aa r X i v : . [ m a t h . C V ] J a n ON THE RESTRICTION FORMULA
XIANKUI MENG AND XIANGYU ZHOU
Abstract.
Let ϕ be a quasi-psh function on a complex manifold X and let S ⊂ X be a complex submanifold. Then the multiplier ideal sheaves I ( ϕ | S ) ⊂I ( ϕ ) | S and the complex singularity exponents c x ( ϕ | S ) c x ( ϕ ) by Ohsawa-Takegoshi L extension theorem. An interesting question is to know whetherit is possible to get equalities in the above formulas. In the present article, weshow that the answer is positive when S is chosen outside a measure zero setin a suitable projective space. Introduction
Let X be a complex manifold. Throughout this paper, our complex manifoldsare always assumed to be second countable. A quasi-plurisubharmonic (quasi-psh)function on X is by definition a function ϕ which is locally equal to the sum ofa plurisubharmonic function and of a smooth function. The multiplier ideal sheaf I ( ϕ ) is the ideal subsheaf of O X defined by I ( ϕ ) x = { f ∈ O X,x ; | f | e − ϕ is integrable in a neighborhood of x } , It’s well-known that I ( ϕ ) is a coherent analytic sheaf.Another related invariant is complex singularity exponent. For any compactset K ⊂ X , we introduce the complex singularity exponent of ϕ on K to be thenonnegative number(1.1) c K ( ϕ ) = sup (cid:8) c > e − cϕ is L on a neighborhood of K (cid:9) . If ϕ = −∞ near some connected component of K , we put of course c K ( ϕ ) = 0.Given a point x ∈ X , we write c x ( ϕ ) instead of c { x } ( ϕ ).Multiplier ideal sheaves and complex singularity exponents associated to quasi-psh functions are basic objects in several complex variables and algebraic geometry.The restriction formulas for multiplier ideal sheaves and for complex singularity ex-ponents are useful in the inductive arguments. Various important and fundamentalproperties about them have been established by [5], [6], [2], [9], [8], [13], etc. In thepresent article, we will discuss the restriction formulas in a more general setting.We first give a generalized version of the Bertini theorem. Theorem 1.1.
Let F be a holomorphic vector bundle over a complex manifold X and let W be a finite dimensional subspace of H ( X, F ) such that W generatesall fibers F x , x ∈ X . Let P ( W ) denote the projective space of W . If W has nonon-vanishing sections, then dim X > rank F and the set Z = n [ s ] ∈ P ( W ) (cid:12)(cid:12) s − (0) is not smooth o has Lebesgue measure zero in P ( W ) . If, moreover, X is compact, then Z is analyticin P ( W ) . The restriction formula on multiplier ideal sheaves is also a Bertini type result.Motived by [2], we shall give a simple approach to this problem. The main idea ofthe proof is to apply the Fubini theorem.
Theorem 1.2.
Let ϕ be a quasi-psh function on a complex manifold X . Let F be a holomorphic vector bundle over X and let W be a finite dimensional subspaceof H ( X, F ) such that W generates all fibers F x , x ∈ X . Let P ( W ) denote theprojective space of W and B = n [ s ] ∈ P ( W ) (cid:12)(cid:12) S = s − (0) is smooth and I ( ϕ | S ) = I ( ϕ ) | S o . If W has no non-vanishing sections, then P ( W ) \ B has measure zero in P ( W ) . A special case of Theorem 1.2 is the following statement.
Corollary 1.3.
Let d be a base point free linear system on a complex manifold X with dim d < + ∞ , and let φ be a quasi-plurisubharmonic function on X . If weput b = n S ∈ d ; S is smooth and I ( ϕ ) | S = I ( ϕ | S ) o , then d \ b has measure zero in d . In case X is compact, Fujino and Matsumura proved that b is dense in d inthe classical topology (cf. Theorem 1.10 in [9]). However, their arguments giveno information on the set d \ b . The main idea of our approach is to use Fubini’stheorem and hence we can prove the set d \ b has measure zero. In fact, there is aproblem due to S´ebastien Boucksom: is d \ b a pluripolar set?The restriction formula on complex singularity exponents is given by the follow-ing result. Theorem 1.4.
Let ϕ be a quasi-psh function on a complex manifold X . Let F be a holomorphic vector bundle over X and let W be a finite dimensional subspaceof H ( X, F ) such that W generates all fibers F x , x ∈ X . Let P ( W ) denote theprojective space of W and Q = n [ s ] ∈ P ( W ) (cid:12)(cid:12) S = s − (0) is smooth and c x ( ϕ | S ) = c x ( ϕ ) , ∀ x ∈ S o . If W has no non-vanishing sections, then P ( W ) \ Q has measure zero in P ( W ) . The restriction formula on multiplier ideal sheaves can be used to deduced animportant exact sequence which is useful in the inductive arguments.
Theorem 1.5.
Let L be a holomorphic line bundle over a complex manifold X andlet W be a finite dimensional subspace of H ( X, L ) such that W generates all fibers L x , x ∈ X . We denote by P ( W ) the projective space of W . Let ϕ be a quasi-pshfunction on X and let Y i , i ∈ I be the analytic subsets associated to O X / I ( ϕ ) .Suppose L is not a trivial line bundle. Then: • The set A = n [ s ] ∈ P ( W ) (cid:12)(cid:12) s − (0) ⊃ Y i for some i ∈ I o is a countable union of proper analytic subsets of P ( W ) . If, moreover, X is compact, then A is analytic in P ( W ) . ESTRICTION FORMULA 3 • Suppose [ s ] ∈ P ( W ) and S = s − (0) , then the sequence −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ ) | S −→ is exact if and only if [ s ] / ∈ A . Corollary 1.6.
Let d be a base point free linear system on a complex manifold X with dim d < + ∞ , and let φ be a quasi-plurisubharmonic function on X . Thenthere exists a measure zero set n ⊂ d such that for each S ∈ d \ n the divisor S issmooth, I ( ϕ | S ) = I ( ϕ ) | S and the sequence (1.2) 0 −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ | S ) −→ is exact. The above corollary can also be obtained by the approach of Cao ([2]). Followingthe arguments of Cao, we can show the following result.
Theorem 1.7.
Let ϕ be a quasi-psh function on a complex manifold X . Suppose σ is a positive continuous function on X such that I ( ϕ ) = I ((1 + σ ) ϕ ) . Let L be a holomorphic line bundle over X . Let W be a finite dimensional subspace of H ( X, L ) such that W generates all fibers L x , x ∈ X . Then there exists a measurezero set N ⊂ P ( W ) such that for each [ s ] ∈ P ( W ) \ Z the multiplier ideal sheaf I ( ϕ ) can be written as (1.3) I ( ϕ ) x = (cid:26) f ∈ O X,x ; ∃ U x such that Z U x | f | | s | − ε ) e − σ ) ϕ d V < + ∞ (cid:27) , for < σ σ and < ε < σ , where dV is a smooth volume form on X , The existence of σ is guaranteed by the solution of the strong openness conjec-ture (cf. [11], [12]). 2. Bertini type theorem
Let us recall a well-known fact.
Theorem 2.1 (cf. [10]) . Let F : X → Y be a surjective holomorphic map betweencomplex manifolds and let X y = F − ( y ) be the ”full” fiber over a point y ∈ Y (i.e. F − ( y ) is equipped with the structure sheaf coming from Im ( F ∗ ( m y ) → O X ) ).Then the set Z = { y ∈ Y ; X y is not smooth } has Lebesgue measure zero in Y , and the tangent map F ∗ : T X → T Y is surjectiveat each point of X \ F − ( Z ) . If, moreover, F is proper, then Z is analytic in Y . Theorem 2.2.
Let F be a holomorphic vector bundle over a complex manifold X and let W be a finite dimensional subspace of H ( X, F ) such that W generatesall fibers F x , x ∈ X . Let P ( W ) denote the projective space of W . If W has nonon-vanishing sections, then dim X > rank F and the set Z = n [ s ] ∈ P ( W ) (cid:12)(cid:12) s − (0) is not smooth o has Lebesgue measure zero in P ( W ) . If, moreover, X is compact, then Z is analyticin P ( W ) . XIANKUI MENG AND XIANGYU ZHOU
Proof.
We denote by r the rank of F and by N the dimension of W . We first notethat dim W > r since F is generated by sections in W . However, if dim W = r , thenone can find a basis s , · · · , s r ∈ W such that F is generated by these r sections. Inthis case s ( x ) , · · · , s r ( x ) ∈ F x are linearly independent at each point x ∈ X andhence s , · · · , s r are non-vanishing sections in W . Therefore, we may assume that N > r + 1.Let W = X × W be the trivial vector bundle of rank N over X . Since F isgenerated by sections in W , we have a surjective bundle morphism W Φ −−→ F −→ . Here, the bundle morphism Φ is given by the evaluation map( x, s ) s ( x ) , x ∈ X, s ∈ W. Then we have the following exact sequence of vector bundles0 → E → W → F → , where E is a holomorphic subbundle of W and the fibers of E are E x = { s ∈ W ; s ( x ) = 0 } , x ∈ X. Let us consider the projectivized bundles P ( W ) = X × P ( W ) . The points of P ( W ) can be identified with the lines in the fibers of W . The elementsof P ( W ) can be written as( x, [ s ]) ∈ P ( W ) , x ∈ X, s ∈ W. If we define P ( E ) = a x ∈ X P ( E x ) , then π : P ( E ) −→ X is the projectivized bundle of E . Since E is a subbundle of W , we have P ( E ) ⊂ P ( W )and the points of P ( E ) can be represented by n ( x, [ s ]) ∈ P ( W ) ; s ∈ W, s ( x ) = 0 . o The natural projection µ : P ( W ) = X × P ( W ) → P ( W ) , ( x, [ s ]) [ s ]induces a holomorphic map µ : P ( E ) −→ P ( W ) . Thus we obtain the following diagram(2.1) P ( E ) −−−−→ µ P ( W ) y π X We claim that µ is surjective. In fact, for any s ∈ W , we can find a point x ∈ X such that s ( x ) = 0 since W has no non-vanishing sections. Therefore, for any ESTRICTION FORMULA 5 point [ s ] ∈ P ( W ), we can find a point ( x , [ s ]) ∈ P ( E ) such that µ ( x , [ s ]) = [ s ].The surjectivity of µ yields dim P ( E ) > P ( W ). The dimensions of P ( E ) and P ( W )are dim X + N − r − N − X > r .We next consider the fibers of µ . If [ s ] ∈ P ( W ), then the fiber µ − ([ s ]) = { ( x, [ s ]) ∈ X × P ( W ); s ( x ) = 0 } , and it is isomorphic to a subvariety S = { x ∈ X ; s ( x ) = 0 } of X . Here, the iso-morphism is given by the projection π : P ( E ) → X . In what follows we do notdistinguish the fiber µ − ([ s ]) and the subvariety S .If X is compact, then the map µ : P ( E ) → P ( W ) is proper. By Theorem 2.1,one can find a proper analytic subset Z ⊂ P ( W ) such that µ : P ( E ) \ µ − ( Z ) −→ P ( W ) \ Z is a submersion and the fibers µ − ([ s ]) ∼ = s − (0), [ s ] ∈ P ( W ) \ Z are smooth.The proper mapping theorem implies Z is an analytic subset of P ( W ) in case µ isproper.In the general case, the holomorphic map µ : P ( E ) → P ( W ) may not be proper.Let C denote the critical set of µ , which is the set of points in X at which thedifferential µ ∗ : T P ( E ) → T P ( W ) of µ is not surjective. It is obvious that C isan analytic subset of P ( E ). By Sard’s theorem, the set µ ( C ) has measure zeroin P ( W ). Moreover, one can show that µ ( C ) is a countable union of nowheredense closed subsets of P ( W ). Let Z = µ ( C ). Then Z has measure zero in P ( W )and s − (0) ⊂ X is smooth for [ s ] ∈ P ( W ) \ Z . In general, Z is not an analyticsubset. (cid:3) Given a globally generated holomorphic vector bundle F , one can always find afinite dimensional subspace W ⊂ H ( X, F ) to generate all fibers of F . This fact isshown by the following theorem. Theorem 2.3 (cf. [4]) . Let F be a holomorphic vector bundle of rank r over acomplex manifold X . If F is globally generated, then there exists a finite dimen-sional subspace W ⊂ H ( X, F ) , dim W dim X + r , such that W generates allfibers F x , x ∈ X .Remark. If F is an ample vector bundle over a complex manifold X . Suppose F canbe generated by sections in W ⊂ H ( X, F ). Then W has no non-vanishing sectionif and only if F has no trivial subbundles. In fact, if F has a trivial subbundle F ,then we have an exact sequence of vector bundles0 → F → F → Q → . Then F ∼ = F ⊕ Q by the vanishing theorem of Le Potier ([15]). In this case W hasnon-vanishing sections because F can be generated by sections in W . Remark.
Let F be a globally generated homomorphic vector bundle over a complexmanifold X . If rank F > dim X , then F has a trivial subbundle of rank (rank F − dim X ). This is a result due to Serre. For a proof, we may refer to [19].3. Restriction formula on multiplier ideal sheaves
Theorem 3.1.
Let ϕ be a quasi-psh function on a complex manifold X . Let F be a holomorphic vector bundle over X and let W be a finite dimensional subspace XIANKUI MENG AND XIANGYU ZHOU of H ( X, F ) such that W generates all fibers F x , x ∈ X . Let P ( W ) denote theprojective space of W and B = n [ s ] ∈ P ( W ) (cid:12)(cid:12) S = s − (0) is smooth and I ( ϕ | S ) = I ( ϕ ) | S o . If W has no non-vanishing sections, then P ( W ) \ B has measure zero in P ( W ) .Proof. Let us consider the diagram constructed in the proof of Theorem 2.2(3.1) P ( E ) −−−−→ µ P ( W ) y π X For any smooth submanifold S ⊂ X , the Ohsawa-Takegoshi extension theoremimplies that I ( ϕ | S ) ⊂ I ( ϕ ) | S . For the other direction, let U ⊂ X be a small open subset and suppose f ∈ O ( U )and Z U | f | e − ϕ dV < + ∞ . Let us consider the bundle π : π − ( U ) → U , where π − ( U ) ⊂ P ( E ). After shrinking U , we may assume π − ( U ) ∼ = U × P N − r − is a trivial bundle over U . Set F = f ◦ π, e ϕ = ϕ ◦ π. Then F ∈ O (cid:0) π − ( U ) (cid:1) and e ϕ is quasi-psh on π − ( U ). Let dV F S be the volume formon P N − r − associated to the Fubini-Study metric. Let dV π − ( U ) be the smoothvolume form on π − ( U ) induced by dV and dV F S . Since F and e ϕ are constantalong the fibers P N − r − ,(3.2) Z π − ( U ) | F | e − e ϕ dV π − ( U ) = Z P N − r − dV F S · Z U | f | e − ϕ dV < + ∞ by Fubini’s theorem.We first assume that X is compact. Then the map µ : P ( E ) → P ( W ) is proper.By Theorem 2.1, one can find a proper analytic subset Z ⊂ P ( W ) such that µ : P ( E ) \ µ − ( Z ) −→ P ( W ) \ Z is a submersion and the fibers µ − ([ s ]) , [ s ] ∈ P ( W ) \ Z are smooth.Let Ω be a simply connected domain in P ( W ) \ Z . Then µ : µ − (Ω) → Ω isa submersion and hence µ − (Ω) ⊂ P ( E ) is diffeomorphic to the product smoothmanifold Ω × S , where S is a complex submanifold of X determined by an el-ement in Ω. Let dV P ( W ) be a smooth volume form on P ( W ) and dV S a smoothvolume form on S . These two measure induce a smooth volume form dV µ − (Ω) on µ − (Ω). However, by shrinking U and Ω smaller, the two volume forms dV π − ( U ) and dV µ − (Ω) are equivalent. So we can conclude(3.3) Z π − ( U ) | F | e − e ϕ dV µ − (Ω) < + ∞ . ESTRICTION FORMULA 7
By Fubini’s theorem, Z [ s ] ∈ Ω Z U ∩ µ − ([ s ]) | f | e − ϕ dV S ! dV P ( W ) = Z π − ( U ) ∩ µ − (Ω) | F | e − e ϕ dV µ − (Ω) Z π − ( U ) | F | e − e ϕ dV µ − (Ω) < + ∞ (3.4)and hence the set N ( U, Ω , f ) = ( [ s ] ∈ Ω; Z U ∩ µ − ([ s ]) | f | e − ϕ dV S = + ∞ ) has measure zero in P ( W ). For [ s ] ∈ Ω \ N ( U, Ω , f ), we have Z U ∩ µ − ([ s ]) | f | e − ϕ dV S < + ∞ and hence f | S ∈ I ( ϕ | S ). After shrinking U , we may assume that I ( ϕ ) | U is globallygenerated by f , · · · , f m ∈ O ( U ) and Z U | f j | e − ϕ dV < + ∞ , j m. The set N ( U, Ω) = ∪ mj =1 N ( U, Ω , f j ) has measure zero. If [ s ] / ∈ N ( U, Ω), then f j | S ∈ I ( ϕ | S ) for all j and hence I ( ϕ ) | S ∩ U = I ( ϕ | S ∩ U ). Let N ( U ) = { [ s ] ∈ P ( W ) \ Z ; I ( ϕ ) | S ∩ U = I ( ϕ | S ∩ U ) } . Then it is easy to see that the measure of N ( U ) is zero and hence the set N = { [ s ] ∈ P ( W ) \ Z ; I ( ϕ ) | S = I ( ϕ | S ) } . has measure 0. Therefore, the set P ( W ) \ B = Z ∪ N has measure zero in P ( W ).In the general case, the holomorphic map µ : P ( E ) → P ( W ) may not be proper.Let C denote the critical set of µ , which is the set of points in X at which thedifferential µ ∗ : T P ( E ) → T P ( W ) of µ is not surjective. It is obvious that C is ananalytic subset of P ( E ). By Sard’s theorem, the set µ ( C ) has measure zero in P ( W ). Moreover, one can show that µ ( C ) is a countable union of nowhere denseclosed subsets of P ( W ). The proper mapping theorem implies µ ( C ) is an analyticsubset of P ( W ) in case µ is proper. But this is no longer true if µ is not proper.Let p ∈ π − ( U ) \ C be a regular point of µ . Then the differential of µµ ∗ : T p P ( E ) → T µ ( p ) P ( W )is surjective. By the inverse function theorem, there is an open neighborhood U p ⊂ π − ( U ) \ C of p , an open neighborhood Ω p of µ ( p ) and a domain G p ⊂ C n − r such that U p is isomorphic to the product space Ω p × G p and the map µ | U p is givenby the natural projection Ω p × G p → Ω p . In other words, µ − ([ s ]) ∩ U p is isomorphic to G p . XIANKUI MENG AND XIANGYU ZHOU
Suppose dV P ( W ) and dV G p are smooth volume forms on P ( W ) and G p respec-tively. These two measure induce a smooth volume form dV U p on U p . By shrinking U p smaller if necessary, the two volume forms dV π − ( U ) and dV U p are equivalent on U p . Then we can conclude(3.5) Z U p | F | e − e ϕ dV U p < + ∞ . By Fubini’s theorem,(3.6) Z Ω p Z G p | F | e − e ϕ dV G p ! dV P ( W ) = Z U p | F | e − e ϕ dV U p < + ∞ . Thus the set N ( U p , f ) = ( [ s ] ∈ P ( W ) \ µ ( C ); Z U p ∩ µ − ([ s ]) | F | e − e ϕ dV G p = + ∞ ) has measure zero in P ( W ). Remark.
One can define a function ψ : Ω p −→ [ −∞ , + ∞ ) by setting ψ ([ s ]) = − log Z U p ∩ µ − ([ s ]) | F | e − e ϕ dV G p , It is easy to see that ψ is upper semi-continuous and N ( U p , f ) ∩ Ω p = ψ − ( −∞ ) . So N ( U p , f ) is a pluripolar set in case ψ is quasi-psh. In fact, this is true if F ≡ ϕ is invariant under the actions of certain Lie groups(cf. [1], [7]).By the second-countability, we can choose a countable collection { U p j } such that S j U p j = π − ( U ) \ C and the set N ( U p j , f ) = ( [ s ] ∈ P ( W ) \ µ ( C ); Z U pj ∩ µ − ([ s ]) | F | e − e ϕ dV G p = + ∞ ) has measure zero in P ( W ) for all j . Let N ( f ) = [ N ( U p j , f ) . It is obvious that N ( f ) has measure zero.Let K ⋐ U be a compact subset. If [ s ] / ∈ µ ( C ), then µ − ([ s ]) is smooth and theanalytic subset µ − ([ s ]) ∩ π − ( K ) is compact. Since µ − ([ s ]) ∩ C = ∅ , the compactset µ − ([ s ]) ∩ π − ( K ) can be covered by finitely many U p j . If, moreover, s / ∈ N ( f ),then [ s ] / ∈ N ( U p j , f ) and hence(3.7) Z U pj ∩ µ − ([ s ]) | F | e − e ϕ dV G p < + ∞ , ∀ j. Let dS be a smooth volume form on S = s − (0) ∼ = µ − ([ s ]). Then we have(3.8) Z S ∩ K | f | e − ϕ dV S = Z µ − ([ s ]) ∩ π − ( K ) | F | e − e ϕ dV S < + ∞ . In other words, | f | e − ϕ is locally integrable on S ∩ U . Therefore, f | S ∈ I ( ϕ | S ∩ U ) . ESTRICTION FORMULA 9
After shrinking U , we may assume that I ( ϕ ) | U is globally generated by f , · · · , f m ∈ O ( U ) and Z U | f k | e − ϕ dV < + ∞ , k m. The set N ( U ) = ∪ mk =1 N ( f k ) has measure zero. If [ s ] / ∈ N ( U ), then f k | S ∈ I ( ϕ | S )for all k and hence I ( ϕ ) | S ∩ U = I ( ϕ | S ∩ U ). Finally, we may take at most countablemany U i covering X so that N ( U i ) ⊂ P ( W ) has measure zero and I ( ϕ ) | s − (0) ∩ U = I ( ϕ | s − (0) ∩ U )for [ s ] / ∈ N ( U k ) S µ ( C ). Let N = { [ s ] ∈ P ( W ) \ µ ( C ); I ( ϕ ) | S = I ( ϕ | S ) } . Then N ⊂ S k N ( U i ) has measure 0. Therefore, the set P ( W ) \ B = µ ( C ) [ N has measure zero in P ( W ). (cid:3) Let X be a complex manifold (not necessary compact). A complete linear systemon X is defined as the set of all effective divisors linearly equivalent to some givendivisor D . It is denoted | D | . Let L be the line bundle associated to D . In the casethat X is compact the set | D | is in natural bijection with (cid:0) H ( X, L ) \ { } (cid:1) / C ∗ and is therefore a projective space.A linear system d is then a projective subspace of a complete linear system, soit corresponds to a vector subspace W of H ( X, L ). The dimension of the linearsystem d is its dimension as a projective space. Hence dim d = dim W − Corollary 3.2.
Let d be a base point free linear system on a complex manifold X with dim d < + ∞ , and let ϕ be a quasi-plurisubharmonic function on X . If weput b = n S ∈ d ; S is smooth and I ( ϕ ) | S = I ( ϕ | S ) o , then d \ b has measure zero in d . It is clear that b is dense in d when d \ b has measure zero. So the above corollaryimplies Theorem 1.10 in [9]. Remark.
One cannot expect that d \ b is a countable union of analytic subsets of d in general. For counterexamples, one can refer to Example 3.12 in [9].4. Restriction formula on complex singularity exponents
The Ohsawa-Takegoshi L extension theorem implies the following importantmonotonicity result. Proposition 4.1 ([6]) . Let ϕ be a quasi-psh function on a complex manifold X ,and let Y ⊂ X be a complex submanifold such that ϕ | Y
6≡ −∞ on every connectedcomponent of Y . Then, if K is a compact subset of Y , we have (4.1) c K ( ϕ | Y ) c K ( ϕ ) . Theorem 4.2.
Let ϕ be a quasi-psh function on a complex manifold X . Let F be a holomorphic vector bundle over X and let W be a finite dimensional subspaceof H ( X, F ) such that W generates all fibers F x , x ∈ X . Let P ( W ) denote theprojective space of W and Q = n [ s ] ∈ P ( W ) (cid:12)(cid:12) S = s − (0) is smooth and c x ( ϕ | S ) = c x ( ϕ ) , ∀ x ∈ S o . If W has no non-vanishing sections, then P ( W ) \ Q has measure zero in P ( W ) .Proof. Let { x j } j ∈ J be a countable dense subset of X . Let g be a complete Rie-mannian metric on X . Then the collection of open balls B jk = (cid:26) x ∈ X ; dist( x, x j ) < k (cid:27) , j ∈ J, k > X and B jk ⋐ X is compact. The set Z = n [ s ] ∈ P ( W ); s − (0) is not smooth o has Lebesgue measure zero by Theorem 2.2. Fix c ∈ Q ∩ [0 , c ¯ B jk ( ϕ )) and set E j,k,c = n [ s ] ∈ P ( W ) \ Z ; e − cϕ | S / ∈ L on S ∩ B jk o , where S = s − (0) is the submanifold of X .We next show that E j,k,c has measure zero. Again, let us consider the diagram(4.2) P ( E ) −−−−→ µ P ( W ) y π X Since c < c ¯ B jk ( ϕ ), the function e − cϕ is integrable on a neighborhood of B jk . If weset e ϕ = π ∗ ϕ , then e ϕ is quasi-psh on P ( E ) and e − c e ϕ is integrable on a neighborhoodof π − (cid:0) B jk (cid:1) . Let C be the critical set of µ . As in the proof of Theorem 3.1, forany point p ∈ π − (cid:0) B jk (cid:1) , there is an open neighborhood U p of p such that U p isisomorphic to a product space Ω p × G p and the map µ | U p is given by the naturalprojection Ω p × G p → Ω p . By shrinking U p smaller if necessary, we may assume e − c e ϕ is integrable on U p .Then we can apply Fubini’s theorem to conclude the set n [ s ] ∈ P ( W ) \ µ ( C ); e − c e ϕ / ∈ L (cid:0) U p ∩ µ − ([ s ]) (cid:1)o has measure zero. By the second-countability, π − (cid:0) B jk (cid:1) \ C can be covered by acountable collection { U p ℓ } . So the set n [ s ] ∈ P ( W ) \ µ ( C ); e − c e ϕ / ∈ L (cid:0) π − (cid:0) B jk (cid:1) ∩ µ − ([ s ]) (cid:1)o has measure zero too. Note that the map π − (cid:0) B jk (cid:1) ∩ µ − ([ s ]) π −−→ B jk ∩ s − (0)is isomorphic. Thus we can conclude N j,k,c = (cid:8) [ s ] ∈ P ( W ) \ µ ( C ); e − cϕ / ∈ L (cid:0) B jk ∩ s − (0) (cid:1)(cid:9) has measure zero. ESTRICTION FORMULA 11
Now we define N = [ j,k,c N j,c,k , j ∈ J, k > , c ∈ Q ∩ h , c ¯ B jk ( ϕ ) (cid:17) . Then N ∪ Z ⊂ P ( W ) has measure zero. Suppose[ s ] ∈ P ( W ) \ ( Z ∪ N ) , S = s − (0) . We claim that c x ( ϕ ) = c x ( ϕ | S ) , ∀ x ∈ S. To see this, let c be a rational number such that 0 c < c x ( ϕ ). Then there is aneighborhood W of x in X such that e − cϕ is integrable on W . We may assumethat the ball B (cid:0) x, k (cid:1) of center x and radius k is contained in W . Since { x j } j ∈ J isdense in X , we can find an index j ∈ J such that dist ( x j , x ) < k . Then x ∈ B jk = B (cid:18) x j , k (cid:19) ⊂ B (cid:18) x, k (cid:19) ⊂ W. It is obvious that e − cϕ is integrable on B jk . Now S is smooth since [ s ] / ∈ Z and e − cϕ | S is integrable on B jk ∩ S because [ s ] / ∈ N j,k,c . It follows that e − cϕ | S isintegrable in a neighborhood of x ∈ S and hence c x ( ϕ | S ) > c . So we can get thedesired inequlity c x ( ϕ | S ) > c x ( ϕ ) . Finally, we have c x ( ϕ | S ) = c x ( ϕ ) because the inequality in the opposite directionis always true by Proposition 4.1. (cid:3) We next discuss the restriction formula on jumping numbers. Let ϕ be a quasi-psh function on a complex manifold X and let I ⊂ O X be a nonzero coherent idealsheaf. The jumping number c I x ( ϕ ) is defined as follows (see [14]): c I x ( ϕ ) = sup n c > | I | e − cϕ is locally integrable at x o , where ( f j ) j N are local generators of I and | I | = P Nj =1 | f j | . When I = O X , the jumping number reduces to the complex singularity exponent. Argumentssimilar to those in the proof of Theorem 3.1 and Theorem 4.3 easily yield Theorem 4.3.
Let ϕ be a quasi-psh function on a complex manifold X and let I ⊂ O X be a nonzero coherent ideal sheaf. Let F be a holomorphic vector bundle over X and let W be a finite dimensional subspace of H ( X, F ) such that W generatesall fibers F x , x ∈ X . Let P ( W ) denote the projective space of W . If W has nonon-vanishing sections, then there is a measure zero set N ⊂ P ( W ) such that forevery [ s ] / ∈ Z , the subvariety S = s − (0) is smooth and c I x ( ϕ ) c I | S x ( ϕ | S ) , ∀ x ∈ S. A short exact sequence
The following theorem is a special case of Theorem 4 in [21]. We follow theexposition of [16].
Theorem 5.1 (Siu) . Let X be a compact space and let F be a coherent analyticsheaf on X . Then there is a locally finite family { Y i , i ∈ I } of irreducible analyticsubsets of X such that for each x ∈ X the associated primes of F x is Ass O X,x F x = { p x, , · · · , p x,r ( x ) } , where p x, , · · · , p x,r ( x ) are the prime ideals of O X,x associated to the irreduciblecomponents of the germs Y i,x i ∈ I with x ∈ Y i . Definition 5.2 (cf. [21]) . The analytic subsets Y i , i ∈ I of the above theoremare called analytic subsets associated to the sheaf F . The index set I is at mostcountable since the family { Y i , i ∈ I } is locally finite. Lemma 5.3.
Let L be a holomorphic line bundle over a complex manifold X . Let F be a coherent analytic sheaf on X and let Y i , i ∈ I be the analytic subsetsassociated to F . Suppose s is a nonzero section of L and S = s − (0) , then thesequence −→ F ⊗ O ( − S ) ⊗ s −−→ F is exact if and only if S Y i for all i ∈ I .Proof. Locally, we may assume the section s is given by a holomorphic function f and the map F ⊗ O ( − S ) ⊗ s −−→ F is given by F f −→ F .Suppose s vanish on Y i for some i ∈ I . Let p x be the prime ideal associatedto an irreducible component of the germ Y i,x . Since s = 0, Y i is a proper analyticsubset of X and hence p x = 0. We may assume p x = Ann( t x ) for some nonzero t x ∈ F x . Since s vanish on Y i , we have f x ∈ p x = Ann( t x ). Thus f x · t x = 0 andhence F x f x −→ F x is not injective.For the other direction, suppose F x f x −→ F x is not injective for some point x ∈ X .There exists a nonzero section ˜ t x ∈ F x such that f x · ˜ t x = 0. Then f x ∈ Ann(˜ t x ).It is easy to show that every maximal element of the family of ideals { Ann( t x ); 0 = t ∈ F x , Ann( t x ) ⊃ Ann(˜ t x ) } is an associated prime of F x (cf. [17]). Suppose p x ⊃ Ann(˜ t x ) is an associatedprime of F x . Then f x ∈ p x . So the section s vanishes on a component of the germ Y i,x . Since Y i is an irreducible analytic subset of X , we have s − (0) ⊃ Y i . (cid:3) Lemma 5.4.
Let L be a holomorphic line bundle over a complex manifold X andlet W be a finite dimensional subspace of H ( X, L ) such that W generates allfibers L x , x ∈ X . Let F be a coherent analytic sheaf on X and let Y i , i ∈ I bethe analytic subsets associated to F . Let P ( W ) denote the projective space of W .Suppose dim P ( W ) > . Then the set A = n [ s ] ∈ P ( W ) (cid:12)(cid:12) s − (0) ⊃ Y i for some i ∈ I o is a countable union of proper analytic subsets of P ( W ) . If, moreover, X is compact,then A is analytic in P ( W ) .Proof. Let E be the holomorphic vector bundle given by the exact sequence0 → E → X × W → L → . Let us consider the following diagram(5.1) P ( E ) −−−−→ µ P ( W ) y π X ESTRICTION FORMULA 13
Then we can write A = [ i ∈ I \ y ∈ Y i µ ( P ( E y )) . By the definition of µ , the set µ ( P ( E y )) = P ( E y ) ⊂ P ( W ) is a hyperplane andhence T y ∈ Y i µ ( P ( E y )) is a linear subspace of P ( W ). Since the family { Y i , i ∈ I } is locally finite, the index set I is at most countable. So A is at most a countableunion of proper analytic subsets of P ( W ). If X is compact, then I is finite andhence A is an analytic subset of P ( W ). (cid:3) Theorem 5.5.
Let L be a holomorphic line bundle over a complex manifold X andlet W be a finite dimensional subspace of H ( X, L ) such that W generates all fibers L x , x ∈ X . We denote by P ( W ) the projective space of W . Let ϕ be a quasi-pshfunction on X and let Y i , i ∈ I be the analytic subsets associated to O X / I ( ϕ ) .Suppose L is not a trivial line bundle. Then: • The set A = n [ s ] ∈ P ( W ) (cid:12)(cid:12) s − (0) ⊃ Y i for some i ∈ I o is a countable union of proper analytic subsets of P ( W ) . If, moreover, X is compact, then A is analytic in P ( W ) . • Suppose [ s ] ∈ P ( W ) and S = s − (0) , then the sequence (5.2) 0 −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ ) | S −→ is exact if and only if [ s ] / ∈ A .Proof. By Lemma 5.4, we only need to prove the second statement. Let J be theideal sheaf of O X defined by s ∈ W . There is a natural exact sequence(5.3) 0 −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ ) ⊗ ( O X / J ) −→ . By definition, the ideal sheaf I ( ϕ ) | S is the image of the map ρ : I ( ϕ ) ⊗ ( O X / J ) → O X ⊗ ( O X / J ) ∼ = −→ O X / J . To obtain the desired exact sequence, we only need to consider the injectivity ofthe map ρ . From the short exact sequence0 −→ I ( ϕ ) −→ O X −→ O X / I ( ϕ ) −→ → Tor ( O X / I ( ϕ ) , O X / J ) → I ( ϕ ) ⊗ ( O X / J ) → O X / J . Thus ρ is injective if and only ifTor ( O X / I ( ϕ ) , O X / J ) = 0 . From the short exact sequence0 −→ J −→ O X −→ O X / J −→ → Tor ( O X / I ( ϕ ) , O X / J ) → ( O X / I ( ϕ )) ⊗ J → O X / I ( ϕ ) . Therefore, the injectivity of ρ is equivalent to the injectivity of the map( O X / I ( ϕ )) ⊗ J → O X / I ( ϕ ) . Since the ideal sheaf J = O ( − S ), the above map can be written as( O X / I ( ϕ )) ⊗ O ( − S ) ⊗ s −−→ O X / I ( ϕ ) . By Lemma 5.3, we can conclude that ρ is injective if and only [ s ] / ∈ A . Therefore,the sequence 0 −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ ) | S −→ s ] / ∈ A . (cid:3) Corollary 5.6.
Let L be a holomorphic line bundle over a complex manifold X .Let W be a finite dimensional subspace of H ( X, L ) such that W generates allfibers L x , x ∈ X . We denote by P ( W ) the projective space of W . Let ϕ be aquasi-psh function on X . Suppose L is not a trivial line bundle. Then there existsa measure zero set N ⊂ P ( W ) such that for each [ s ] ∈ P ( W ) \ Z the analytic subset S = s − (0) is smooth, I ( ϕ | S ) = I ( ϕ ) | S and the sequence (5.4) 0 −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ | S ) −→ is exact. The following result is essentially due to Cao ([2]) and Guan-Zhou ([11]).
Theorem 5.7.
Let L be a holomorphic line bundle over a complex manifold X . Let W be a finite dimensional subspace of H ( X, L ) such that W generates all fibers L x , x ∈ X . Then there exists a measure zero set N ⊂ P ( W ) such that for each [ s ] ∈ P ( W ) \ Z the following statements holda) the multiplier ideal sheaf I ( ϕ ) can be written as (5.5) I ( ϕ ) x = (cid:26) f ∈ O X,x ; ∃ U x such that Z U x | f | | s | − ε ) e − σ ) ϕ d V < + ∞ (cid:27) , for < σ σ and < ε < σ , where dV is a smooth volume form on X ,b) the divisor S = s − (0) is smooth,c) the following sequence −→ I ( ϕ ) ⊗ O ( − S ) −→ I ( ϕ ) −→ I ( ϕ | S ) −→ is exact.Proof. We can choose s , s , · · · , s N to be a basis of W so that P Nj =1 | s j ( x ) | = 0for any x ∈ X . Let ( τ , τ , · · · , τ N ) be the coordinate of C N . Suppose f ∈ I ( ϕ ) x = I ((1 + σ ) ϕ ) x , < σ < σ . Then Z P Nj =1 | τ j | =1 d τ Z U x | f | | P Nj =1 τ j s j | − ε ) e − σ ) ϕ d V = Z U x | f | (cid:12)(cid:12)(cid:12)P Nj =1 | s j ( y ) | (cid:12)(cid:12)(cid:12) − ε e − σ ) ϕ d V Z P Nj =1 | τ j | =1 d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)P Nj =1 τ j s j ( y ) qP Nj =1 | s j ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ε ) = Z U x | f | (cid:12)(cid:12)(cid:12)P Nj =1 | s j ( y ) | (cid:12)(cid:12)(cid:12) − ε e − σ ) ϕ d V Z P Nj =1 | τ j | =1 d τ | τ | − ε ) < + ∞ . ESTRICTION FORMULA 15
For the last equality, one can change coordinate via unitary transformation so that Z P Nj =1 | τ j | =1 d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)P Nj =1 τ j s j ( y ) qP Nj =1 | s j ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ε ) = Z P Nj =1 | τ j | =1 d τ | τ | − ε ) < + ∞ . By Fubini’s theorem, we can choose ( τ , τ , · · · , τ N ) ∈ C N outside a measure zeroset such that the section s = P Nj =1 τ j s j satisfy Z U x | f | | P Nj =1 τ j s j | − ε ) e − σ ) ϕ d V < + ∞ . Then we can choose [ τ : τ : · · · : τ N ] ∈ C P N − outside a measure set in P ( W ).The first statement is proved.By Theorem 2.2 and Theorem 3.1, we can choose S outside a measure zero setso that S is smooth and the restriction I ( ϕ ) → I ( ϕ | S ) is well-defined.If f ∈ I ( ϕ | S ) x , then the Ohsawa-Takegoshi extension theorem implies that thereexists a F ∈ I ( ϕ ) x such that F | S = f near x . So the the restriction I ( ϕ ) → I ( ϕ | S )is surjective. For the exactness of the middle term, let f ∈ I ( ϕ ) x whose restrictionin I ( ϕ | S ) x is zero. Then f vanish along S near x , hence fs ∈ O x . So we canconclude that(5.6) Z U x | f | | s | − η d V = Z U x (cid:12)(cid:12)(cid:12)(cid:12) fs (cid:12)(cid:12)(cid:12)(cid:12) · | s | − η d V < + ∞ , η > . By H¨older’s inequality, we have Z U x | f | | s | e − δ ) ϕ d V (cid:18)Z U x | f | | s | − ε ) e − σ ) ϕ d V (cid:19) δ σ (cid:18) | f | | s | α d V (cid:19) σ − δ σ , where α = (cid:20) − − ε ) 1 + δ σ (cid:21) σσ − δ . If we choose 0 < δ < σ − ε ε < σ , then α < Z U x | f | | s | e − δ ) ϕ d V < + ∞ . This shows that fs ∈ I ((1 + δ ) ϕ ) x = I ( ϕ ) x . Thus f ∈ I ( ϕ ) x ⊗ O ( − S ) x . (cid:3) References [1] B. Berndtsson,
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Xiankui Meng: School of Science, Beijing University of Posts and Telecommunica-tions, Beijing 100876, People’s Republic of China.
Email address : [email protected] Xiangyu Zhou: Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratoryof Mathematics, Chinese Academy of Sciences, Beijing, 100190, China
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