Let be an ample line bundle on a non singular projective fold It is 64257rst shown that 2 mL is very ample for 2 1 The proof developes an original idea of YT Siu and is based on a combination of th e RiemannRoch theorem together with an improved N ID: 35512 Download Pdf

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Let be an ample line bundle on a non singular projective fold It is 64257rst shown that 2 mL is very ample for 2 1 The proof developes an original idea of YT Siu and is based on a combination of th e RiemannRoch theorem together with an improved N

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Eﬀective Bounds for Very Ample Line Bundles Jean-Pierre Demailly Institut Fourier, Universite de Grenoble I Summary. Let be an ample line bundle on a non singular projective -fold . It is ﬁrst shown that 2 mL is very ample for 2+ +1 . The proof developes an original idea of Y.T. Siu and is based on a combination of th e Riemann-Roch theorem together with an improved Noetherian induction tec hnique for the Nadel multiplier ideal sheaves. In the second part, an eﬀective ve rsion of the big Matsusaka theorem is obtained, reﬁning an earlier

version of Y.T. Siu: there is an explicit polynomial bound ,L ) of degree in the arguments, such that mL is very ample for . The reﬁnement is obtained through a new sharp upper bound for the dualizing sheaves of algebraic varietie s embedded in projective space. 0. Introduction In the last six or seven years, considerable progress has bee n achieved in the under- standing of adjoint linear systems mL associated with an ample line bundle on a smooth projective manifold . When is a surface, I. Reider [Rei88] ob- tained a quasi-optimal criterion for the global generation and very ampleness

of , showing in particular that + 3 is always generated by global sections and + 4 very ample. Around the same period, T. Fujita [Fuj87] raised the following interesting conjecture. (0.1) Conjecture (Fujita). Let be a smooth projective -fold over and let be an ample line bundle on . Then + ( + 1) is generated by global sections and + ( + 2) is very ample. One of the ﬁrst results proved in dimension 3 is the very ampleness of + 12 , using an analytic method based on the solution of a Monge-Am p`ere equation (see [Dem93]). Slightly later, J. Kollar [Kol93] obtained an

eﬀective version of the base point free theorem, while a major step was made in s mall dimension by L. Ein and R. Lazarsfeld [EL93], with the solution of the glob al generation part of Fujita’s conjecture for = 3. Other related works are [EL92], [Fuj94], [EKL94], [Ein94] (see also [Laz93] and [Dem94] for survey exposition s). Recently, Y.T. Siu [Siu94a] introduced a simple algebraic method for proving t he very ampleness of mL . His method is based on a combination of the Riemann-Roch for mula with the Kawamata-Viehweg vanishing theorem, in the genera lized form given by A. Nadel

[Nad89]. Our ﬁrst goal is to develope a more eﬃcient N oetherian induction process for the Nadel multiplier sheaves associated with si ngular hermitian metrics, along the lines of Siu’s method. The new induction process is simpler and allows us to reﬁne further Siu’s original bounds. In the sequel the int ersection numbers of over -dimensional subvarieties are denoted

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2 Eﬀective Bounds for Very Ample Line Bundles We say that is numerically eﬀective (nef for short) if 0 for every algebraic curve . By [Dem90], is nef if and only if for each

ε> 0 there is a hermitian metric on of curvature , where is a given Kahler form on (0.2) Theorem. Let be a smooth projective -fold and let be an ample line bundle over . Then a) 2 mL is very ample for 2 + +1 b) 2 generates simultaneous jets of order ,...,s at arbitrary points ,...,x provided that the intersection numbers of over all -dimensional algebraic subsets of satisfy Y > n/d + 1)(4 + 2 + 1) n. c) + ( + 2) is very ample for +1 All results still hold true by adding any nef line bundle to the line bundles under consideration. Our method of proof is sharp enough to yield as a

by-product th e well-known result that + ( + 1) is numerically eﬀective if is ample (a result originally proved as a consequence of Mori theory). A basic problem woul d be to ﬁnd an ana- logue of Th. (0.2 a, b) with in place of 2 . For the global generation question, the answer has been settled in the aﬃrmative recently by U. An gehrn and Y.T. Siu [AS94], who showed that +2) is always generated by global sections; their method is again based on Nadel’s vanishing theorem, us ing a diﬀerent idea for the construction of the required singular hermitian metric s. The

result of Angehrn- Siu implies that + 2 + 2) ) is very ample for 2 (by the elementary observation that +2 nF is always very ample if is ample and gener- ated by sections); the bound obtained in (0.2 c) can then be im proved into In a related paper [Siu94b], Y.T. Siu obtains a variant of (0. 2 b) in which the numer- ical condition for is replaced by ( /d +2 + 2 pn this bound, which has a rather involved proof, is sharper tha n ours for (ln( )) but weaker for larger values of . At the time these lines are written, it seems to be unknown whether there is a bound ) depending only on the dimension

such that mL is very ample for larger than ). Also it seems to be unknown whether polynomial bounds ) exist for 2 mL (the bound given by (0.2 a) is of the order of magnitude of (27 4) and seems to be the best presently known). Another important question is to ﬁnd eﬀective bounds such that mL be- comes very ample for . From a theoretical point of view, this problem is solved by Matsusaka [Mat72] and Kollar-Matsusaka [KoM83] . Their result states that there is a bound n,L ,L ) depending only on the dimension and on the ﬁrst two coeﬃcients and in the Hilbert

polynomial of Unfortunately, the original proof does not tell much on the a ctual dependence of

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1. Nadel’s Vanishing Theorem 3 in terms of these coeﬃcients. In a ground-breaking paper [Si u93], Y.T. Siu in- troduced new techniques leading to eﬀective bounds for . The published version of [Siu93] incorporates an induction argument which we deve loped in collaboration with the author after the preprint version circulated, enab ling us to obtain much better ﬁnal estimates. Our goal in the last sections 3, 4 is to present a further substantial reﬁnement

of this method. The main point is that a crucial technical lemma used in [Siu93] to deal with dualizing sheaves can be ma de optimal by using a diﬀerent idea based on the Ohsawa-Takegoshi extension theorem [OT87]. (0.3) Theorem. Let be a very ample line bundle on a projective algebraic man- ifold , and let be a -dimensional irreducible algebraic subvariety. Denote by the dualizing sheaf of . If is the degree of with respect to , the sheaf om (( 2) has a nontrivial section. Using this sharp “upper estimate” on dualizing sheaves and s ome other reﬁne- ments of the inductive method

explained in [Siu93], we obtai n the following improved bounds. (0.4) Theorem. If is an ample line bundle on a projective -fold , then mL is very ample for + 2 + n/ 2+3 4)+1 where depends only on , e.g., = (2 (3 1) + 1 n/ 2+3 4)+1 The bound (0.4) turns out to be essentially optimal for = 2 (apart from a small multiplicative constant), as was shown recently by F ernandez del Busto [FdB94] by means of Reider’s theorem and an example of Gang Xi ao. Our bound is probably not optimal for 3, and we strongly believe that there should exist an optimal bound of the form +2+ /L , involving

exponents of the order of magnitude of or instead of 1. Nadel’s Vanishing Theorem We recall here brieﬂy a few basic ideas developed in [Dem90, 9 3], which will be equally useful in this paper. Let be a projective algebraic manifold equipped with a Kahler metric , and let be a holomorphic line bundle over . We assume that is equipped with a (possibly singular) hermitian metric . In each open set where is trivial, the metric is given by a weight such that for all , where is the trivialization map. If is supposed to be locally integrable on , the curvature form of can be

deﬁned to

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4 Eﬀective Bounds for Very Ample Line Bundles be the closed (1 1)-current ) = . Here, we will only consider the case of nonnegative curvature currents 0, i.e., we suppose that the weights are plurisubharmonic. Following Nadel [Nad89], we associate t the ideal sheaf (1 1) ) = ∈O X,x x, dV where dV /n ! is the Kahler volume form and is an arbitrary open neigh- borhood of . Of course, ) does not depend on the choice of the trivialization, and thus we get a global ideal sheaf ) on depending only on . By [Nad89] and [Dem93], ) is a coherent

ideal sheaf in , and we have the following fun- damental vanishing theorem. (1.2) Nadel vanishing theorem. Assume that for some ε> . Then X, ⊗I = 0 for all The proof is a straightforward consequence of the Bochner-K odaira-Nakano identity ([AN54], [Nak55]) and of Hormander’s estimates for the operator (see [Hor65], [AV65], [Nad89], [Dem93]). In the present paper, we only need “algebraic metrics of the form (1 3) )) / where ,..., X,F ) are non zero algebraic sections of F , and is the local trivialization of induced by a local trivialization of . The

corresponding weight is (1 4) log )) In this case, (1.2) is equivalent to the Kawamata-Viehweg va nishing ([Kaw82], [Vie82]), and the proof can be reduced to the usual Kodaira va nishing theorem by purely algebraic means. Now, recall that the Lelong numbe r of a plurisubhar- monic function at a point is ϕ,x ) = lim sup x,r ϕ/ log . In the special case (1.4) under consideration, we simply have ϕ,x ) = min ord where ord ) is the vanishing order of at (1.5) Corollary. Let X, and be as in (1 2) and let ,...,x be isolated points in the zero variety )) . Then there is a surjective map

X,K

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2. Some Results Around the Fujita Conjecture 5 In particular, if ϕ,x , then X,K generates simultaneously all jets of order at Proof. Consider the long exact sequence of cohomology associated t o the short exact sequence 0 →I →O →O 0 twisted by ), and apply Th. (1.2) to obtain the vanishing of the ﬁrst group. The asserted surjectivity property follows. The last statement follows from the fact t hat implies +1 . Indeed, we then have ) log (1) , c> as is obvious in the “algebraic case” (in general, the inequa lity follows from the standard

logarithmic convexity property of plurisubharmo nic functions). (1.6) Remark. As is well known, Corollary (1.6) can be proved by a direct app lica- tion of Hormander’s estimates, namely by solving a -equation ∂u for forms of type ( n, 1), where is a ﬁnite holomorphic Taylor expansion achieving the desired jet at , and where is a cut-oﬀ function with support in a neighbor- hood of . In this way, we see that Cor. (1.6) still holds if we only have and in a neighborhood of each 2. Some Results Around the Fujita Conjecture This section is devoted to a proof of various

results related to the Fujita conjec- ture. The main ideas occuring here are inspired by a very rece nt work of Y.T. Siu [Siu94a]. His method, which is algebraic in nature and quite elementary, consists in a combination of the Riemann-Roch formula together with Nad el’s vanishing theo- rem (in fact, only the algebraic case is needed, thus the orig inal Kawamata-Viehweg vanishing theorem would be suﬃcient). In the sequel, denotes a projective al- gebraic -dimensional manifold. The ﬁrst observation is the followi ng well-known consequence of the Riemann-Roch formula. (2.1) Special

case of Riemann-Roch. Let J⊂O be a coherent ideal sheaf on such that the subscheme has dimension with possibly some lower dimensional components . Let ] = be the eﬀective algebraic cycle of dimension associated to the dimensional components of taking into ac- count multiplicities given by the ideal . Then for any line bundle , the Euler characteristic Y, mL ) = X, mL ⊗O is a polynomial of degree and leading coeﬃcient /d The second fact is an elementary lemma about numerical polyn omials (polyno- mials with rational coeﬃcients, mapping into ).

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Eﬀective Bounds for Very Ample Line Bundles (2.2) Lemma. Let be a numerical polynomial of degree d > and leading coeﬃcient /d . Suppose that for . Then a) For every integer , there exists ,m Nd such that b) For every , there exists ,m kd such that c) For every integer , there exists ,m such that Proof. a) Each of the equations ) = 0, ) = 1, ... ) = 1 has at most roots, so there must be an integer ,m dN ] which is not a root of these. b) By Newton’s formula for iterated diﬀerences ∆P ) = +1) ), we get ) = 1) ) = Hence if ,..., d/ [0 ,d ] is the even integer achieving

the maximum of ) over this ﬁnite set, we ﬁnd ) = ... whence the existence of an integer ,m ] with . The case = 1 is thus proved. In general, we apply the above case to the po lynomial ) = km 1) ), which has leading coeﬃcient /d c) If = 1, part a) already yields the result. If = 2, a look at the parabola shows that max ,m 8 if is even, 1) 8 if is odd; thus max ,m whenever 8. If 3, we apply b) with equal to the smallest integer such that , i.e. 2( N/ 2) /d , where e denotes the round-up of . Then kd (2( N/ 2) /d + 1) whenever , as a short computation shows. We now apply

Nadel’s vanishing theorem pretty much in the sam e way as Siu [Siu94a], but with substantial simpliﬁcations in the techn ique and improvements in the bounds. Our method yields simultaneously a simple proof of the following basic result. (2.3) Theorem. If is an ample line bundle over a projective -fold , then + ( + 1) is nef. By using Mori theory and the base point free theorem ([Mor82] , [Kaw84]), one can even show that + ( + 1) is semiample, i.e., there exists a positive integer such that + ( + 1) ) is generated by sections (see [Kaw85] and [Fuj87]). The proof rests on the observation

that + 1 is the maximal length of extremal rays of smooth projective -folds. Our proof of (2.3) is diﬀerent and will be given simultaneously with the proof of Th. (2.4) below.

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2. Some Results Around the Fujita Conjecture 7 (2.4) Theorem. Let be an ample line bundle and let be a nef line bundle on a projective -fold . Then the following properties hold. a) 2 mL generates simultaneous jets of order ,...,s at arbitrary points ,...,x , i.e., there is a surjective map X, mL (2 mL ⊗O X,x +1 X,x provided that 2 + + 2 In particular mL is very ample for 2 + + 1 b) 2 +( +1)

generates simultaneous jets of order ,...,s at arbitrary points ,...,x provided that the intersection numbers of over all -dimensional algebraic subsets of satisfy Y > n/d + 2 Proof. The proofs of (2.3) and (2.4 a, b) go along the same lines, so we deal with them simultaneously (in the case of (2.3), we simply agree th at ,...,x ). The idea is to ﬁnd an integer (or rational number) and a singular hermitian metric on with strictly positive curvature current , such that )) is 0-dimensional and the weight of satisﬁes ,x for all . As and are nef, ( has for all a metric whose

curvature has arbitrary small negative part (see [Dem90]), e.g., Then is again positive deﬁnite. An application of Cor (1.5) to mL = ( ) + (( ) equipped with the metric implies the existence of the desired sections in = 2 mL for Let us ﬁx an embedding L 0, given by sections ,..., X,L ), and let be the associated metric on of positive deﬁnite curvature form ). In order to obtain the desired metric on , we ﬁx and use a double induction process to construct singular met rics ( k, on aK for a non increasing sequence of positive integers ... ... . Such a

sequence much be stationary and will just be the stationary limit = lim /a . The metrics k, are taken to satisfy the following properties: k, is an algebraic metric of the form k, ν, +1) a +1) am +1) deﬁned by sections X, + 1) ), +1 , 1 where 7 ) is an arbitrary local trivialization of aK ; note that a +1) am is a section of

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8 Eﬀective Bounds for Very Ample Line Bundles a (( + 1) ) + (( + 1) am L = ( + 1) aK ) ord + 1)( ) for all i,j k, +1 ⊃I k, ) and k, +1 k, ) whenever the zero variety k, )) has positive dimension. The weight k, 2( +1) log

+1) a +1) am of k, is plurisubhar- monic and the condition +1 implies ( +1) am 1, thus the diﬀerence k, 2( +1) log is also plurisubharmonic. Hence k, aK ) = k, +1) . Moreover, condition ) clearly implies k, ,x ). Finally, condition ) combined with the strong Noetherian property of coherent sheaves ensures that the sequence ( k, will ﬁnally produce a zero dimensional subscheme k, )). We agree that the sequence ( k, stops at this point, and we denote by k, the ﬁnal metric, such that dim )) = 0. For = 1, it is clear that the desired metrics ( , exist if is taken large enough

(so large, say, that ( + 1) + ( 1) generates jets of order + 1)( + max ) at every point; then the sections ,..., can be chosen with ... 1). Suppose that the metrics ( k, and have been constructed and let us proceed with the construction of ( +1 , . We do this again by induction on , assuming that +1 , is already constructed and that dim +1 , )) 0. We start in fact the induction with = 0, and agree in this case that +1 ) = 0 (this would correspond to an inﬁnite metric of weight identically equal to ). By Nadel’s vanishing theorem applied to aK mL = ( aK ) + ( with the metric , we get X,

(( + 1) mL ⊗I )) = 0 for 1, As )) is 0-dimensional, the sheaf ) is a skyscraper sheaf, and the exact sequence 0 →I →O →O 0 twisted with the invertible sheaf (( + 1) mL ) shows that X, (( + 1) mL )) = 0 for 1, Similarly, we ﬁnd X, (( + 1) mL ⊗I +1 , )) = 0 for 1, +1 (also true for = 0, since +1 ) = 0), and when max( ,b +1 ) = , the exact sequence 0 →I +1 , →O →O +1 , 0 implies X, (( + 1) mL ⊗O +1 , )) = 0 for 1, In particular, since the group vanishes, every section of ( + 1) mL on the subscheme +1 , )) has an extension to . Fix a

basis ,...,u of the sections on +1 , )) and take arbitrary extensions ,...,u to . Look at the linear map assigning the collection of jets of order ( +1)( 1 at all points 7 +1)(

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2. Some Results Around the Fujita Conjecture 9 Since the rank of the bundle of -jets is , the target space has dimension + ( + 1)( In order to get a section +1 satisfying condition ) with non trivial restriction +1 to +1 , )), we need at least + 1 independent sections ,...,u This condition is achieved by applying Lemma (2.2) to the num erical polynomial ) = X, (( + 1) mL ⊗O +1 , )) X, (( + 1) mL

⊗O +1 , )) , m The polynomial has degree = dim +1 , )) 0. We get the existence of an integer ,b ] such that + 1 with some explicit integer (for instance + 1) always works by (2.2 a), but we will also use the other possibilities to ﬁnd an optimal choice in each case). T hen we ﬁnd a section +1 X, + 1) mL ) with non trivial restriction +1 to +1 , )), vanishing at order + 1)( ) at each point . We just set +1 , and the condition +1 +1 +1 is satisﬁed if η< +1 +1 . This shows that we can take inductively +1 + 1 + 1 By deﬁnition, +1 , +1 +1 , , hence +1 , +1

⊃I +1 , ). We necessar- ily have +1 , +1 +1 , ), for +1 , +1 ) contains the ideal sheaf as- sociated with the zero divisor of +1 , whilst +1 does not vanish identically on +1 , )). Now, an easy computation shows that the iteration of +1 +1 + 1 stops at + 1) + 1 for any large initial value . In this way, we obtain a metric of positive deﬁnite curvature on aK +( +1)+1) with dim )) = 0 and ,x ) at each point Proof of (2.3). In this case, the set is taken to be empty, thus = 0. By (2.2 a), the condition 1 is achieved for some ,b ] and we can take As L is very ample, there

exists on L a metric with an isolated logarithmic pole of Lelong number 1 at any given point (e.g., the algebraic metric deﬁned with all sections of L vanishing at ). Hence aK + ( + 1) + 1) nL has a metric such that )) is zero dimensional and contains . By Cor (1.5), we conclude that = ( + 1) + ( + 1) + 1 + n is generated by sections, in particular +1)+1+ n +1 is nef. As tends to + we infer that + ( + 1) is nef. Proof of (2.4a). Here, the choice = 1 is suﬃcient for our purposes. Then

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10 Eﬀective Bounds for Very Ample Line Bundles +

2 If }6 , we have + 1 + 1 for 2. Lemma (2.2 c) shows that + 1 for some ,b ] with + 1. We can start in fact the induction procedure 7 + 1 with + 1 = + 2, because the only property needed for the induction step is the vanishing property X, mL ) = 0 for 1, which is true by the Kodaira vanishing theorem and the amplen ess of (here we use Fujita’s result (2.3), observing that > n + 1). Then the recursion formula +1 + 1 yields + 1 = + 2 for all , and (2.4 a) follows. Proof of (2.4b). Quite similar to (2.4 a), except that we take = 1 and +1 for all . By Lemma (2.2 b), we have for some integer ,m kd

], where 0 is the coeﬃcient of highest degree in . By Lemma (2.1) we have inf dim . We take n/d . The condition +1 can thus be realized for some ,m kd ,m ] as soon as inf dim n/d >δ, which is equivalent to the condition given in (2.4 b). Theorem (0.2 a) is a special case of Th. (2.4 a). Theorem (0.2 b ) can be derived from (2.4 b) by using the following simple lemma. (2.5) Lemma. Assume that for some integer the line bundle F gene- rates simultaneously all jets of order ) + 1 at any point in a subset ,...,x } . Then generates simultaneously all jets of order at Proof. Take

the algebraic metric on deﬁned by a basis of sections ,..., of F which vanish at order ) +1 at all points . Since we are still free to choose the homogeneous term of degree ) + 1 in the Taylor expansion at , we ﬁnd that ,...,x are isolated zeroes of (0). If is the weight of the metric of near , we thus have ) log in suitable coordinates. We replace in a neighborhood of by ) = max + ( ) log and leave elsewhere unchanged (this is possible by taking C > 0 very large). Then ) = + ( ) log near , in particular is strictly plurisubharmonic near . In this way, we get a metric on

with semipositive curvature everywhere on , and with positive deﬁnite curvature on a neighborhood of ,...,x . The conclusion then follows from Cor. (1.5) and Rem. (1.6). Proof of Theorem (0.2b). By Lemma (2.5) applied with and +1, the desired jet generation of 2 occurs if ( + 1)( ) generates jets of

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3. An Estimate for Dualizing Sheaves 11 order ( +1)( )+1 at . By Lemma (2.5) again with aK +( +1) and = 1, we see by backward induction on that we need the simultaneous generation of jets of order ( + 1)( ) + 1 + ( + 1 )( + 1) at . In particular, for + ( + 1) we need the

generation of jets of order ( + 1)(2 1) + 1. Theorem (2.4 b) yields the desired condition. Proof of Theorem (0.2c). Apply Th. (2.4 a) with + ( + 1) ) + , so that mL = ( + 2)( + ( + 2) ) + ( G, and take + 2 + 4 2 + +1 3. An Estimate for Dualizing Sheaves If is a complex -dimensional analytic space with arbitrary singularities , we deﬁne the dualizing sheaf of to be the sheaf of holomorphic -forms on the regular part reg which are locally near sing , that is, for any open set W, ) = reg , reg ) ; W, x, reg u< where is an arbitrary neighborhood of . It is easily seen that is the direct

image of the dualizing sheaf of a desingularization of , thus is a coherent sheaf on is just the usual dualizing sheaf of algebraic geometers). T hen we have the following optimal “upper estimate” for (3.1) Theorem. Let be a very ample line bundle on a projective algebraic man- ifold , and let be a -dimensional irreducible algebraic subvariety. If is the degree of with respect to , the sheaf om (( 2) has a nontrivial section. Observe that if is a smooth hypersurface of degree in ( X,H ) = ( +1 (1)), then 2) and the estimate is optimal. On the other hand, if is a smooth complete intersection of

multidegree ( ,..., ) in , then ... whilst ... 1) ; in this case, Th. (3.1) is thus very far from being sharp. Proof. Let be the embedding given by , so that (1). There is a linear projection +1 whose restriction +1 to is a ﬁnite and regular birational map of onto an algebraic hypersurface of degree in +1 . Let +1 )) be the polynomial of degree deﬁning . We claim that for any small Stein open set +1 and any holomorphic form on , there is a holomorphic ( + 1)-form on with values in ) such that ds . In fact, this is precisely the conclusion of the Ohsawa-Takegoshi extension theorem

[OT87], [Ohs88] (see a lso [Man93] for a more

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12 Eﬀective Bounds for Very Ample Line Bundles general version); one can also invoke more standard local al gebra arguments (see Hartshorne [Har77], Th. III-7.11). As +1 2), the form can be seen as a section of 2) on , thus the sheaf morphism 7 ds extends into a global section of om 2) . The pull-back by yields a section of om (( 2) . Since is ﬁnite and generically 1 : 1, it is easy to see that . The Theorem follows. 4. An Eﬀective Version of Matsusaka’s Big Theorem Let be an ample line bundle on a projective

algebraic manifold . We look for an explicit value of such that mL is very ample for . As in [Siu93], our starting point is the following lemma. (4.1) Lemma. Let and be nef line bundles over . If > nF , all large positive multiples , have non trivial sections. Proof. This is a special case of the holomorphic Morse inequalities (see [Dem85], [Tra95], [Siu93]). Here is a simple proof, following a sugge stion of F. Catanese. We can suppose that and are very ample (otherwise, we replace and by pF and pG with very ample and large enough, and p > 0 very large). Then )) 'O kF ... ) for arbitrary

members ,...,G in the linear system , and the Lemma follows from Riemann-Roch by looking at the restriction morphism X, kF )) kF ). (4.2) Corollary. Let and be nef line bundles over . If is big and m > nF G/F , then mF can be equipped with a possibly singular hermitian metric with positive deﬁnite curvature form mF ε> , for some Kahler metric Proof. In fact, if is ample and small enough, Lemma (4.1) implies that some multiple mF εA ) has a section. Let be the divisor of this section and let ) be a Kahler form. Then mF εA can be equipped with a singular

metric of curvature form mF ) = ) + We now consider the question of obtaining a nontrivial secti on in mL . The idea, more generally, is to obtain a criterion for the ampleness of mL when is nef. In this way, one is able to subtract from mL any undesirable multiple of which otherwise gets added to by the application of Nadel’s vanishing theorem (for this, we simply replace by plus a multiple of + ( + 1) ). (4.3) Proposition. Let be an ample line bundle over a projective -fold and let be a nef line bundle over . Then mL has a nonzero section for some integer such that

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4. An

Eﬀective Version of Matsusaka’s Big Theorem 13 + 1 Proof. Let be the smallest integer > n . Then can be equip- ped with a singular hermitian metric of positive deﬁnite curvature. By Nadel’s vanishing theorem, we have X, mL ⊗I )) = 0 for 1, thus ) = X, mL ⊗I )) is a polynomial for . Since is a polynomial of degree and is not identically zero, there must be an integer ,m ] which is not a root. Hence there is a nontrivial section in X, mL )) X, mL ⊗I )) for some ,m ], as desired. (4.4) Corollary. If is ample and is nef, mL has a nonzero section for some integer + 1

Proof. By Fujita’s result (2.3 a), + ( + 1) is nef. We can thus replace by + ( + 1) in the result of Prop. (4.3). Corollary (4.4) follows. (4.5) Remark. We do not know if the above Corollary is sharp, but it is certai nly not far from being so. Indeed, for = 0, the initial constant cannot be replaced by anything smaller than n/ 2 : take to be a product of curves of large genus and = 0; our bound for ]) ... ⊗O ]) to have mL |6 becomes (2 2) /a + 1), which fails to be sharp only by a factor 2 when ... = 1 and ... . On the other hand, the additive constant + 1 is already best possible when

= 0 and So far, the method is not really sensitive to singularities ( Lemma (4.1) is still true in the singular case as is easily seen by using a desingul arization of ). The same is true with Nadel’s vanishing Theorem (1.2), provided that ⊗I ) is replaced by the sheaf ) of -forms which are locally near sing with respect to the weight of (according to that notation, the dualizing sheaf is associated with = 0 or with any nonsingular weight ). Then Prop. (4.3) can be generalized as (4.6) Proposition. Let be an ample line bundle over a projective -fold and let be a nef line bundle over .

For every -dimensional reduced algebraic subvariety of , there is an integer + 1 such that the sheaf ⊗O mL has a nonzero section.

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14 Eﬀective Bounds for Very Ample Line Bundles By an appropriate induction process based on the above resul ts, we can now improve Siu’s eﬀective version of the Big Matsusaka Theorem [Siu93]. Our version depends on a constant such that + ( + 2) ) + is very ample for and every nef line bundle . Theorem (0.2 c) shows that +1 and a similar argument involving the recent results of Angeh rn-Siu [AS94] implies 1 for 2. Of course, it

is expected that = 1 in view of the Fujita conjecture. (4.7) Eﬀective version of the Big Matsusaka Theorem. Let and be nef line bundles on a projective -fold . Assume that is ample and set + ( + 2) . Then mL is very ample for (2 (3 1) )) (3 +1) n/ 4) n/ 4)+1 In particular mL is very ample for + 2 + n/ 2+3 4)+1 with = (2 (3 1) n/ 2+3 4)+1 Proof. We use Th. (3.1) and Prop. (4.6) to construct inductively a se quence of (non necessarily irreducible) algebraic subvarieties ... such that p,j is -dimensional, and is obtained for each 2 as the union of zero sets of sections p,j p,j p,j p,j ))

with suitable integers p,j 1. We proceed by induction on decreasing values of the dimension , and ﬁnd inductively upper bounds for the integers p,j By Cor. (4.4), an integer for to have a section can be found with + ( + 1) Now suppose that the sections ... +1 ,j have been constructed. Then we get inductively a -cycle p,j p,j deﬁned by = sum of zero divisors of sections +1 ,j in +1 ,j , where the mutiplicity p,j on p,j +1 ,k is obtained by multiplying the corresponding multiplicity +1 ,k with the vanishing order of +1 ,k along p,j . As cohomology classes, we ﬁnd +1 ,k +1 ,k

+1 ,k +1 +1 Inductively, we thus have the numerical inequality +1 ...m Now, for each component p,j , Prop. (4.6) shows that there exists a section of p,j ⊗O p,j p,j ) for some integer

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4. An Eﬀective Version of Matsusaka’s Big Theorem 15 p,j p,j p,j + 1 pm +1 ...m + 1 Here, we have used the obvious lower bound p,j 1 (this is of course a rather weak point in the argument). The degree of p,j with respect to admits the upper bound p,j := p,j +1 ...m We use the Hovanski-Teissier concavity inequality ([Hov79], [Tei79, 82], see also [Dem93]) to express our boun ds in

terms of the inter- section numbers and only. We then get p,j +1 ...m By Th. (3.1), there is a nontrivial section in om p,j p,j (( p,j 2) Combining this section with the section in p,j ⊗O p,j p,j ) already cons- tructed, we get a section of p,j p,j + ( p,j 2) ) on p,j . Since we do not want to appear at this point, we replace with + ( p,j 2) and thus get a section p,j of p,j p,j ) with some integer p,j such that p,j pm +1 ...m + ( p,j 2) ) + + 1 pm +1 ...m p,j +1 ...m Therefore, by putting nL ), we get the recursion relation +1 ...m for 2 with initial value M/L . If we let ( ) be the

sequence obtained by the same recursion formula with equalities instead of inequali ties, we get with and +1 +1 for 2 2. We then ﬁnd inductively 2)+1 2)+1 We next show that is nef for = max , m ,...,m , m ...m

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16 Eﬀective Bounds for Very Ample Line Bundles In fact, let be an arbitrary irreducible curve. Either ,j for some or there exists an integer = 2 ,...,n such that is contained in but not in If p,j , then p,j does not vanish identically on . Hence ( p,j has nonnegative degree and p,j On the other hand, if ,j , then ...m By the deﬁnition of (and the

proof of (0.2 c) that such a constant exists), is very ample for every nef line bundle , in particular is very ample. We thus replace again with . This has the eﬀect of replacing with + 2 and with = max , m ,...,m ,m ...m The last term is the largest one, and from the estimate on , we get (3 1) (3 1)( 2) 2+( 2) )) (3 1)( 2) 2+( 2) 2+1 (2 (3 1) )) (3 +1) n/ 4) n/ 4)+1 (4.8) Remark. In the surface case = 2, one can take = 1 and our bound yields mL very ample for + 4 )) If one looks more carefully at the proof, the initial constan t 4 can be replaced by 2. In fact, it has been shown

recently by Fernandez del Busto th at mL is very ample for m> + 4 ) + 1) + 3 and an example of G. Xiao shows that this bound is essentially optimal (see [FdB94]). References [AN54] Akizuki, Y., Nakano, S. Note on Kodaira-Spencer’s proof of Lefschetz the- orems , Proc. Jap. Acad., 30 ), 266–272. [AV65] Andreotti, A., Vesentini, E. Carleman estimates for the Laplace-Beltrami equation in complex manifolds , Publ. Math. I.H.E.S., 25 ), 81–130. [AS94] Angehrn, U., Siu, Y.T. Eﬀective freeness and point separation for adjoint bundles , Preprint December 1994.

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Eﬀective Bounds for Very Ample Line Bundles [Siu93] Siu, Y.T. An eﬀective Matsusaka big theorem , Ann. Inst. Fourier., 43 ), 1387–1405. [Siu94a] Siu, Y.T. Eﬀective Very Ampleness , Preprint 1994, to appear in Inventiones Math. [Siu94b] Siu, Y.T. Very ampleness criterion of double adjoint of ample line bun dles Preprint 1994, to appear in Annals of Math. Studies, volume in h onor of Gunning and Kohn, Princeton Univ. Press. [Tei79] Teissier, B. Du theor`eme de l’index de Hodge aux inegalites isoperi metriques C. R. Acad. Sc. Paris,

Ser. A,, 288 ), 287–289. [Tei82] Teissier, B. Bonnesen-type inequalities in algebraic geometry , Sem. on Diﬀ. Geom. edited by S.T. Yau, 1982, Princeton Univ. Press, 85–105. [Tra95] Trapani, S. Numerical criteria for the positivity of the diﬀerence of am ple divisors , Math. Zeitschrift, 219 ), 387-401. [Vie82] Viehweg, E. Vanishing theorems , J. Reine Angew. Math., 335 ), 1–8. Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier, BP74 F–38400 Saint-Martin d’H`eres, France e-mail : demailly@fourier.ujf-grenoble.fr

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