by [12, Thm. 4.22]. Indeed, let W be the nomalization of the fibe poduct X Y Y, and let µ: W X and ν : W Y be the pojections. Then = µ ν is fully faithful. We egad D b CohY as a full subcategoy of D b CohX though this functo. Let E i = φ D i be the pime divisos on Y coesponding to the edges v i fo 1 i n. These E i fo 1 i α ae the divisos that contain the cente F of the blow-up φ, and D = D n+1 is the exceptional diviso. Let E i be the pime divisos on Y coesponding to the E i. The following fomuls poved in the poof of [12, Thm. 4.22]: n O Y k i E i = O X k i D i, k n+1 = n+1 b n+1 n fo any integes k i with 1 i n, whee we put b n+1 = a n+1 > 0.

11 Deived Categoies of Toic Vaieties 527 Let be a positive intege such that s divisible by fo 1 i n + 1. We set = s i. Let s = s 1,..., s n+1 be the geatest common diviso, and set s i = s s i. Then the factional pat of the ational numbe n+1 n n = k i s i b n+1 can take an abitay value in the set { 1 0, s n+1,..., s } n+1 1 s n+1 when we vay the sequence k, because s 1,..., s n+1 = 1. The Moi fibe space φ : D F is descibed as follows. The lattice of 1- paamete subgoups fo D is given by N = N X /Zv n+1. We wite v i mod Zv n+1 = t i v i fo 1 i n, whee the t i ae positive integes and the v i ae pimitive vectos in N. Let t = a 1 t 1,..., a n t n be the geatest common diviso, and set t i = tā i ; then ā 1 v 1 + +ā n v n = 0. We define a Q-diviso B on D by putting coefficients t i 1 t i to the pime divisos D i = D i D fo 1 i n. We also define a Q-diviso C on the base space of the Moi fibe space F using B as in the pevious section. Let D and F be the smooth stacks associated to the pais D, B and F, C, espectively. Then thee ae induced mophisms of stacks ψ : D F and j : D X. Let D i be the pime divisos on D coesponding to the D i fo 1 i n. Then we have j O X D i = O D D i. We note that D i D = 1 t i D i in the usual language. Remak 5.1. If n+1 > 1, then the action of the stabilize goup at the geneic point of D n+1 is nontivial. Hence j O X kd n+1 = 0onD if k is not divisible by n+1. Indeed, we have Homj O X kd n+1, A = HomO X kd n+1, j A = 0 fo any sheaf A on D in this case. Fo example, let X be an affine line with a point P, and let X be the stack associated to the pai X, 1 P with a point P above P. Then we have j O X kp = 0ifk is not divisible by, whee j : P X is the natual mophism. Fom a esolution 0 O X k 1P O X kp O P kp 0 it follows that L q j O P kp is isomophic to O P if q = 0 and k 0 modulo o if q = 1 and k 1 modulo, and is zeo othewise. Thus Hom q O P kp, O P = Hom q O P kp, j O P = Hom q Lj O P kp, O P is nonzeo if and only if q = 0 and k 0 modulo o q = 1 and k 1 modulo. s n+1

12 528 Yujio Kawamata Theoem The functo j ψ : D b CohF D b CohX is fully faithful. Let D b CohF k denote the full subcategoy of D b CohX defined by D b CohF k = j ψ D b CohF O X k i D i fo a sequence of integes k = k i with 1 i n If 0 > n+1 n+1, then Hom q D b CohY, D b CohF k = 0 fo all q. 3 If n+1 > n+1 > n+1 k i 1, then Hom q D b CohF k, D b CohF k = 0 fo all q, whee k = k i is anothe sequence of integes. 4 If n+1 = n+1 k i but j O n+1 X k i k i D i = 0 o j O X k i k i D i / ψ D b CohF, then Hom q D b CohF k, D b CohF k = 0 fo all q. 5 The subcategoies D b CohY and the D b CohF k fo 0 > geneate the tiangulated categoy D b CohX. i Poof. 1 It is sufficient to pove that the natual homomophism Hom q L, L Hom q j ψ L, j ψ L is bijective fo all q and all locally fee sheaves L and L on F, because these sheaves span the categoy D b CohF. We have an exact sequence 0 O X D n+1 O X O Dn+1 0 with an isomophism O Dn+1 = j O D. Hence O D fo q = 0, L q j j O D = j O X D n+1 fo q = 1, 0 othewise, whee j O X D n+1 is an invetible sheaf on D if n+1 = 1 and is zeo othewise.

13 Deived Categoies of Toic Vaieties 529 If n+1 > 1, then Hom q j ψ L, j ψ L = Hom q Lj j ψ L, ψ L = Hom q ψ L, ψ L = Hom q L, L as equied. If n+1 = 1 then we know that j O X D n+1 is negative fo ψ, while j O n+1 X D n+1 is ample fo ψ because n+1 > 0. Since α α ω D/F = OD D i = j O X D i, we may calculate Hom q L 1 j j ψ L, ψ L = Hom q ψ L, ψ L j O X D n+1 = 0 by the elative vanishing theoem fo ψ [13]. Theefoe, we have also ou assetion in this case. 2 It is sufficient to pove Hom q O X k i D i, j ψ A O X k i D i = 0 fo all integes q, fo all sheaves A on F, and fo the sequences k and k unde the additional conditions that k n+1 = n n+1 k i, b n+1 0 > By the fist condition, we have hence 0 > 0. k i < b n+1 n+1 ; k i k i > n By the elative vanishing theoem fo ψ,wehave Hom q O X k i D i, j ψ A O X k i D i. = Hom q j O X k i k id i, ψ A = 0. 3 is similaly poved as in 1. Since 0 > n+1 k i k i > n + b n+1 n+1, it follows that

14 530 Yujio Kawamata R ψ j O X k i k id i = R ψ j O X D n+1 + k i k id i = 0 by the elative vanishing theoem fo ψ. Thus Hom q j ψ L, j ψ L O X k i k id i = 0 fo all q and fo all locally fee sheaves L and L on F. 4 is simila to 3. 5 We shall pove that the left othogonal T to the tiangulated subcategoy T of D b CohX geneated by these subcategoies consists of zeo objects as in the poof of Theoem 4.3. Let A be an abitay skyscape sheaf of length 1 on X suppoted at a point P. If P/ D n+1 then A T ; othewise, thee is a point P on D such that P = j P. Then, by Theoem 3.6, thee exists a skyscape sheaf B of length 1 on F suppoted at Q = ψ Psuch that A is contained in the subcategoy geneated by the sheaves of the fom j ψ B O n+1 X k id i fo b n+1 > n+1 i. If b n+1 n+1 > n+1 0, then k n+1 = n+1 n b n+1. Theefoe, A is contained in T and hence T = 0. Coollay 5.3. Assume that D b CohY has a complete exceptional collection consisting of sheaves. Then so has D b CohX. 6. Log Flip We conside a toic small contaction φ : X Y with the log flip φ + : X + Y. Hee K X + B is negative fo φ and K X + + B + is ample fo φ +, whee B + = φ + 1 φ B is the stict tansfom. The agument fo log flips in this section is supisingly simila to that fo the divisoial contactions in the pevious section. Let X + be the smooth Deligne Mumfod stack associated to the pai X +, B +. Then thee is a fully faithful functo : D b CohX + D b CohX by [12, Thm. 4.23]. Indeed, let W be the nomalization of the fibe poduct X Y X +, and let µ: W X and ν : W X + be the pojections. Then = µ ν is fully faithful. We egad D b CohX + as a full subcategoy of D b CohX though this functo. Let D + i = φ + 1 φ D i be the pime divisos on X + coesponding to the edges v i fo 1 i n + 1, and let D + i be the coesponding pime divisos on X +. The following fomuls poved in the poof of [12, Thm. 4.23]: O X + k i D + i = OX k i D i if

15 Deived Categoies of Toic Vaieties < i=β+1 whee we put b i = fo β + 1 i n + 1. Let D be the exceptional locus of the contaction φ. Then we have D = n+1 i=β+1 D i, and the estiction φ : D F = φd is a Moi fibe space descibed as follows. The lattice of 1-paamete subgoups fo D is given by N = N X / n+1 i=β+1 Zv i. We wite v i mod n+1 i=β+1 Zv i = t i v i fo 1 i β, whee the t i ae positive integes and the v i ae pimitive vectos in N. Let t = a 1 t 1,..., a β t β be the geatest common diviso and let t i = tā i. Then b i, ā 1 v 1 + +ā β v β = 0. We define a Q-diviso B on D by putting coefficients t i 1 t i to the pime divisos D i = D i D fo 1 i β. We also define a Q-diviso C on the base space of the Moi fibe space F using B as befoe. Let D and F be the smooth Deligne Mumfod stacks associated to the pais D, B and F, C, espectively. Then thee ae induced mophisms of stacks ψ : D F and j : D X. Let D i be the pime divisos on D coesponding to the D i fo 1 i β. Then We note that D i D = 1 t i j O X D i = O D D i. D i in the usual language. Theoem The functo j ψ : D b CohF D b CohX is fully faithful. Let D b CohF k denote the full subcategoy of D b CohX defined by D b CohF k = j ψ D b CohF O X k i D i fo a sequence of integes k = k i with 1 i n If 0 > n+1 n+1, then Hom q D b CohX +, D b CohF k = 0 fo all q. 3 If n+1 > n+1 > n+1 k i 1, then Hom q D b CohF k, D b CohF k = 0 fo all q, whee k = k i is anothe sequence of integes. 4 If n+1 = n+1 k i but j O n+1 X k i k i D i = 0 o j O X k i k i D i / ψ D b CohF, then

16 532 Yujio Kawamata Hom q D b CohF k, D b CohF k = 0 fo all q. 5 The subcategoies D b CohX + and the D b CohF k fo 0 > geneate the tiangulated categoy D b CohX. i Poof. 1 We shall pove that the natual homomophism Hom q L, L Hom q j ψ L, j ψ L is bijective fo all q and fo all locally fee sheaves L and L on F. We have an exact sequence Hence 0 O X n+1 i=β+1 i=β+1 D i 2 n+1 i=β+1 O X D i O X j O D 0. q n+1 L q j j O D = i=β+1 j O X D i. O X D i The sheaf j O X i I D i fo any subset I {β +1,..., n +1} is eithenvetible o zeo, and it is negative fo ψ if it is not a zeo sheaf. Since α α ω D/F = OD D i = j O X D i, we may calculate Hom q L p j j ψ L, ψ L = Hom q ψ L, ψ L p n+1 i=β+1 j O X D i = 0 fo p>0and fo any q by the elative vanishing theoem fo ψ, because n+1 > 0. Hence Hom q j ψ L, j ψ L = Hom q j j ψ L, ψ L = Hom q ψ L, ψ L = Hom q L, L fo any q, as equied.

17 Deived Categoies of Toic Vaieties It is sufficient to pove Hom q O X k i D i, j ψ A O X k i D i = 0 fo all integes q, fo all sheaves A on F, and fo the sequences k and k unde the additional conditions that It follows that 0 > 0 0 > k i < k i k i i=β+1 b i,. > α By the elative vanishing theoem fo ψ,wehave Hom q O X k i D i, j ψ A O X k i D i. = Hom q j O X k i k id i, ψ A = 0. 3 is similaly poved as in 1. Since 0 > n+1 b i, it follows that n+1 i=β+1 k i k i R ψ j O X D i + k i k id i = 0 i I > α + fo any subset I {β + 1,..., n + 1} by the elative vanishing theoem fo ψ. Thus Hom q j ψ L, j ψ L O X k i k id i = 0 fo all q and fo all locally fee sheaves L and L on F. 4 is simila to 3. 5 We shall pove that the left othogonal T to the tiangulated subcategoy T of D b CohX geneated by these subcategoies consists of zeo objects. Let A be an abitay skyscape sheaf of length 1 on X suppoted at a point P. If P is not above a point in D then A T ; othewise, thee is a point P on D such that P = j P. Then, by Theoem 3.6, thee exists a skyscape sheaf B of

18 534 Yujio Kawamata length 1 on F suppoted at Q = ψ Psuch that A is contained in the subcategoy geneated by the sheaves of the fom j ψ B O X n+1 k id i fo i=β+1 b i > i Theefoe, A is contained in T and hence T = 0.. Coollay 6.2. Assume that D b CohX + has a complete exceptional collection consisting of sheaves. Then so has D b CohX. Refeences [1] A. A. Beilinson, Coheent sheaves on P n and poblems of linea algeba, Funct. Anal. Appl , [2] A. I. Bondal, Repesentations of associative algebas and coheent sheaves, Izv. Akad. Nauk SSSR Se. Mat , Russian; English tanslation in Math. USSR-Izv , [3] A. I. Bondal and M. M. Kapanov, Repesentable functos, See functos, and econstuctions, Izv. Akad. Nauk SSSR Se. Mat , , 1337 Russian; English tanslation in Math. USSR-Izv , [4] A. I. Bondal and D. O. Olov, Reconstuction of a vaiety fom the deived categoy and goups of autoequivalences, Compositio Math , [5] T. Bidgeland, Equivalences of tiangulated categoies and Fouie Mukai tansfoms, Bull. London Math. Soc , [6], T -stuctues on some local Calabi Yau vaieties, J. Algeba , [7] A. L. Goodentsev, S. A. Kuleshov, and A. N. Rudakov, t-stabilities and t-stuctues on tiangulated categoies, Izv. Ross. Akad. Nauk Se. Mat , Russian; English tanslation in Izv. Math , [8] M. M. Kapanov, Deived categoy of coheent sheaves on Gassmann manifolds, Izv. Akad. Nauk SSSR Se. Mat , [9], On the deived categoy and K-functo of coheent sheaves on intesections of quadics, Izv. Akad. Nauk SSSR Se. Mat , Russian; English tanslation in Math. USSR-Izv , [10], On the deived categoies of coheent sheaves on some homogeneous spaces, Invent. Math , [11] Y. Kawamata, E[q]uivalences of deived categoies of sheaves on smooth stacks, Ame. J. Math , [12], Log cepant biational maps and deived categoies, J. Math. Sci. Univ. Tokyo , [13] Y. Kawamata, K. Matsuda, and K. Matsuki, Intoduction to the minimal model poblem, Algebaic geomety Sendai, 1985, Adv. Stud. Pue Math., 10, pp , Kinokuniya and Noth-Holland, Tokyo and Amstedam, [14] E. Maci, Some examples of moduli spaces of stability conditions on deived categoies, pepint, math.ag/ [15] K. Matsuki, Intoduction to the Moi pogam, Spinge-Velag, New Yok, 2002.

19 Deived Categoies of Toic Vaieties 535 [16] D. O. Olov, Pojective bundles, monoidal tansfomations, and deived categoies of coheent sheaves, Russian Acad. Sci. Izv. Math , [17] M. Reid, Decomposition of toic mophisms, Aithmetic and geomety, vol. II, Pog. Math., 36, pp , Bikhäuse, Boston, [18] A. N. Rudakov, Rigid and exceptional vecto bundles and sheaves on a Fano vaiety, Poceedings of the Intenational Congess of Mathematicians, vol. 1 Zuich, 1994, pp , Bikhäuse, Basel, Depatment of Mathematical Sciences Univesity of Tokyo Komaba, Meguo Tokyo, Japan kawamata@ms.u-tokyo.ac.jp