Effective birationality

Transcription

1 Chapter 1 Effective birationality Jingjun Han (Peking University) & Chen Jiang (IPMU, Tokyo) 1.1 Introduction In this chapter, we consider the effective birationality of anti-pluricanonical systems of ɛ-lc Fano varieties, i.e. [8, Theorem 1.2]. Instead of proving [8, Theorem 1.2], we will prove two special cases of [8, Theorem 1.2] under additional assumptions, which is crucial for the proof of other main theorems. The main results are Propositions and We briefly explain the strategy of showing the effective birationality. Given an ɛ-lc Fano variety X of dimension d, consider m N to be the smallest number such that mk X defines a birational map. The goal is to show that m is bounded from above. One important idea is that, in order to show that m is bounded from above, we first show that m n is bounded from above, where n N is the smallest number such that vol( nk X ) > (2d) d. Once m n is bounded from above, then vol( mk X) is bounded from above. Since mk X defines a birational map, this implies that X is birationally bounded, and we can then construct a nice log bounded family P, see Lemma Now we can work on the log bounded family P. From the assumption that X is ɛ-lc, we can construct a subɛ-lc pair (W, Λ W ) whose support is in P. On the other hand, under some additional assumptions, we may construct an effective Q-divisor 1

2 2 CHAPTER 1. EFFECTIVE BIRATIONALITY L such that (W, Λ W +L) is not sub-klt. By comparing the singularities of this two pairs, we know that numerically L cannot be too small, see Proposition On the other hand, by the construction of L, we know that L can be arbitrarily small as long as m is unbounded, which shows the boundedness of m. Then it remains to show that m n is bounded from above. The idea is to construct isolated non-klt centers by nk X, that is, for a general point x X, we need to construct an effective Q-divisor Q knk X where k is independent of X, so that (X, ) has an isolated non-klt center at x. Once this is done, then by using vanishing theorem, it is easy to see that 2knK X defines a birational map and we can then conclude that m n 2k. From the definition of n, it is easy to construct an effective Q-divisor Q nk X, so that (X, ) has a non-klt center G containing x. The problem here is that the dimension of G could be positive. If vol( mk X G ) is bounded from below, then it is easy to construct a new non-klt center with dimension strictly smaller than G by standard methods, and after finitely many steps, we get isolated non-klt centers So finally we need to consider the case that vol( mk X G ) is not bounded from below, which in fact implies that vol( mk X G ) is bounded from above. Along with the fact that mk X G defines a birational map on G as x is general, such G is log birationally bounded, and we can then construct a nice log bounded family P by applying Lemma Similarly to the proof of boundedness of m, we can work on the log bounded family P. From the assumption that X is ɛ-lc, we can construct a sub-ɛ-lc pair (F, Λ F ) whose support is in P. On the other hand, under some additional assumptions, we may construct an effective Q-divisor L such that (F, Λ F + L ) is not sub-klt. Here note that we work on F, which is a birational model of G. In order to consider the singularities of F, we need to consider the singularities of G which is induced from This method are also used to show the effective birationality of pluricanonical systems on algebraic varieties of general type, see [24, 61, 64, 27]. The difference is that for varieties of general type, the center G is again of general type since x is general, and one may apply induction on G to conclude that vol( mk X G) is indeed bounded from below. In our case, this induction step fails, so we need further discussions.

3 1.2. SINGULARITIES IN BOUNDED FAMILIES 3 that of X. This requires a nice adjunction theory, which is highly nontrivial, see [8, Section 3]. By comparing the singularities of this two pairs, we know that numerically L cannot be too small, see Proposition On the other hand, by the construction of L, we know that L can be arbitrarily small as long as m n is unbounded, which shows the boundedness of m n. 1.2 Singularities in bounded families Proposition Let ɛ R >0 and let P be a bounded set of couples. Then there is λ R >0 depending only on ɛ, P satisfying the following. Let (X, B) be a projective sub-pair and let T be a reduced divisor on X. Assume (X, B) is sub-ɛ-lc and (X, Supp(B >0 + T )) P, L is an effective R-Cartier R-divisor on X, L R N for some R-divisor N on X, Supp(N >0 ) T, and the coefficients of N are at most λ. Then (X, B + L) is sub-klt. Remark Proposition here is slightly more generalized than [8, Proposition 4.2]. The difference is that we do not assume that B is effective. This turns out to be very useful in applications. Proof of Proposition Since P is a bounded set of couples, we can find a log resolution φ : W X of (X, Supp(B >0 + T )) and write {proposition 4.2} K W + B W = φ (K X + B) + E, such that (W, Supp(B W + T W )) belongs to a bounded set of couples depending only on P, where B W 0 and E 0 have no common components, and T W is the sum of the birational transform of T and all φ-exceptional divisors. Now (W, B W ) is ɛ-lc. Let L W = φ L, and N W = φ N. Then there is an integer m > 0 depending only on P so

4 4 CHAPTER 1. EFFECTIVE BIRATIONALITY that the coefficients of N W >0 is at most mλ. It suffices to prove that (W, B W + L W ) is klt. By the boundedness there exists a very ample divisor H W on W such that, T W H d 1 < M for some number M > 0 depending only on P. Let λ = mult w L W L W H d 1 W ɛ mm W, then for any point w W, we have = N W H d 1 W By Exercise 1.6.2, (W, B W + L W ) is klt. mλt W H d 1 W < ɛ. lemma bdd family 1} 1.3 Construction of bounded families Lemma Let d N and v 1, v 2 R >0. Let P be a set of (Y, C, D) satisfying the following: Y is a normal projective variety of dimension d, C is a (possibly zero) reduced integral divisor on Y, and D is an effective nef R- Cartier R-divisor on Y, there exists a log resolution φ : Z Y and a base point free divisor H Z on Z such that H Z defines a birational map and φ D H Z, write Σ Z to be the support of φ (B + D) and all φ-exceptional divisors, then vol(k Z + Σ Z + 2(2d + 1)H Z ) v 1, vol(d) v 2. Then P is log birationally bounded. More precisely, there exists a log bounded family P of couples such that for each (Y, D) P, there exists a couple (Z, Σ Z ) P satisfying the following. Y is birational to Z, (Z, Σ Z ) is log smooth, where we may take a higher model of Z such that the induced map ψ : Z Z is a morphism. Σ Z consists of the support of the strict transform of B + D and divisors exceptional over Y,

5 1.3. CONSTRUCTION OF BOUNDED FAMILIES 5 the coefficients of ψ φ D are bounded from above by a number depending only on d, v 1, and v 2. Proof. This is just [26, Lemmas 3.2 and 2.4.2(4)]. To be more precise, consider the birational morphism Z Z defined by H Z, then ( Z, Σ Z) is log bounded, where Σ Z is the pushforward of Σ Z. Taking a log resolution of ( Z, Σ Z), we get a log smooth couple (Z, Σ Z ) which is still in a log bounded family, say P, where Σ Z is the sum of the strict transform of Σ Z and all exceptional divisors over Z, in other words, Σ Z consists of the support of strict transform of D and divisors exceptional over Y. We may take a higher model of Z such that the induced map ψ : Z Z is a morphism. By the boundedness, there exists a very ample divisor H Z on Z and a number b depending only on P such that bψ H Z H Z is big. Recall that ψ ψ H Z = H Z by the construction. Now the coefficients of ψ φ D are bounded by the following intersection number: ψ φ D H d 1 Z = φ D ψ H d 1 Z vol(φ D + ψ H Z ) vol(φ D + bh Z ) vol((1 + b)φ D) (1 + b) d v 2. Here the first inequality holds since both D and H Z are nef. Notation Let X be a klt Fano variety and m N such that mk X defines a birational map. Then we may take a resolution φ : W X such that φ ( mk X ) A W + R W where A W is base point free and R W is the fixed part of φ ( mk X ). We may pick a A W general in its linear system. Denote m := φ A W + φ R W mk X. Here we remark that R W is in general a Q-divisor, but φ R W is an integral Weil divisor, Lemma Let d N and ɛ, v R >0. Let P be a set of varieties such that for each X P, the following hold: {notation AW} {lemma bdd family 2}

6 6 CHAPTER 1. EFFECTIVE BIRATIONALITY X is an ɛ-lc Fano variety of dimension d, m N is the smallest number such that mk X defines a birational map, n N is the smallest number such that vol( nk X ) > (2d) d, n > 1, and m n < v. Then there exists a log bounded family P of couples such that for each X P, there exists a couple (W, Σ W ) P satisfying the following. X is birational to W, (W, Σ W ) is log smooth, where we may take a higher model of W such that the induced map ψ : W W is a morphism. Σ W consists of the support of the strict transform of m and divisors exceptional over X where m is defined in Notation 1.3.2, the coefficients of ψ φ m are bounded from above by a number depending only on d, ɛ, and v. Proof. By the assumption, mk X defines a birational map. Take φ : W X, A W, R W, m as in Notation By the minimality of n, vol( (n 1)K X ) (2d) d. Hence ( m ) d ( 2m ) d vol( mk X ) vol( (n 1)KX ) vol( (n 1)KX ) (4vd) d. n 1 n Write Σ W to be the support of A W +R W and all φ-exceptional divisors, then vol(k W + Σ W + 2(2d + 1)A W ) vol(k X + φ Σ W + 2(2d + 1)φ A W ) vol(k X + φ A W + φ R W + 2(2d + 1)φ A W ) (4d + 3) d vol(φ R W + φ A W ) = (4d + 3) d vol( mk X ) (4vd(4d + 3)) d. Now we may apply Lemma to (Y, C, D, Z, H Z ) = (X, 0, m, W, A W ) to finish the proof.

7 1.4. EFFECTIVE BIRATIONALITY FOR FANO VARIETIES WITH GOOD Q-COMPLEMENTS7 tion 4.4} 1.4 Effective birationality for Fano varieties with good Q- complements Proposition Let d N and ɛ, δ R >0. Then there exists a number v depending only on d, ɛ, and δ satisfying the following. Assume X is an ɛ-lc Fano variety of dimension d, m N is the smallest number such that mk X defines a birational map, n N is a number such that vol( nk X ) > (2d) d, and nk X + N Q 0 for some Q-divisor N with coefficients δ. Then m n < v. Proof. Step 1. Construct a family of non-klt centers with bounded volumes. By [8, 2.16(2)], since vol( nk X ) > (2d) d, there is a bounded family of subvarieties of X such that for two general points x, y in X, there is a member G of the family and an effective Q-divisor Q (n+1)k X such that (X, ) is lc near x with a unique non-klt place whose center contains x, that center is G, (X, ) is not klt at y. Recall that this family is given by finitely many morphisms V j T j of projective varieties with surjective morphisms V j X and G is a general fiber of one of V j T j. Denote k := max{dim V j dim T j }. We do induction on k. If k = 0, that is, dim G = 0 for all general G, then (n + 2)K X is potentially birational, hence K X (n + 2)K X defines a birational map. By the minimality of m, m n + 1 2n, which implies that m n 2. Now assume that k > 0. Define l N to be the smallest number such that vol( lk X G ) > d d for all general G with dim G > 0. Then there exists j such that if G is a general fiber of V j T j, then dim G > 0

8 8 CHAPTER 1. EFFECTIVE BIRATIONALITY and vol( (l 1)K X G ) d d. Now since vol( lk X G ) > d d, by [8, 2.16(2)], after replacing n by n + (d 1)l, we may construct a new bounded family of subvarieties of X corresponding to non-klt centers of (X, Q (n+(d 1)l+1)K X ) while k is strictly decreased. Hence by the induction, there exists a number v depending only on d, ɛ, and m δ such that n+(d 1)l+1 < v. If m > 2v (d 1)l, then 2m < 2v (n + (d 1)l + 1) < 2v n + m + 2v which implies that m < 4v n and we are done. Hence from now on, we may assume that m 2v (d 1)l. If l = 1, then m 2v (d 1) 2v (d 1)n and again we are done. Hence we may assume that l > 1 and therefore m 4v d(l 1). In this case, we have vol( mk X G ) (4v d) k vol( (l 1)K X G ) (4v d) k d d if G is a general fiber of V j T j In summary, we constructed a family V j T j of projective varieties with a surjective morphism V j X such that if G is a general fiber of V j T j, then dim G = k > 0, there exists an effective Q-divisor Q (n + 1)K X and there is a unique non-klt place of (X, ) whose center is G, and vol( mk X G ) v 2 for a number v 2 depending only on d, ɛ, and δ. Step 2. Construct a bounded family. Take F to be the normalization of G. By [8, Theorem 3.9] and ACC for LCT [27, Theorem 1.1], there is an effective Q-divisor Θ F with coefficients in a DCC set Φ depending only on d such that we may write (K X + ) F = K F + F := K F + Θ F + P F where P F is pseudo-effective. By increasing n and adding to, we may assume that P F is effective and big. Since G is general, by [8, Lemma 3.11], we may write K X F = K F + Λ F for some sub-boundary Λ F such that (F, Λ F ) is sub-ɛ-lc and Λ F Θ F F.

9 1.4. EFFECTIVE BIRATIONALITY FOR FANO VARIETIES WITH GOOD Q-COMPLEMENTS9 By the assumption, mk X defines a birational map. Take φ : W X, A W, R W, m as in Notation Take a log resolution f : F F of (F, F ) such that the induced map F W (which is well-defined since G is general) is a morphism. Denote A F := A W F which is base point free and defines a birational map on F. Denote M F := m F. Note that f M F = (A W + R W ) F A F. Take Σ F to be sum of the strict transforms of M F, Supp Θ F, and f-exceptional divisors. Fix a rational number ɛ (0, ɛ) such that ɛ < min Φ >0. By the definition of Φ, Supp(Θ F ) Θ F ɛ. Note that by [8, Lemma 3.10], Supp(M F ) Θ F + M F since m is an integral Weil divisor. Recall that by [8, Lemma 2.30], K F +(2k+1)A F is big. Hence vol(k F + Σ F + 2(2k + 1)A F ) vol(k F + Σ F + 2(2k + 1)A F + ɛ 1 (K F + (2k + 1)A F )) vol(k F + Σ F + 2(2k + 1)A F + ɛ 1 (K F + (2k + 1)A F )) vol((1 + ɛ 1 )K F + Supp(M F ) + Supp(Θ F ) + (2 + ɛ 1 )(2k + 1)A F ) vol((1 + ɛ 1 )K F + Θ F + M F + ɛ 1 Θ F + (2 + ɛ 1 )(2k + 1)A F ) vol((1 + ɛ 1 )(K F + Θ F + P F ) + m(2 + ɛ 1 )(2k + 2)( K X F )) vol(((1 + ɛ 1 )n + (2 + ɛ 1 )(2k + 2)m)( K X F )). We may always assume that n m otherwise there is nothing to prove. Hence, by Step 1, vol(k F + Σ F + 2(2d + 1)A F ) v 1 for a number v 1 depending only on d, ɛ, and δ. Now we may apply Lemma to (Y, C, D, Z, H Z ) = (F, Supp Θ F, M F, F, A F ) to construct a log bounded family P of couples such that there exists a couple (F, Σ F ) P satisfying the following. F is birational to F, (F, Σ F ) is log smooth, Σ F consists of the support of M F + Θ F and divisors exceptional over F,

10 10 CHAPTER 1. EFFECTIVE BIRATIONALITY the coefficients of M F are bounded from above by a number, say u, depending only on d, ɛ, and δ. Here we may take a higher model of F such that the induced map g : F F is a morphism, M F := g f M F, and Θ F := g f Θ F. Step 3. Apply Proposition Write K F + Λ F := g f (K F + Λ F ), K F + F := g f (K F + F ). Since Λ F Θ F, Supp(Λ >0) Σ F F by the construction. Moreover, the coefficients of Λ F is at most 1 ɛ since (F, Λ F ) is sub-ɛ-lc. Hence, (F, Λ F ) is again sub-ɛ-lc. Consider J F = 1 δ N F and let D be a component of J F. Then by [8, Lemma 3.10], µ D ( F + J F ) µ D (Θ F + J F ) 1. {proposition 4.5} In particular, (F, F + J F ) is not sub-klt. Note that K F + F + J F Q (1+ 1 δ )nk X F is nef. Hence (F, F +g f J F ) is not sub-klt by Exercise Rewrite this sub-pair as (F, Λ F + F Λ F + g f J F ). Applying Proposition to the sub-pair (F, Λ F ), L = F Λ F + g f J F and N = ( n+1 m + n mδ )M F Q L, there is λ R >0 depending only on ɛ, P such that u( n+1 m + n mδ ) > λ, which implies that m n < u(2+δ) λδ. Proposition Let d N and ɛ, δ R >0. Then there exists a number m N depending only on d, ɛ, and δ satisfying the following. Assume that X is an ɛ-lc Fano variety of dimension d such that K X + B Q 0 for some Q-divisor B with coefficients δ. Then mk X defines a birational map. Proof. Take m N the smallest number such that mk X defines a birational map, and n N the smallest number such that vol( nk X ) > (2d) d. By the assumption, the assumptions of Proposition are

11 1.5. EFFECTIVE BIRATIONALITY FOR NEARLY CANONICAL FANO VARIETIES11 satisfied, hence there exists a number v depending only on d, ɛ, and δ such that m/n < v. We may assume that n > 1, otherwise m < v and there is nothing to prove. Now the assumptions of Lemma are satisfied. Hence there exists a log bounded family P of couples such that there exists a couple (W, Σ W ) P satisfying the following. X is birational to W, (W, Σ W ) is log smooth, where we may take a higher model of W such that the induced map ψ : W W is a morphism. Σ W consists of the support of the strict transform of m and divisors exceptional over X where m is defined in Notation 1.3.2, the coefficients of ψ φ m are bounded from above by a number, say u, depending only on d, ɛ, and v. Write K W + Λ W := ψ φ K X. Note that Supp(Λ W ) Σ W by the construction since Λ W is exceptional over X. Since X is ɛ-lc, the coefficients of Λ W are at most 1 ɛ. Hence (W, Λ W ) is sub-ɛ-lc. By the assumption, the coefficients of 1 δ B is at least 1, hence (X, 1 δ B) is not klt. Also note that K X + 1 δ B Q 1 δ δ K X is ample where without loss of generality we may assume that δ < 1. Hence (W, Λ W + 1 δ ψ φ B) is not sub-klt by Exercise Note that 1 δ ψ φ 1 B Q mδ ψ φ ( m ). Applying Proposition to the sub-pair (W, Λ W ), L = 1 δ ψ φ B and N = 1 mδ ψ φ ( m ), there is λ R >0 depending only on ɛ, P such that u mδ > λ, which means that m < u λδ. 1.5 Effective birationality for nearly canonical Fano varieties Proposition Let d N. Then there exist numbers τ (0, 1) and m N depending only on d satisfying the following. If X is a τ-lc Fano variety of dimension d, then mk X defines a birational map. Proof. Fix τ ( 1 2, 1) which is sufficiently closed to 1. (We may get the restriction on τ which depending only on d during the proof and state them in the end). Take m N the smallest number such that {proposition 4.7}

12 12 CHAPTER 1. EFFECTIVE BIRATIONALITY mk X defines a birational map, and n N the smallest number such that vol( nk X ) > (2d) d. Step 1. Similar to Proposition 1.4.1, we show that there exists a number v depending only on d such that m/n < v. Step 1.1. Construct a bounded family. Note that we may apply Steps 1 and 2 of Proof of Proposition here (take ɛ = 1 2 and ignore δ). We will keep the notation and constructions from there. Recall that we can construct a family V j T j of projective varieties with a surjective morphism V j X such that if G is a general fiber of V j T j, then dim G = k > 0, there exists an effective Q-divisor Q (n + 1)K X and there is a unique nonklt place of (X, ) whose center is G, and vol( mk X G ) v 2 for a number v 2 depending only on d. Take F to be the normalization of G. Then we can construct a log bounded family P of couples such that there exists a couple (F, Σ F ) P satisfying the following. F is birational to F, (F, Σ F ) is log smooth, Σ F consists of the support of M F + Θ F and divisors exceptional over F, the coefficients of M F are bounded from above by a number, say u, depending only on d. Here we may take a higher model of F such that the induced map g : F F is a morphism, M F := g f M F, and Θ F := g f Θ F. Recall that by the construction, (F, Λ F ) is sub-τ-lc. Recall that by the construction, Λ F Θ F F and the coefficients of Θ F belong to a DCC set Φ depending only on d. Denote min Φ >0 = b. We may assume that τ > 1 b 4 and fix τ = 1 b 2. Step 1.2. Reduce to the case that (F, F ) is τ -lc and Λ F Θ F = 0. To the contrary, if (F, F ) is not τ -lc, then (F, F + c( F Λ F )) is not lc for c = τ τ τ 2τ b by the discrepancy computation. Since K F + F +c( F Λ F ) Q (n+c(n 1))K X F is nef, (F, F +c( F Λ F )) is

13 1.5. EFFECTIVE BIRATIONALITY FOR NEARLY CANONICAL FANO VARIETIES13 not klt by Exercise where Λ F is defined by K F + Λ F = g f (K F + Λ F ). Rewrite this sub-pair as (F, Λ F + (1 + c)( F Λ F )). Recall that (F, Λ F ) is a sub-τ-lc (which is also sub- 1 2-lc) pair with Supp(Λ>0 ) Σ F F. Hence we may apply Proposition to L = (1 + c)( F Λ F ) and N = (c+1)(n+1) m M F Q L, there is λ R >0 depending only on P such that u (c+1)(n+1) m > λ, which implies that m n < 2u(1+c) λ which is a number depending only on d. Hence in this case we can complete Step 1. So we may assume that (F, F = Θ F + P F ) is τ -lc for any choice of P F 0 from now on. Note that by the construction, Λ F Θ F F and the coefficients of Θ F belong to a DCC set Φ depending only on d. On the other hand, the coefficients of F are at most 1 τ. Then the coefficients of Θ F are at most 1 τ < b = min Φ >0, which implies that Λ F Θ F = 0. Step 1.3. Reduce to the case that K F is pseudo-effective with κ σ (K F ) = 0. Assume that K F is not pseudo-effective, by the boundedness of P and [8, Lemma 2.21], there exists a number λ depending only on P such that K F + λ Σ F is not pseudo-effective. By the construction, K F + Λ F + 1 m M F Q 0. In particular, K F + Λ >0 + 1 F m M F is pseudo-effective. Note that the support of Λ >0 + 1 F m M F is contained in Σ F by the construction and its coefficients are at most 1 τ + u m. Hence 1 τ + u m > λ. We may assume that τ > 1 λ 2u 2, which implies that m λ and we are done in this case. Hence we may assume that K F is pseudo-effective from now on. Assume that κ σ (K F ) > 0. Recall that by the construction, vol( mk X F ) v 2. By the boundedness of P and [8, Lemma 2.24], there is a number p (dependent only on v 2 ) such that vol(pk F + g A F ) > 2 k v 2 which is equivalent to vol(pk F + A F ) > 2 k v 2 since F is smooth. On the other

14 14 CHAPTER 1. EFFECTIVE BIRATIONALITY hand, ( m ) vol n K F + A F ( m ) vol n K F mk X F ( m ) vol n (K F + F ) mk X F ( m ) = vol n ( nk X F ) mk X F = vol( 2mK X F ) 2 k v 2. Hence m n p in this case and we complete Step 1. Hence we may assume that κ σ (K F ) = 0 from now on. Step 1.4 Reduce to Proposition Since κ σ (K F ) = 0, by [8, Lemma 2.19], there is r N depending only on P such that h 0 (rk F ) 0. Then h 0 (rk F ) 0 and rk F T F for some integral divisor T F 0. Suppose that T F 0, then (F, (1 + r) F + T F ) is not klt. Since K F + (1 + r) F + T F Q (1 + r)(k F + F ) Q (1 + r)nk X F is nef, (F, Λ F + F Λ F + g f (r F + T F )) is not sub-klt by Exercise Recall that (F, Λ F ) is a sub-τ-lc (which is also sub- 1 2-lc) subpair with Supp(Λ >0) Σ F F. Hence we may apply Proposition to L = F Λ F + g f (r F + T F ) and N = n+1+rn m M F Q L, there is λ R >0 depending only on P such that u n+1+rn m > λ, which implies that m n < u(2+r) λ. Hence in this case we complete Step 1. Suppose that T F = 0, then h 0 ( rk X F ) = h 0 ( r(k F + Λ F )) = h 0 ( rλ F ) > 0, since Λ F 0 by Step 1.2. By Step 1.2 and [8, Proposition 3.13], after replacing r with a multiple depending only on P, h 0 ( rnk X ) 0. Hence nk X +N Q 0 for some Q-divisor N with coefficients at least 1 r. Now we can apply Proposition to show that m n < v for a number v depending only on d. Step 2. Similar to Proposition 1.4.2, we show that m is bounded from above by a number depending only on d.

15 1.6. EXERCISES 15 By Step 1, m/n < v. We may assume that n > 1, otherwise m < v and there is nothing to prove. Now the assumptions of Lemma are satisfied. Hence there exists a log bounded family P of couples such that there exists a couple (W, Σ W ) P satisfying the following. X is birational to W, (W, Σ W ) is log smooth, where we may take a higher model of W such that the induced map ψ : W W is a morphism. Σ W consists of the support of the strict transform of m and divisors exceptional over X where m is defined in Notation 1.3.2, the coefficients of ψ φ m are bounded from above by a number, say u, depending only on d, ɛ, and v. Note that K W is not pseudo-effective as X is Fano, by the boundedness of P and [8, Lemma 2.21], there exists a number λ depending only on P such that K W + λ Σ W is not pseudo-effective. Write K W + Λ W := ψ φ K X. Note that Supp(Λ W ) Σ W by the construction since Λ W is exceptional over X. Since X is τ-lc, the coefficients of Λ W are at most 1 τ. By the construction, K W + Λ W + 1 m ψ φ m Q ψ φ (K X + 1 ) m m Q 0. which implies that K W +Λ >0+ 1 W m ψ φ m is pseudo-effective. Note that Λ >0+ 1 W m ψ φ m is supported on Σ W with coefficients at most 1 τ + u m, hence 1 τ + u m > λ. We may assume that τ > 1 λ 2, which implies that m < 2u λ and we are done. Finally, we summarize the restrictions on τ. We need τ > min Φ>0 in Step 1.1, τ > 1 λ λ 2 in Step 1.3, τ > 1 2 in Step 2. All these restrictions depends only on d. 1.6 Exercises Exercise Let X be a projective variety of dimension d, H a very ample divisor, and L an effective R-divisor. Then mult x L {exercise eff bir 4}

16 16 CHAPTER 1. EFFECTIVE BIRATIONALITY exercise eff bir 3} exercise eff bir 2} exercise eff bir 1} L H d 1 for any point x X. Exercise Let X be a smooth variety, B an effective R-divisor with simple normal crossing support, and L an effective R-divisor. Fix ɛ > 0. Suppose that (X, B) is ɛ-lc (or equivalently, the coefficients of B are at most 1 ɛ) and mult x L < ɛ for any point x X. Then (X, B + L) is klt. Exercise Let φ : W Y be a birational contraction between normal projective varieties. Let D be an R-divisor on W. Show that vol(d) vol(φ D). In particular, if D is big, so is φ D. Exercise Let X and Y be two birational equivalent normal projective varieties. Take a common resolution φ : W X and ψ : W Y. Assume that (X, ) is a sub-pair such that (X, ) is not sub-klt and K X + is nef. Assume that K Y + Y := ψ φ (K X + ) is R-Cartier. Show that (Y, Y ) is not sub-klt.

17 Chapter 2 Special BAB Jingjun Han (Beijing University) & Chen Jiang (IPMU, Tokyo) 2.1 Introduction The goal of this chapter is to prove the following theorem, which is known as the Special BAB. Theorem (Special BAB). Let d N and ɛ, δ R >0. Consider projective varieties X satisfying: {theorem 1.4} (X, B) is ɛ-lc of dimension d for some boundary B, B is big and K X + B R 0, and the coefficients of B are at least δ. Then the set of such X forms a bounded family. Hacon and Xu, [28, Theorem 1.3], proved Theorem assuming the coefficients of B belong to a fixed DCC set of rational numbers, relying on the special case when K X is ample [27, Corollary 1.7]. We will need their result in the proof of the theorem. To be specific, we will show that K X has a klt n-complement. The theorem can be viewed as a special case of the Borisov Alexeev Borisov conjecture. In fact, if we take δ = 0 in Theorem 2.1.1, it is equivalent to the Borisov Alexeev Borisov conjecture, which is the main theme of this book. 17

18 18 CHAPTER 2. SPECIAL BAB Now, we briefly explain the idea of the proof of Theorem Run a ( K X )-MMP and take the canonical model, by applying [28, Theorem 1.3], we reduce to the case that X is Fano. By [27, Theorem 1.6], we only need to show the log birational boundedness of (X, B). By Proposition 1.4.2, mk X defines a birational map for some m depending only on d. In order to show the log birational boundedness, we can apply Lemma The only remain thing is to show that vol( K X ) is bounded from above. {proposition 5.2} Finally, in order to show that vol( K X ) is bounded from above, the idea is to construct isolated non-klt centers by K X. In fact, if vol( K X ) is large enough, then we can construct an effective Q-divisor Q ak X where a > 0 is sufficiently small, so that (X, ) has a non-klt center G containing x. By connectedness principle [44, 17.4], the dimension of G is positive. Then we can use the method similar to the proof of Proposition to conclude an upper bound of vol( K X ). The last part is the most technical result in this chapter. Proposition Let d N, and ɛ, δ R >0. Then there is a number v depending only on d, ɛ, δ such that for any X as in Theorem 2.1.1, vol( K X ) < v. 2.2 Boundedness of volumes {lemma 5.1} In this section we will prove Proposition Lemma Let (X, B),, G, F, Θ F, P F be as in [8, 3.9], where G is not necessarily a general member of a covering family. Assume that X is Q-factorial near the generic point of G, P F is big, and (X, B + ) has a non-klt center intersecting G. Then for any ɛ (0, 1), we can choose P F 0 so that (F, Θ F + P F ) is not ɛ-lc.

19 2.2. BOUNDEDNESS OF VOLUMES 19 Proof. Recall (Y, Γ Y + N Y ) and (S, Γ S + N S ) of the proof of [8, 3.10]. Let I G be a maximal non-klt center intersecting G. We can assume that some component of Γ Y S maps onto I. Since Γ Y is the non-klt locus of (Y, Γ Y + N Y ), by the connectedness principle, Γ Y is connected near the fiber of Y X over any closed point x I G. Thus, some component of Γ Y S intersects the fiber of S G over x, hence ( Γ Y S) S 0 which is vertical over F which in turn implies (S, Γ S + N S ) has a divisorial non-klt center vertical over F. By [8, 3.7], there is a resolution F F, such that if Ω F and R F are the discriminant and moduli parts of adjunction on F for (S, Γ S + N S ) over F, then there is a component of Ω F with coefficient 1 and R F is pseudo-effective. Since Ω F Θ F and since P F is big, replacing Θ F + P F by ((1 t)ω F + tθ F ) + ((1 t)r F + tp F ) for some 0 < t 1, we may assume that P F 0 in its R-linear equivalence class, so that (F, Θ F + P F ) is not ɛ-lc. Proof of Proposition Take a > 0 to be the real number such that vol( ak X ) = (4d) d + 1. It suffices to prove that a is bounded from below. Step 1. Reduce to the case when X is Fano of dimension d 2 and B is a Q-boundary. When d = 1, the proposition clearly holds. We may assume that d 2. Let X X be a Q-factorialization. Note that X is of Fano type. Running a ( K X )-MMP, we get a ( K)-minimal model X min. Since B is big, K Xmin is nef and big. By the Basepoint-free Theorem (cf. [47, Theorem 3.3]), K Xmin is semiample, and defines a birational contraction X min X can, where K Xcan is ample. By Exercise 2.4.1, (X can, B can ) is ɛ-lc. By Exercise 2.4.2, vol( K X ) = vol( K Xcan ), hence by replacing (X, B) by (X can, B can ), we may assume that X is Fano. Moreover, by modifying B, we may assume that B is a Q-boundary (cf. Exercise 2.4.3). Step 2. Construct a family of non-klt centers with bounded volumes.

20 20 CHAPTER 2. SPECIAL BAB Fixed a smooth point y X, since vol( a 2 K X) > d d, by [42, 6.1], we may find 1 Q a 2 K X, such that mult y 1 > d, and (X, 1 ) is not lc at y. Since vol( a 2 K X) > (2d) d, by [8, 2.16(2)], there is a bounded family of subvarieties of X such that for a general point x in X, there is a member G of the family and an effective Q-divisor 2 Q a 2 K X such that (X, 2 ) is lc near x with a unique non-klt place whose center contains x and that center is G. Here we always consider general x such that x Supp( 1 ), hence (X, := ) is lc near x with a unique non-klt place whose center contains x and that center is G. Recall that this family is given by finitely many morphisms V j T j of projective varieties with surjective morphisms V j X and G is a general fiber of one of V j T j, and we can assume that for each j the points on T j corresponding to the G are dense. Denote k := max{dim V j dim T j }. We show that a is bounded from below by the induction on k. When k = 0, then G = {x} is an isolated non-klt center and x, y belong to two disconnected non-klt centers. By connectedness principle [44, 17.4], (K X + ) (1 a)k X cannot be ample. Hence a 1. Now we may assume that k > 0, let b N to be the smallest number such that vol( bk X G ) d d + 1 for all general G with dim G > 0. We can assume that the equality is obtained for G which are general fibers of the morphism V j T j for some j. By [8, 2.16(2)], after replacing a by a + (d 1)b and replacing, we can replace each family of G with one having strictly smaller dimension. By the induction, we may assume that a + (d 1)b > 2(d 1)µ for some µ > 0 depending only on d, ɛ, δ. We may assume that a < (d 1)µ, otherwise we are done. Therefore b µ. From now on, we consider general G as a general fiber of the morphism V j T j and by the construction, vol( K X G ) = 1 (d d + 1) 1 (d d + 1) which b k µ d is bounded from above. Step 3. Construct a bounded family. Take F to be the normalization of G. By [8, Theorem 3.9] and ACC for LCT [27, Theorem 1.1], there is an effective Q-divisor Θ F with

21 2.2. BOUNDEDNESS OF VOLUMES 21 coefficients in a DCC set Φ depending only on d such that we may write (K X + ) F = K F + F := K F + Θ F + P F where P F is pseudo-effective. By increasing a and adding to, we may assume that P F is effective and big. Since G is general, by [8, Lemma 3.11], we may write K X F = K F + Λ F for some sub-boundary Λ F such that (F, Λ F ) is sub-ɛ-lc and Λ F Θ F F. By Proposition 1.4.2, there exists a natural number m depending only on d, ɛ, δ, such that mk X defines a birational map. Take φ : W X, A W, R W, m as in Notation Take a log resolution f : F F of (F, F ) such that the induced map F W (which is well-defined since G is general) is a morphism. Denote A F := A W F which is base point free and defines a birational map on F. Denote M := 1 δ B, M F := M F. Note that mδf M F R (A W + R W ) F A F. Take Σ F to be the sum of the strict transforms of M F, Supp Θ F, and f-exceptional divisors. Fix a rational number ɛ (0, ɛ) such that ɛ < min Φ >0. By the definition of Φ, Supp(Θ F ) Θ F ɛ. Note that by [8, Lemma 3.10], Supp(M F ) Θ F + M F since the coefficients of M are at least 1. Recall that by [8, Lemma 2.30], K F + (2k + 1)A F is big. Hence vol(k F + Σ F + 2(2k + 1)A F ) vol(k F + Σ F + 2(2k + 1)A F + ɛ 1 (K F + (2k + 1)A F )) vol(k F + Σ F + 2(2k + 1)A F + ɛ 1 (K F + (2k + 1)A F )) vol((1 + ɛ 1 )K F + Supp(M F ) + Supp(Θ F ) + (2 + ɛ 1 )(2k + 1)A F ) vol((1 + ɛ 1 )K F + Θ F + M F + ɛ 1 Θ F + (2 + ɛ 1 )(2k + 1)A F ) vol((1 + ɛ 1 )(K F + Θ F + P F ) + δ 1 ( K X F ) + m(2 + ɛ 1 )(2k + 1)( K X F )) vol(((1 + ɛ 1 )(a 1) + δ 1 + (2 + ɛ 1 )(2k + 1)m)( K X F )). Hence, by Step 2, vol(k F + Σ F + 2(2d + 1)A F ) v 1 for a number v 1 depending only on d, ɛ, and δ.

22 22 CHAPTER 2. SPECIAL BAB Now we may apply Lemma to (Y, C, D, Z, H Z ) = (F, Supp Θ F, mδm F, F, A to construct a log bounded family P of couples such that there exists a couple (F, Σ F ) P satisfying the following. F is birational to F, (F, Σ F ) is log smooth, Σ F consists of the support of M F + Θ F and divisors exceptional over F, the coefficients of M F are bounded from above by a number, say u, depending only on d, ɛ, and δ. Here we may take a higher model of F such that the induced map g : F F is a morphism, M F := g f M F, and Θ F := g f Θ F. Step 4. Apply Proposition We may assume that a < 1. Then (K X + ) Q (1 a)k X is ample. By the connectedness principle [44, 17.4], the non-klt locus of (X, ) is connected. By the construction, since x is not contained in Supp( 1 ), (X, ) is not lc at y with a non-klt center not equal to G. By the connectedness, (X, ) has a non-klt center intersecting G, but not equal to G. By Lemma 2.2.1, we can choose P F 0 such that (F, F ) is not ɛ 2 -lc. Since (F, Λ F ) is sub-ɛ-lc, (F, F + ( F Λ F )) is not klt by the discrepancy computation. Now since (X, B) is ɛ-lc and G is general, by [8, Lemma 3.11], (F, B F ) is sub-ɛ-lc where K F + B F = (K X + B) F Q 0 and B F = Λ F + B F. Moreover, B F + 2( F Λ F ) = F + ( F Λ F ) + B F, and hence (F, B F + 2( F Λ F )) is not klt. In addition, K F + B F + 2( F Λ F ) Q 2aK X F is ample. Write K F + B F := g f (K F + B F ). Then by Exercise 1.6.4, (F, B F + 2g f ( F Λ F )) is not sub-klt and by Exercise 2.4.3, (F, B F ) is sub-ɛ-lc.

23 2.3. PROOF OF THE SPECIAL BAB 23 Since Λ F Θ F, we have Supp(B >0 F ) Supp(Θ F + B F ) = Supp(Θ F + M F ) which implies that Supp(B >0) Σ F F. Applying Proposition to the sub-pair (F, B F ), L = 2g f ( F Λ F ) and N = 2aδM F Q L, there is λ R >0 depending only on ɛ, P such that 2aδu > λ, which implies that a > λ 2uδ. 2.3 Proof of the special BAB The following special case of [28, Theorem 1.3] will be used to reduce Theorem to the Fano case. Theorem ([28, Theorem 1.3]). Let d, m N. Consider projective varieties X satisfying: {thm:hx1.3} (X, B) is klt of dimension d for some boundary B, B is big and K X + B Q 0, and mb is an integral Weil divisor. Then the set of such X forms a bounded family. The following result will will be used to further reduce Theorem to showing the log birational boundedness. Theorem ([27, Theorem 1.6]). Let d N and δ, ɛ R >0. Consider the set of log pairs (X, ) such that {thm:hmx1.6} (X, ) is ɛ-lc, K X + is ample, and the coefficients of are at least δ. If the set of such (X, ) is log birationally bounded, then it is log bounded.

24 24 CHAPTER 2. SPECIAL BAB Proof of Theorem Let X X be a Q-factorialization. Running a ( K X )-MMP, we get a ( K)-minimal model X min. Since B is big, K Xmin is nef and big. By the Basepoint-free Theorem (cf. [47, Theorem 3.3]), K Xmin is semiample, and defines a birational contraction X min X can, where K Xcan is ample. Denote by B can the birational transform of B on X can. Note that K Xcan + B can R 0 and (X can, B can ) is again ɛ-lc by Exercise If the set of such X can is bounded, then there is a natural number n, such that nk Xcan is Cartier (cf. Exercise 2.4.4). Moreover, by the Effective Basepoint-free Theorem [41], we may assume that nk Xcan is base point free. This implies that there is a klt n-complement of K Xcan, which in turn gives a klt n-complement of K X (cf. Exercise 2.4.5). Hence, by Theorem 2.3.1, X forms a bounded family. Finally, in order to show the boundedness of X can, we show that (X can, B can ) forms a log bounded family. We may take can = (1 + t)b can for some sufficiently small t > 0, such that (X can, can ) is ɛ 2 -lc where K Xcan + can Q tk Xcan is ample. By Theorem 2.3.2, it suffices to prove that (X can, can ) forms a log birationally bounded family, or equivalently, (X can, B can ) is log birationally bounded. By Proposition and Proposition 2.1.2, there exist m N and v depending only on d, ɛ, and δ, such that mk Xcan defines a birational map and vol( K Xcan ) < v. Take φ : W X can, A W, R W as in Notation Write Σ W to be the support of φ 1 B can and all φ-exceptional divisors, then vol(k W + Σ W + 2(2d + 1)A W ) vol(k Xcan + φ Σ W + 2(2d + 1)φ A W ) vol(k Xcan + δ 1 B can + 2(2d + 1)φ A W ) vol( (δ 1 + (4d + 2)m)K Xcan ) (δ 1 + (4d + 2)m) d v. Now we may apply Lemma to (Y, C, D, Z, H Z ) = (X can, 0, mb can, W, A W ) to finish the proof.

25 al bab 1} 2.4. EXERCISES Exercises Exercise Let (X, B) be a ɛ-lc pair, and f : (X, B) (X, B ) a birational map. If K X + B is numerically trivial, then (X, B ) is also ɛ-lc. Exercise Let f : X Y be a D-non-positive birational contraction of normal projective variety. Show that vol(d) = vol(f D). Exercise Let (X, B) be a ɛ-lc pair, and K X + B R 0. For any ɛ (0, ɛ), we may find a Q-divisor B, such that (X, B ) is a ɛ -lc pair, and K X + B Q 0. Exercise Suppose that X forms a bounded family and X is projective with klt singularities. Then there is a natural number n, such that nk X is Cartier. Exercise Let f : X Y be a K X -non-positive birational contraction of normal projective variety. If K Y has a klt n-complement for some natural number n, show that K X also has a klt n-complement. {exercise special ba {exercise special ba {exercise special ba {exercise special ba

26 26 CHAPTER 2. SPECIAL BAB

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