RobertoPaoletti Someremarksonmultiplieridealsandvectorbundles

Adv. Geom.3(2003), 101–104 Advances in Geometry (de Gruyter 2003

Some remarks on multiplier ideals and vector bundles

Roberto Paoletti

(Communicated by A. Sommese)

In this paper, we give two applications of the theory of multiplier ideals to vector bundles over complex projective manifolds, generalizing to higher rank results already established for line bundles. The first addresses the existence of sections of (suitable twists) of symmetric powers of a very ample vector bundle, vanishing on a given subvariety. The second is a vanishing theorem of Gri‰ths type, adjusted with an additional multiplier ideal term as in the standard Nadel vanishing theorem.

To begin with, we recall that, starting with work of Bombieri and Skoda, there has been considerable interest in going from hypersurfaces inPⁿhighly singular at a given set of pointsSto hypersurfaces throughSof relatively low degree; there is now a very direct and terse approach based on multiplier ideals [3], [4], [5], [6].

Here, the same method is shown to yield a statement in the same spirit about sec- tions of very ample vector bundles on a projective manifold. One says that a rank r holomorphic vector bundle on a complex projective manifold is very ample if the relative hyperplane line bundleOPEð1Þon the projectivised dualPE is very ample.

IfXis a projective manifold andZJXan irreducible subvariety, for every integer pd1 the p-th symbolic powerI_Z^hpiJOZ of the ideal sheaf ofZ is the ideal sheaf of the holomorphic functions vanishing with multiplicitydp along Z(that is, at a generic point ofZ).

Theorem 1.Let X be an n-dimensional complex projective manifold.LetEbe a rank r very ample vector bundle over X.Let ZJX be a codimension e irreducible subvariety.

If H⁰ðX;Sym^dEnI_Z^htiÞ00,then

H⁰ðX;KXndetðEÞnSym^nþlEnIZÞ00 as soon asld ^de_t .

SetY¼PE!^p X, the projectivised dual, and letOYð1Þbe the relative hyperplane bundle onY. LetAbe any divisor inY, not necessarily e¤ective, such thatOYðAÞ ¼ OYð1Þ. IfD¼P

iaiDiADivQðXÞ, with theDiHXirreducible divisors, the pull-back ofDtoY is theQ-divisorpðDÞ ¼P

iaipðDÞ. We shall say thatEðDÞis nef and

big if the rational divisor ApðDÞADivQðYÞis nef and big. The following state- ment generalizes the Gri‰ths vanishing theorem for ample vector bundles [7]:

Theorem 2. Suppose that DADivQðXÞ,E is a holomorphic vector bundle on X, and EðDÞis nef and big.Then

HⁱðX;KXndetðEÞnSym^mEnJðDÞÞ ¼0 if i>0;md0:

Vanishing theorems for vector bundles involving multiplier ideals (or multiplier subsheaves) were obtained also in [1], based on the di¤erential-theoretic notions of t-nefness and singular hermitian metrics (see also [2]).

Proof of Theorem1. LetY¼PE!^p X be the projectivized dual, so that Sym^kE¼pOYðkÞ

for every kd0. We shall denote the natural isomorphism H⁰ðX;Sym^kðEÞÞG H⁰ðY;OYðkÞÞbys7!ss.~

Suppose that 00sAH⁰ðX;Sym^dEnI_Z^htiÞ: for every xAZ and in any trivi- alization of E in an open neighbourhood U of x, s is represented by an ^rþd_d

- tuple of functions f_I AI_Z^htiðUÞ. Here I runs over the set Ad of all multiindices I ¼ ði1;. . .;i_rÞd0 withjIj ¼d.

The trivialization ofEonU induces isomorphisms

p¹ðUÞGUP^r1 and OYðkÞj_p1ðUÞGp₂O_P^r1ðkÞ

for every k, where p2:UP^r1!P^r1 is the projection onto the second factor.

In terms of these isomorphisms, ss~ is then given in homogeneous coordinates on UP^r1 by

s~

sðy⁰;XÞ ¼ X

IAAd

fIðy⁰ÞX^I:

Thus, ifZZ~¼p¹ðZÞJY and

Dss~:¼divð~ssÞAjH⁰ðY;OYðdÞÞj;

we have multZZ~ðDss~Þdt. In other words,ss~AH⁰ðY;I_ZZ~^htinOYðdÞÞ.

If D¼ ðe=tÞ Dss~, then multZZ~ðDÞde¼codimðZZ;~ YÞ, and therefore the multiplier ideal satisfiesJðDÞJIZZ~.

Set d¼ ^de_t . Then if AAjOYð1Þj the Q-divisor lAD is ample for lddþ1.

Therefore, K_YnOYðlþnþr1ÞnJðDÞis globally generated. Since the canoni- cal line bundle ofY isK_Y ¼pðKXndetðEÞÞnOYðrÞ,

pðKXndetðEÞÞnOYðl1þnÞnJðDÞ Roberto Paoletti

102

is globally generated. In particular,

H⁰ðY;pðKXndetðEÞÞnOYðl1þnÞnJðDÞÞ00;

and therefore H⁰ðY;pðKXndetðEÞÞnOYðl1þnÞnIZZ~Þ00. By pushing for- ward, this implies

H⁰ðX;K_XndetðEÞnSym^l1þnEnIZÞ00

forl1dd. r

Proof of Theorem2. By the Nadel vanishing theorem,

HⁱðY;K_YnOYðmÞnJðpDÞÞ ¼0; for alli;m>0:

By the projection formula and and the arguments in Chapter V of [7], HⁱðX;KXndetðEÞnpðOYðmrÞnJðpðDÞÞÞ ¼0; if i;m>0:

Lemma 1.JðpðDÞÞ ¼pðJðDÞÞ.

Assuming the lemma,

HⁱðX;KXndetðEÞnpðOYðmrÞnJðpðDÞÞÞ GHⁱðX;K_XndetðEÞnSym^mrðEÞnJðDÞÞ;

again invoking the projection formula, and the theorem follows. r Proof of Lemma 1. Letm:X⁰!X be a log-resolution of D; recall that this means thatX⁰is non-singular,mis a proper birational map, andmðDÞ þExcðmÞhas simple normal crossing support, where ExcðmÞdenotes the sum of the exceptional divisors ofm[6].

SetE⁰¼mðEÞ,Y⁰¼PE⁰¼YXX⁰. We have a commutative diagram Y^{0 m}! Y

p⁰

??

?y

??

?y^p

X^{0 m}! X

Since p⁰ is smooth, p⁰ðExcðmÞÞ ¼Excðm⁰Þ, m⁰:Y⁰!Y is a log-resolution of pðDÞ, K_Y⁰_=Y ¼p⁰ðKX⁰=XÞ, ½p⁰mðDÞ ¼p⁰½mðDÞ. Therefore, since m⁰ðpðDÞÞ ¼ p⁰ðmðDÞÞ,

JðpðDÞÞ ¼m⁰O_Y⁰ðKY⁰=Y ½m⁰pðDÞÞ ¼m⁰p⁰O_X⁰ðKX⁰=X ½mðDÞÞ: Some remarks on multiplier ideals and vector bundles 103

Let CohðZÞdenote the category of coherent sheaves ofOX-modules on a projective manifoldZ. Lemma 1 is now an immediate consequence of the following:

Lemma 2.m⁰p⁰¼pm:CohðX⁰Þ !CohðYÞ.

Proof of Lemma2. LetFbe a coherent sheaf ofOX⁰-modules. A basis for the Zariski topology ofYis given by a‰ne open subsetsU¼SpecðBÞJYsuch thatV¼pðUÞ ¼ SpecðAÞJXis also a‰ne. The restricted projection SpecðBÞ !SpecðAÞcorresponds to a morphism of rings A!B. Let V⁰¼m¹ðVÞJX⁰. Then mFðVÞ ¼FðV⁰Þ, viewed as anA-module by the morphismA¼OXðVÞ !OX⁰ðV⁰Þ. Thus,pmFðUÞ ¼ FðV⁰Þn_AB.

Next,m⁰p⁰FðUÞ ¼p⁰FðU⁰Þ, whereU⁰¼m¹ðUÞandp⁰FðU⁰Þis regarded as a B-module via the homomorphismB!OY⁰ðU⁰Þ. On the other hand, p⁰Fis the sheafification of the presheaf onY⁰

F~

FðSÞ ¼Fðp⁰ðSÞÞn

OX0ðp⁰ðSÞÞOY⁰ðSÞ ðSJY⁰ openÞ: Asp⁰ðm⁰¹ðUÞÞ ¼m¹ðpðUÞÞ ¼V⁰JX⁰, we have

F~

FðU⁰Þ ¼FðV⁰Þn

OX0ðV⁰ÞOY⁰ðU⁰Þ:

Since m and m⁰ are projective morphisms, OX⁰ðV⁰ÞGA and OY⁰ðU⁰Þ ¼A. Thus, FðV⁰Þn_ABGFðV⁰Þn_O

X0ðV⁰ÞOY⁰ðU⁰Þ. r

References

[1] M. A. A. de Cataldo, Singular Hermitian metrics on vector bundles. J. Reine Angew.

Math.502(1998), 93–122. MR 2000c:32067 Zbl 0902.32012

[2] J.-P. Demailly, Estimations L² pour l’ope´rateurqd’un fibre´ vectoriel holomorphe semi- positif au-dessus d’une varie´te´ ka¨hle´rienne comple`te. Ann. Sci. E´ cole Norm. Sup. (4)15 (1982), 457–511. MR 85d:32057 Zbl 0507.32021

[3] L. Ein, Multiplier ideals, vanishing theorems and applications. In: Algebraic geometry—

Santa Cruz 1995, 203–219, Amer. Math. Soc. 1997. MR 98m:14006 Zbl 0978.14004 [4] H. Esnault, E. Viehweg, Sur une minoration du degre´ d’hypersurfaces s’annulant en cer-

tains points.Math. Ann.263(1983), 75–86. MR 84m:14023 Zbl 0505.14029

[5] H. Esnault, E. Viehweg,Lectures on vanishing theorems. Birkha¨user 1992. MR 94a:14017 Zbl 0779.14003

[6] R. Lazarsfeld, Multiplier ideals for algebraic geometers. Unpublished notes.

[7] B. Shi¤man, A. J. Sommese,Vanishing theorems on complex manifolds. Birkha¨user 1985.

MR 86h:32048 Zbl 0578.32055

Received 6 December, 2001; revised 10 January, 2002

Dipartimento di Matematica ‘‘E. De Giorgi’’, Universita´ di Lecce, Via per Arnesano, 73100 Lecce, Italy

Email: [email protected]

Roberto Paoletti 104