A note on multiplier ideal sheaves on complex spaces with singularities
aa r X i v : . [ m a t h . C V ] M a r A NOTE ON MULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITHSINGULARITIES
ZHENQIAN LIA bstract . The goal of this note is to present some recent results of our research con-cerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonicfunctions. We firstly introduce multiplier ideal sheaves on complex spaces ( not necessar-ily normal) via Ohsawa’s extension measure, as a special case of which, it turns out to bethe so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As appli-cations, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to theanalytic setting and establish some extension theorems on Kähler manifolds from singular hypersurfaces. Relying on our multiplier and adjoint ideals, we also give characteriza-tions for several important classes of singularities of pairs associated to plurisubharmonicfunctions.Moreover, we also investigate the local structure of singularities of log canonical locusof plurisubharmonic functions. Especially, in the three-dimensional case, we show that forany plurisubharmonic function with log canonical singularities, its associated multiplierideal subscheme is weakly normal, by which we give a complete classification of multiplierideal subschemes with log canonical singularities. C ontents
1. Introduction 21.1. Multiplier and adjoint ideal sheaves on complex spaces with singularities 21.2. Applications to extension theorems and singularities of pairs 31.3. Singularities of log canonical locus of plurisubharmonic functions 42. Multiplier and adjoint ideal sheaves 62.1. Ohsawa’s extension measure 62.2. Multiplier ideal sheaves on complex spaces 72.3. Adjoint ideal sheaves along closed complex subspaces 82.4. The restriction theorem of multiplier ideals 103. Extension theorems on Kähler manifolds from singular hypersurfaces 103.1. Extension theorems for adjoint ideals on Kähler manifolds 103.2. Siu’s extension theorem and deformation invariance of plurigenera 114. Singularities of pairs associated to plurisubharmonic functions 125. Log canonical singularities of plurisubharmonic functions 135.1. Finite generation of graded system of ideal sheaves 135.2. The relation between log canonical locus and symbolic powers 155.3. A solution to Question 1.1 in dimension three 16Appendix A. Constructibility of log canonical threshold of graded system ofideal sheaves from symbolic powers 17Appendix B. Semi-log-canonical hypersurface singularities of dimension two 20References 21
Date : March 27, 2020.2010
Mathematics Subject Classification.
Key words . Multiplier ideal sheaf, Plurisubharmonic function, Ohsawa-Takegoshi L extension theorem,Graded system of ideal sheaves, Log canonical threshold, Semi-log-canonical singularity, Weakly normal com-plex space.E-mail: [email protected].
1. I ntroduction
The multiplier ideal sheaves together with a variant of them, the adjoint ideal sheaves,which measure the singularities of plurisubharmonic functions, turn out to be a powerfultool in complex geometry and algebraic geometry in recent years; one can refer to [14, 26,54]&[37, 32, 55] for a general exposition to the analytic and algebro-geometric side of thetheory respectively. Throughout this article, all complex spaces are always assumed to bereduced and paracompact unless otherwise specified; we refer to [20, 31, 42, 50] for mainreferences on the theory of complex spaces.In the present article, we firstly extend the concept of multiplier ideal sheaves to thesingular case, by which, we then make the adjoint ideal sheaves to be well-defined in theanalytic setting successfully and prove some interesting properties related to them. Sinceour multiplier ideals sheaves can be defined on any complex space of pure dimension ( not necessarily normal or Q -Gorenstein), we also study the singularities of complex spacestogether with plurisubharmonic functions on them by both of ideals; one can refer to [40]for a definition of multiplier ideal sheaves on Q -Gorenstein complex spaces.1.1. Multiplier and adjoint ideal sheaves on complex spaces with singularities.
In[28], Guenancia generalized the notion of adjoint ideal sheaves (along SNC divisors) to theanalytic setting by means of the Ohsawa-Takegoshi-Manivel extension theorem, and thenproved the coherence for the locally Hölder continuous plurisubharmonic weights (see also[32]). However, the analytic adjoint ideal sheaves defined in [28, 32] are not coherent ingeneral and one can find an explicit example given by Guan and the author in [24]. In theevent that we would like to construct analytic adjoint ideals along arbitrary closed complexsubspace of pure codimension as in the algebraic setting [55, 19] and establish an analo-gous adjunction exact sequence, it is necessary for us to understand what is the meaning ofmultiplier ideals on complex spaces (may be singular), i.e., the last non-trivial term in theadjunction exact sequence.Let X be a complex space of pure dimension n , ω X the dualizing sheaf of X and ϕ ∈ L ( X reg ) with respect to the Lebesgue measure. Then, we can define the Nadel-Ohsawamultiplier ideal sheaf b I ( ϕ ) ⊂ M X associated to the weight ϕ on X (see Definition 2.2),via Ohsawa’s extension measure. Moreover, we have the following: Theorem 1.1. (Theorem 2.4).
Let ϕ ∈ Psh( X ) be a plurisubharmonic function on X suchthat ϕ . −∞ on every irreducible component of X. Then, b I ( ϕ ) ⊂ O X is a coherent idealsheaf and satisfies the strong openness property, i.e., b I ( ϕ ) = b I + ( ϕ ) : = [ ε> b I (cid:0) (1 + ε ) ϕ (cid:1) . Remark . If the embedding dimension emb x X ≤ n + x ∈ X , an analogous resulthas been established in [39]. When X is smooth, b I ( ϕ ) is nothing but the usual multiplierideal sheaf I ( ϕ ).In the algebro-geometric setting, the authors in [13, 18] introduced the so-called Mather-Jacobian multiplier ideals via the Nash blow-up, which turned out to be very helpful in thestudy of higher-dimensional algebraic geometry. In fact, we can show that the analyticcorresponding of Mather-Jacobian multiplier ideals is nothing but the multiplier idealsassociated to analytic weights defined above, i.e., Theorem 1.3. (Theorem 2.7).
Let X be a complex space of pure dimension, a ⊂ O X anonzero ideal sheaf on X and c ∈ R ≥ . Then, the Mather-Jacobian multiplier ideal sheafassociated to a c coincides with the Nadel-Ohsawa multiplier ideal sheaf associated to ϕ c · a ,i.e., I MJ ( a c ) = b I ( ϕ c · a ) , where ϕ c · a = c log( P k | f k | ) and ( f k ) is any local system of generators of a . ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 3
Let M be an ( n + r )-dimensional complex manifold and X ⊂ M a closed complexsubspace of pure codimension r with g = ( g , . . . , g m ) a system of generators of I X near x ∈ M ( m may depend on x ). Let ϕ ∈ Psh( M ) be a plurisubharmonic function such that ϕ | X . −∞ on every irreducible component of X . Then, we can obtain the following: Theorem 1.4. (Theorem 2.9).
There exists an ideal sheafAd j X ( ϕ ) ⊂ O M , called the analytic adjoint ideal sheaf associated to ϕ along X, sitting in an exact sequence: −→ I ( ϕ + r log | I X | ) ι −→ Ad j X ( ϕ ) ρ −→ i ∗ b I ( ϕ | X ) −→ , ( ⋆ ) where i : X ֒ → M , ι and ρ are the natural inclusion and restriction map respectively, and log | I X | : = log | g | = log( | g | + · · · + | g m | ) near every point x ∈ M.Remark . (1) Similar to the proof of Proposition 2.11 in [28], it follows that Ad j X ( ϕ )coincides with the algebraic adjoint ideal sheaf defined by Takagi and Eisenstein in [55, 19]whenever ϕ has analytic singularities.(2) If ϕ is a Hölder plurisubharmonic function and X is a smooth divisor, our definitionof Ad j X ( ϕ ) is the same as that given by Guenancia (see Theorem 2.16 in [28]). Similarly,we always have Ad j X ( ϕ ) ⊂ I ( ϕ ), and Ad j X ( ϕ ) x = I ( ϕ ) x for each x < X as well.1.2. Applications to extension theorems and singularities of pairs.
Using our notion ofmultiplier and adjoint ideal sheaves, we firstly establish the following extension theoreminvolved multiplier and adjoint ideals on Kähler manifolds from singular hypersufaces.
Theorem 1.6.
Let ( M , ω ) be a weakly pseudoconvex Kähler manifold and H ⊂ M a (re-duced) complex hypersurface. Let L be a holomorphic line bundle on M equipped with a(possibly singular) Hermitian metric e − ϕ L such that ϕ L | H . −∞ on each irreducible com-ponent of H and √− ∂ ¯ ∂ϕ L ≥ εω for some positive continuous function ε on M. Then, thenatural restriction map induces a surjectionH (cid:0) M , ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ L ) (cid:1) −→ H (cid:0) H , ω H ⊗ O H ( L | H ) ⊗ b I ( ϕ L | H ) (cid:1) , and H q (cid:0) M , ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ L ) (cid:1)g −→ H q (cid:0) H , ω H ⊗ O H ( L | H ) ⊗ b I ( ϕ L | H ) (cid:1) for all q ≥ .Remark . In fact, we can further deduce H q (cid:0) M , ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ L ) (cid:1) = q ≥
1, thanks to a singular version of Nadel vanishing theorem in [14].Combining the above result and Siu’s construction of metric, we obtain a singular ver-sion of Takayama’s extension theorem in [56] as follows.
Theorem 1.8. (Theorem 3.3).
Let M be a smooth complex projective variety and H ⊂ Ma (reduced) hypersurface. Let L be a holomorphic line bundle over M equipped with a(possibly singular) Hermitian metric e − ϕ L such that √− ∂ ¯ ∂ϕ L ≥ εω with ε > for somesmooth Hermitian metric ω on M and b I ( ϕ L | H ) = O H .Then, the natural restriction mapH (cid:0) M , (cid:0) ω M ⊗ O M ( L ⊗ [ H ]) (cid:1) ⊗ m (cid:1) −→ H (cid:0) H , (cid:0) ω H ⊗ O H ( L ) (cid:1) ⊗ m (cid:1) is surjective for every m > . Moreover, we can also establish the following singular version of extension theoremfor projective family due to Siu [51, 52, 53] (see also [48]), by which we generalize Siu’stheorem on plurigenera to the case of singular fibers with log terminal singularities (seeCorollary 3.6).
ZHENQIAN LI
Theorem 1.9. (Theorem 3.4).
Let π : M → ∆ be a projective family and L a holomorphicline bundle over M endowed with a (possibly singular) Hermitian metric e − ϕ L such that √− ∂ ¯ ∂ϕ L ≥ . Assume that the restriction of ϕ L to the central fiber M is well defined andM has at most log terminal singularities.Then, the natural restriction mapH (cid:0) M , ω ⊗ mM ⊗ O M ( L ) (cid:1) −→ H (cid:0) M , ω ⊗ mM ⊗ O M ( L ) ⊗ b I ( ϕ L | M ) (cid:1) is surjective for every m > . Whereas the multiplier and adjoint ideals encode much information on the singularitiesof the underlying complex space together with the plurisubharmonic weights on them, wegive the following characterization of singularities of pairs associated to plurisubharmonicfunctions.
Theorem 1.10. (Theorem 4.2).
Let X be a complex space of pure dimension with x ∈ X apoint and ϕ ∈ Psh( X ) . Then, (1) If ( X , ϕ ) is log terminal at x, then x is a rational singularity of X. (2) If ( X , ϕ ) is log canonical at x, then the complex subspace ( A , ( O X / b I ( ϕ )) | A ) is re-duced near x; in addition, if ϕ has analytic singularities, then ( A , ( O X / b I ( ϕ )) | A ) is weaklynormal at x, where A : = N ( b I ( ϕ )) denotes the zero-set of coherent ideal sheaf b I ( ϕ ) . When X is locally a complete intersection, analogous to the characterization of rational-ity of hypersurface singularities in [39], we can establish the following Theorem 1.11.
Let X ⊂ M be locally a complete intersection and ϕ ∈ Psh( M ) such thatthe slope ν x ( ϕ | X ) = for every x ∈ X. Then, the analytic adjoint ideal sheaf Ad j X ( ϕ ) = O M i ff b I ( ϕ | X ) = O X i ff X is normal and has only rational singularities i ff ( X , x ) is canonicalfor any x ∈ X.Remark . Note that our multiplier (resp. adjoint) ideals measure both the singularitiesof the plurisubharmonic weights and associated complex spaces (resp. subspaces) together.The above result also implies that if a locally complete intersection has at most rationalsingularities, the plurisubharmonic weights with zero slope do not add the singularities inthe sense of multiplier or adjoint ideals.
Remark . If M is a smooth complex algebraic variety and ϕ is trivial, the above resultcoincides with Proposition 9.3.48 (ii) in [37], which turned out to be very useful to studythe singularities of theta divisors on principally polarized abelian varieties.1.3. Singularities of log canonical locus of plurisubharmonic functions.
Let Ω ⊂ C n be a domain with o ∈ Ω the origin and u ∈ Psh( Ω ) a plurisubhamonic function on Ω .The subscheme (or complex subspace) V ( u ) : = ( A , ( O Ω / I ( u )) | A ) of Ω cut out by themultiplier ideal sheaf I ( u ) is called a multiplier ideal subscheme associated to u , where A : = N ( I ( u )) denotes the zero-set of coherent ideal sheaf I ( u ); see e.g. [14, 17, 44].The log canonical threshold (or complex singularity exponent ) c o ( u ) or LCT o ( u ) of u at o is defined to be c o ( u ) : = sup { c ≥ | exp( − cu ) is integrable near o } . It is convenient to put c o ( −∞ ) =
0. If c o ( u ) =
1, we say that ( A , o ) is the germ of logcanonical locus of u at o .Motivated by (2) of Theorem 1.10, it is natural to raise the following question on theweak normality of log canonical locus of plurisubharmonic functions. Question 1.1.
Let o ∈ Ω ⊂ C n be the origin, u ∈ Psh( Ω ) with c o ( u ) = . It is natural towonder whether we are able to show that the multiplier ideal subscheme V ( u ) is weaklynormal at o. ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 5
In [22], we answered the above Question a ffi rmatively in dimension two. Owing tothe absence of desingularization theorem for plurisubharmonic functions, we cannot dealwith the above question like the algebraic situation, and so it is reasonable to find anotherplurisubharmonic weight with mild singularities whose multiplier ideal cuts out the samesubscheme as V ( u ). Thanks to Demailly’s analytic approximation of plurisubharmonicfunctions via Bergman kernels, we can establish the following Theorem 1.14. (Theorem 5.13).
Let a • = { a m } be the graded system of ideal sheaves on Ω given by m Λ -th symbolic powers a m = I < m Λ > A , where m k = e k m for each k and e k is thecodimension of A k in Ω . Then, it follows that LCT o ( a • ) = and I ( a • ) o = I ( u ) o . As an application of above result, we can confirm Question 1.1 in dimension three, i.e.,
Theorem 1.15.
Let Ω ⊂ C be a domain containing the origin o and u ∈ Psh( Ω ) withc o ( u ) = . Then, the multiplier ideal subscheme V ( u ) is weakly normal at o.Remark . As shown in Corollary 5.15 and Corollary 5.17, the answer of Question 1.1is positive if ( A , o ) is a complete intersection, or o is an isolated singularity of A .In view of Theorem 1.15, we can in fact establish a strengthening of the statement asbelow. Corollary 1.17.
With the same hypotheses as in Theorem 1.15, we can derive that anyunion of irreducible components of V ( u ) with the reduced complex structure is weaklynormal near o.Remark . By the same argument as in the proof of Corollary 1.17, we can obtain thatif the answer of Question 1.1 is positive, then any union of irreducible components of themultiplier ideal subscheme V ( u ) is weakly normal near o .Furthermore, as a consequence of Theorem 1.15, we are able to characterize the multi-plier ideal subschemes with log canonical singularities as follows. Theorem 1.19.
Let o ∈ Ω ⊂ C be the origin and u ∈ Psh( Ω ) with c o ( u ) = . Then, themultiplier ideal subscheme V ( u ) at o is exactly defined by one of the following ideals: (1) ( h ) · O , where h ∈ O is the minimal defining function of a germ of hypersurfacewith semi-log-canonical singularity at o; (2) ( z z , z z ) · O ; (3) ( z , z ) · O ; (4) ( z , z z ) · O ; (5) ( z z , z z , z z ) · O ; (6) m , up to a change of variables at o; where m is the maximal ideal of O .Remark . For the sake of convenience, we provide a complete list on the classificationof semi-log-canonical hypersurface singularities of dimension two in Table 1 of AppendixB; one can refer to [36] for more details (see also [41]).Moreover, all the cases in Theorem 1.19 will occur when we take u to be the followingweights respectively:(1) u = log | h | ;(2) u = log( | z z | + | z z | );(3) u = log( | z | + | z | );(4) u = log( | z | + | z z | );(5) u = log( | z z z | + | z z | + | z z | + | z z | );(6) u = log( | z | + | z | + | z | ).Based on Theorem 1.19, in a subsequent paper [23], we are able to answer the Ques-tion posed in [22] for the three-dimensional case, by combining the Ohsawa-Takegoshi ZHENQIAN LI L extension theorem proved in [47] with the restriction formula established in [27]. Inparticular, we can prove the following result in [23]: Theorem 1.21.
Let o ∈ Ω ⊂ C × C n − ( n ≥ be the origin, u ∈ Psh( Ω ) and H = { z = · · · = z n = } ⊂ Ω a three-dimensional plane through o. If c o ( u | H ) = , then there exists anew local coordinates ( w , w , w ; z , ..., z n ) near o such that I ( u ) o is exactly equal to oneof the following ideals: (1) O n ; (2) (cid:0) w , w , w (cid:1) · O n ; (3) (cid:0) w (cid:1) · O n ; (4) (cid:0) w , w w + f ( w , ..., z n ) (cid:1) · O n ; (5) (cid:0) w w , w w , w w (cid:1) · O n ; (6) (cid:0) w w , w w (cid:1) · O n ; (7) (cid:0) h ( w , w , w ) + f ( w , ..., z n ) (cid:1) · O n , where h ∈ O is as in Theorem 1.19 and f ∈ I H , o . Finally, relying on Theorem 1.15, we give the following characterization of log canoni-cal locus of pair ( X , a • ) associated to graded system of ideals in the algebraic setting. Corollary 1.22.
Let X be a smooth complex quasi-projective threefold and a • = { a m } beany graded system of ideals on X.If the log canonical threshold LCT( a • ) = on X, then the multiplier ideal subschemeV ( a • ) associated to a • is weakly normal. In further, all the singularities of V ( a • ) are pre-cisely those presented in Theorem 1.19.
2. M ultiplier and adjoint ideal sheaves
In this section, we firstly present the Ohsawa’s extension measure arising from the re-search of so-called Ohsawa-Takigoshi L extension theorem. Next, we will construct theanalytic multiplier and adjoint ideal sheaves on complex spaces via the measure and thenprove some basic properties of them.2.1. Ohsawa’s extension measure.
In order to establish a general L extension theorem,Ohsawa [46] introduced a positive measure on regular part of the associated closed com-plex subspace. Afterwards, associated with the same measure, Guan and Zhou [25] estab-lished two very general L extension theorems with optimal estimates; see also [16, 10, 11]for various L extension theorems related to Ohsawa’s extension measure. In the subse-quent parts, we will use the Ohsawa’s extension measure to study the multiplier and adjointideal sheaves on complex spaces with singularities.First of all, let’s recall some notations in [46] (see also [25]). Let M be an ( n + r )-dimensional complex manifold and X a (closed) complex subspace of M . Let dV M bea continuous volume form on M . Then, we consider a class of upper semi-continuousfunctions Ψ from M to the interval [ −∞ , A ) with A ∈ ( −∞ , + ∞ ], such that(1) Ψ − ( −∞ ) ⊃ X and(2) If X is k -dimensional around a point x ∈ X reg ( the regular part of X ), there exists alocal coordinate ( z , · · · , z n + r ) on a neighborhood U of x such that z k + = · · · = z n + r = X ∩ U and sup U \ X (cid:12)(cid:12)(cid:12) Ψ ( z ) − ( n + r − k ) log n + r X j = k + | z j | (cid:12)(cid:12)(cid:12) < + ∞ . The set of such functions Ψ will be denoted by A ( X ). For each Ψ ∈ A ( X ), one can as-sociate a positive measure dV X [ Ψ ] on X reg as the minimum element of the partially orderedset of positive measures d µ satisfying Z X k f d µ ≥ lim sup t → + ∞ n + r − k ) σ n + r − k ) − Z M f e − Ψ {− t − < Ψ < − t } dV M ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 7 for any nonnegative continuous function f with Supp f ⊂⊂ M \ X sing , where X k denotes the k -dimensional component of X reg , σ m denotes the volume of the unit sphere in R m + and {− t − < Ψ < − t } denotes the characteristic function of the set { z ∈ M | − t − < Ψ ( z ) < − t } . Remark . If X ⊂ M is a complex subspace of pure codimension r such that I X isglobally generated by holomorphic functions g , . . . , g m on M , by taking Ψ = r log( | g | + · · · + | g m | ) , one can check that the measure dV X [ Ψ ] on X reg can be defined by Z X reg f dV X [ Ψ ] = lim sup t → + ∞ r σ r − Z M f e − Ψ {− t − < Ψ < − t } dV M for any nonnegative continuous function f with Supp f ⊂⊂ M \ X sing .By Theorem 2.0.2 in [62] (see also [19], Corollary 3.2), we obtain a strong factorizingdesingularization π : e M → M of X such that π ∗ ( I X ) = I e X · I R X , where e X is the stricttransform of X in e M and R X is an e ff ective divisor supported on the exceptional divisorEx( π ) of π . Then, one can check that the measure dV X [ Ψ ] is the direct image of measuresdefined upstairs by f Z e X (cid:12)(cid:12)(cid:12) f ◦ π | e X (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Jac( π ) h rR X (cid:12)(cid:12)(cid:12)(cid:12) e X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dV e X , up to equivalence.In particular, if M is a domain in C n + r and X ⊂ M is a complete intersection, then onecan check that dV X [ Ψ ] is the measure on X reg such that dV X [ Ψ ] = | Λ r ( dg ) | dV X , where dV X = n ! ω n | X reg and ω = √− n + r X k = dz k ∧ d ¯ z k . Multiplier ideal sheaves on complex spaces. Di ff erent from the algebraic multiplierideals, the usual analytic multiplier ideals are constructed by the integrability of multipliersassociated to plurisubharmonic weights with respect to the Lebesgue measure. In the ana-lytic setting, the main di ffi culty of extending the notion of multiplier ideals to the singularcase is how to choose a suitable measure in the construction by means of integrability.For Q -Gorenstein complex spaces (e.g., complete intersections), it is natural to choosean adapted measure through the associated pluricanonical forms (cf. [40]). Unfortunately,the approach is out of work for the general case. In the following, we will construct themultiplier ideals on complex spaces of pure dimension, via the Ohsawa’s extension mea-sure. Definition 2.2.
Let X be a complex space of pure dimension n and ϕ ∈ L ( X reg ) withrespect to the Lebesgue measure. Let X ֒ → Ω ⊂ C n + r be a local embedding such that I X is generated by holomorphic functions g , . . . , g m on Ω .The Nadel-Ohsawa multiplier ideal sheaf b I ( ϕ ) on X is defined to be the fractional idealsheaf of germs of meromorphic functions f ∈ M X , x such that | f | e − ϕ is locally integrableat x on X with respect to the measure dV X [ Ψ ], where Ψ = r log( | g | + · · · + | g m | ). One cancheck that b I ( ϕ ) is independent of the local embedding of X and the choice of generatorsof I X . Remark . (1) By a suitable choice of the polar function Ψ , we can also define multiplierideal sheaves on complex spaces not necessarily of pure dimension. ZHENQIAN LI (2) If the complex space X ⊂ M is locally a complete intersection (not necessarilynormal), thanks to the adjunction formula, it yields that ω GR X ( ϕ ) = ω X ⊗ b I ( ϕ ), where ω GR X ( ϕ ) is the O X -sheaf on X defined by Γ ( U , ω GR X ( ϕ )) = { σ ∈ Γ ( U ∩ X reg , ω X reg ) | ( √− n e − ϕ σ ∧ σ ∈ L ( U ) } for any open subset U ⊂ X .Analogous to the smooth case, we have the following: Theorem 2.4.
With the same notations as above and ϕ ∈ Psh( X ) , it follows that (1) (Coherence). b I ( ϕ ) ⊂ O X is a coherent ideal sheaf. (2) (Strong openness). b I ( ϕ ) = b I + ( ϕ ) : = S ε> b I (cid:0) (1 + ε ) ϕ (cid:1) . For the proof, we may assume that X is a complex subspace of some domain Ω in C n + r and π : e Ω → Ω is a strong factorizing desingularization of X such that π ∗ ( I X ) = I e X · I R X ,where e X is the strict transform of X in e Ω and R X is an e ff ective divisor supported on Ex( π ).Then, we can show that b I ( ϕ ) = π ∗ (cid:16) O ( K e Ω / Ω − rR X ) | e X ⊗ I ( ϕ ◦ π | e X ) (cid:17) . Remark . In fact, for the strong openness of multiplier ideal sheaves, if ( ϕ k ) is a se-quence of plurisubharmonic functions converging to ϕ with ϕ k ≤ ϕ on X , then we have b I ( ϕ ) = b I + ( ϕ ) = [ k b I ( ϕ k ) . Remark . If ϕ A is a (quasi)-plurisubharmonic function on X with analytic singularitiesand ϕ A . −∞ on every irreducible component of X , by combining with an argument of logresolution, we can deduce that the fractional ideal sheaf b I ( ϕ − ϕ A ) ⊂ M X is coherent andsatisfies the strong openness property (cf. [40]).Combining with Lemma 4.4 in [19], we achieve the equivalence of Mather-Jacobianand Nadel-Ohsawa multiplier ideal sheaves for analytic weights as follows. Theorem 2.7.
Let X be a complex space of pure dimension, a ⊂ O X a nonzero ideal sheafon X and c ∈ R ≥ . Then, the Mather-Jacobian multiplier ideal sheaf associated to a c coincides with the Nadel-Ohsawa multiplier ideal sheaf associated to ϕ c · a , i.e., I MJ ( a c ) = b I ( ϕ c · a ) , where ϕ c · a = c log( P k | f k | ) and ( f k ) is any local system of generators of a .Remark . In fact, similar to the argument in the proof of Theorem 2.7, we can derivethe following characterization of Ohsawa’s extension measure.Let ( M , ω ) be a Hermitian manifold and X ⊂ M a (closed) complex subspace of puredimension d with the volume element dV X ,ω = d ! ω d | X reg . Let Ψ ∈ A ( X ) be a polar functionon M determined by the ideal sheaf I X . Then, there exists a locally bounded positivefunction ρ on X such that dV X ,ω [ Ψ ] = ρ · dV X ,ω |J ac X | , where |J ac X | is globally defined via a partition of unity on the coordinate charts coveringof X .2.3. Adjoint ideal sheaves along closed complex subspaces.
In [28], Guenancia definedan analytic adjoint ideal sheaf associated to plurisubharmonic weight ϕ along SNC divisors(but not coherent for general ϕ ; see [24] for more details), and established the so-calledadjunction exact sequence for the locally Hölder continuous plurisubharmonic weights andsmooth divisors. Later, the author [39] obtained a well-defined generalization of adjointideal sheaves to the analytic setting in the case of divisors. In this subsection, owing to our ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 9 multiplier ideals introduced as above, we will make a reasonable version of the definitionof analytic adjoint ideal sheaves along closed complex subspaces of higher codimension.
Theorem 2.9. (Theorem 1.4).
Let M be an ( n + r ) -dimensional complex manifold andX ⊂ M a closed complex subspace of pure codimension r with g = ( g , . . . , g m ) a system ofgenerators of I X near x ∈ M (m may depend on x) and ϕ ∈ Psh( M ) such that ϕ | X . −∞ on every irreducible component of X.Then, there exists an ideal sheaf Ad j X ( ϕ ) ⊂ O M , called the analytic adjoint ideal sheaf associated to ϕ along X, sitting in an exact sequence: −→ I ( ϕ + r log | I X | ) ι −→ Ad j X ( ϕ ) ρ −→ i ∗ b I ( ϕ | X ) −→ , ( ⋆ ) where i : X ֒ → M , ι and ρ are the natural inclusion and restriction map respectively, and log | I X | : = log | g | =
12 log( | g | + · · · + | g m | ) near every point x ∈ X.Sketch of the proof . As the statement is local, without loss of generality, we may assumethat M is a bounded Stein domain in C n + r , g = ( g , . . . , g m ) is a system of generators of I X . Case (i).
When X ⊂ M is a closed complex submanifold of pure codimension r.
Let J ⊂ O M be an ideal sheaf such that J | X = b I ( ϕ | X ), which implies that J + I X is independent of the choice of J . Set Ad j X ( ϕ ) : = [ ε> Ad j X (cid:16) (1 + ε ) ϕ (cid:17) ∩ (cid:16) J + I X (cid:17) , where Ad j X ( ϕ ) ⊂ O M is an ideal sheaf of germs of holomorphic functions f ∈ O M , x suchthat | f | e − ϕ | g | r log | g | is locally integrable with respect to the Lebesgue measure near x on M . Case (ii).
When X ⊂ M is a closed complex subspace with singularities.
Let π : e M → M be a strong factorizing desingularization of X such that π ∗ ( I X ) = I e X · I R X , where e X is the strict transform of X in e M and R X is an e ff ective divisor supportedon Ex( π ). Thanks to Case (i), we have the following adjunction exact sequence0 −→ I ( ϕ ◦ π + r log | I e X | ) −→ Ad j e X ( ϕ ◦ π ) −→ i ∗ b I ( ϕ ◦ π | e X ) −→ . Twist the exact sequence by O e M ( K e M / M − rR X ), and then we deduce that0 −→ O e M ( K e M / M ) ⊗ I ( ϕ ◦ π + r log | I X | ◦ π ) −→ Ad j e X ( ϕ ◦ π ) ⊗ O e M ( K e M / M − rR X ) −→ i ∗ I ( ϕ ◦ π | e X ) ⊗ O e M ( K e M / M − rR X ) −→ . Then, we can take
Ad j X ( ϕ ) : = π ∗ (cid:16) Ad j e X ( ϕ ◦ π ) ⊗ O e M ( K e M / M − rR X ) (cid:17) . Here, in order to establish the exact sequence ( ⋆ ), a local vanishing theorem is necessary(cf. [43], Corollary 1.5). Remark . (1) The adjunction exact sequence ( ⋆ ) yields the coherence and uniqueness(independent of the desingularization π ) of Ad j X ( ϕ ).(2) If X ⊂ M is a complex hypersurface, we can also get the following short exactsequence by a similar discussion as in Proposition 9.3.48 in [37]:0 −→ O M ( K M ) −→ O M ( K M + X ) ⊗ Ad j X ( ϕ ) −→ π ∗ O e X ( K e X ) −→ , where π : e M → M is an embedded resolution of ( M , X ) such that the proper transform e X ⊂ e M of X is non-singular.(3) When M is an algebraic variety over C , X ⊂ M is a Q -Gorenstein reduced equidi-mensional subscheme and ϕ has analytic singularities, the above result is nothing but themain theorem (Theorem 5.1) in [19], which answered a question of Takagi in [55].2.4. The restriction theorem of multiplier ideals.
This part is devoted to discuss therelation of multiplier ideal sheaf b I ( ϕ | X ) and the restriction of Ad j X ( ϕ ) and I ( ϕ ) to X .Concretely, we can prove the following: Theorem 2.11.
With hypotheses as above, it follows that b I ( ϕ | X ) ⊂ I (cid:0) ϕ + ( r − δ ) log | g | (cid:1) · O X for any < δ ≤ r; and b I ( ϕ | X ) = Ad j X ( ϕ ) · O X . Proof.
The first inclusion is a direct consequence of Theorem 2.4 (strong openness) and alocal version of Theorem 2.2 in [25], with weights (1 + ε ) ϕ .Twist the exact sequence ( ⋆ ) through by O X , and then we deduce Ad j X ( ϕ ) ·O X = b I ( ϕ | X )from the right exactness of tensor functor. (cid:3) Remark . Note that
Ad j X ( ϕ ) ⊂ I (cid:0) ϕ + ( r − δ ) log | g | (cid:1) for any 0 < δ ≤ r by the definitionof Ad j X ( ϕ ). Then, we can also infer the first inclusion from b I ( ϕ | X ) = Ad j X ( ϕ ) · O X .3. E xtension theorems on K¨ ahler manifolds from singular hypersurfaces Extension theorems for adjoint ideals on Kähler manifolds.
In this subsection,we turn to obtain a global extension theorem related to adjoint ideals on Kähler manifoldsfrom singular hypersuraces by combining the adjunction exact sequence ( ⋆ ) with the Nadelvanishing theorem. Proof of Theorem
It follows from the adjunction formula for the hypersurface H that ω H = (cid:0) ω M ⊗ O M ([ H ]) (cid:1) | H . Then, we consider the long exact sequence of cohomologyassociated to the short exact sequence ( ⋆ ) twisted by ω M ⊗ O M ( L ⊗ [ H ]), i.e.,0 −→ ω M ⊗ O M ( L ) ⊗ I ( ϕ ) −→ ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ ) −→ i ∗ (cid:0) ω H ⊗ O H ( L | H ) ⊗ b I ( ϕ | H ) (cid:1) −→ . Thereupon, the desired result follows immediately from the Nadel vanishing theorem. (cid:3)
In addition, if M is compact and ( L , e − ϕ ) is only a pseudo-e ff ective line bundle (notnecessarily big), by replacing the Nadel vanishing theorem in the proof of above theoremby a Kawamata–Viehweg–Nadel-type vanishing theorem established in [9, 26], i.e., H q (cid:0) M , ω M ⊗ O M ( L ) ⊗ I ( ϕ ) (cid:1) = q ≥ n − nd( L , ϕ ) +
1, where nd( L , ϕ ) is the numerical dimension of ( L , ϕ ) as definedin [9], we can obtain the following: Theorem 3.1.
Let M be a compact Kähler manifold of dimension n and H ⊂ M a (reduced)complex hypersurface. Let ( L , e − ϕ ) be a pseudo-e ff ective line bundle on M such that ϕ | H . −∞ on each irreducible component of H. Then, the natural restriction map induces asurjectionH q − (cid:0) M , ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ ) (cid:1) −→ H q − (cid:0) H , ω H ⊗ O H ( L | H ) ⊗ b I ( ϕ | H ) (cid:1) , and H q (cid:0) M , ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ ) (cid:1)g −→ H q (cid:0) H , ω H ⊗ O H ( L | H ) ⊗ b I ( ϕ | H ) (cid:1) ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 11 for all q ≥ n − nd( L , ϕ ) + .Remark . In particular, if H ⊂ M is a smooth complex hypersurface in Theorem 3.1,then we have H q (cid:0) M , ω M ⊗ O M ( L ⊗ [ H ]) ⊗ Ad j H ( ϕ ) (cid:1) = q ≥ n − nd( L , ϕ ) +
1, by the corresponding vanishing theorems.Combining Theorem 1.6 with Siu’s construction of the metric in [51, 53] (see also [60]),we can derive the following generalization of Takayama’s extension theorem from singularhypersurface.
Theorem 3.3. (Takayama’s extension theorem, [56]).
Let M be a smooth complex projec-tive variety and H ⊂ M a (reduced) hypersurface. Let L be a holomorphic line bundle overM equipped with a (possibly singular) Hermitian metric e − ϕ L such that √− ∂ ¯ ∂ϕ L ≥ εω with ε > for some smooth Hermitian metric ω on M and b I ( ϕ L | H ) = O H .Then, the natural restriction mapH (cid:0) M , (cid:0) ω M ⊗ O M ( L ⊗ [ H ]) (cid:1) ⊗ m (cid:1) −→ H (cid:0) H , (cid:0) ω H ⊗ O H ( L ) (cid:1) ⊗ m (cid:1) is surjective for every m > . Siu’s extension theorem and deformation invariance of plurigenera.
Let M be acomplex manifold and π : M → ∆ = { t ∈ C | | t | < } a proper surjective holomorphicmap with reduced analytic fibers M t = π − ( t ). The holomorphic family π : M → ∆ iscalled projective if there is a positive holomorphic line bundle over M . Then, we are ableto establish the following singular version of extension theorem for projective family dueto Siu [51, 52, 53] (see also [48]), Theorem 3.4.
Let π : M → ∆ be a projective family and L a holomorphic line bundle overM endowed with a (possibly singular) Hermitian metric e − ϕ L such that √− ∂ ¯ ∂ϕ L ≥ .Assume that the restriction of ϕ L to the central fiber M is well defined and M has at mostlog terminal singularities.Then, the natural restriction mapH (cid:0) M , ω ⊗ mM ⊗ O M ( L ) (cid:1) −→ H (cid:0) M , ω ⊗ mM ⊗ O M ( L ) ⊗ b I ( ϕ L | M ) (cid:1) is surjective for every m > . Analogous to the proof of Theorem 3.4, we deduce the following extension theoremfor projective family with singular fibers; see [12] for an argument on the case of smoothfibers.
Corollary 3.5.
Let π : M → ∆ be a projective family and ( L , h L ) a holomorphic linebundle over M endowed with a (possibly singular) Hermitian metric e − ϕ L such that thecurvature current √− ∂ ¯ ∂ϕ L ≥ . Assume that the restriction of ϕ L to the central fiber M is well defined and b I ( ϕ L | M ) = O M .Then, the natural restriction mapH (cid:0) M , (cid:0) ω M ⊗ O M ( L ) (cid:1) ⊗ m (cid:1) −→ H (cid:0) M , (cid:0) ω M ⊗ O M ( L ) (cid:1) ⊗ m (cid:1) is surjective for every m > . As a straightforward application of Theorem 3.4, we obtain the following singular ver-sion of deformation invariance of plurigenera established by Siu in [51, 52]:
Corollary 3.6. (Siu’s theorem on plurigenera).
Let π : M → ∆ be a projective family andassume that every fiber M t has at most log terminal singularities. Then for each m ≥ , theplurigenus P m ( M t ) : = dim C H ( M t , ω ⊗ mM t ) is independent of t. Remark . In fact, in our situation, we have the equivalence of log terminal singularitiesand canonical singularities by Theorem 1.11. Thus, the above invariance of plurigenerahas been established in [57] by combining a complicated L extension theorem with somealgebraic techniques.4. S ingularities of pairs associated to plurisubharmonic functions In this section, we study the singularities of complex spaces ( not necessarily normal or Q -Gorenstein). Concretely, we will characterize several important classes of singularitiesof pairs associated to plurisubharmonic functions in terms of the triviality of our multiplierand adjoint ideals. We refer to [13, 18] and [3, 34] for some algebraic counterparts onsingularities defined by discrepancies. Definition 4.1.
Let X be a complex space of pure dimension with x ∈ X a point and ϕ ∈ Psh( X ) a plurisubharonic function on X .The pair ( X , ϕ ) is log terminal at x if b I ( ϕ ) x = O X , x . The point x is called a log terminalsingularity of X if trivial pair ( X ,
0) is log terminal at x .The pair ( X , ϕ ) is log canonical at x if b I ((1 − ε ) · ϕ ) x = O X , x for all 0 < ε <
1. ( X , ϕ ) is log terminal (resp. log canonical ) if it is log terminal (resp. log canonical) at every pointin X . Theorem 4.2.
Let X be a complex space of pure dimension with x ∈ X a point and ϕ ∈ Psh( X ) . Then, (1) If ( X , ϕ ) is log terminal at x, then x is a rational singularity of X. (2) If ( X , ϕ ) is log canonical at x, then the complex subspace ( A , ( O X / b I ( ϕ )) | A ) is re-duced near x; in addition, if ϕ has analytic singularities, then ( A , ( O X / b I ( ϕ )) | A ) is weaklynormal at x, where A : = N ( b I ( ϕ )) denotes the zero-set of coherent ideal sheaf b I ( ϕ ) . For our proof of above result, we need the following local vanishing theorem, which isa straightforward consequence of Corollary 1.5 in [43] and the arguments as in Theorem3.5 in [18].
Theorem 4.3.
Let X be a complex space of pure dimension and π : e X → X a log reso-lution of J ac X such that J ac X · O e X = O e X ( − J e X / X ) . Let ( L , e − ϕ L ) be a (possibly singular)Hermitian line bundle on e X with semi-positive curvature current. Then,R q π ∗ (cid:16) O e X ( b K e X / X − J e X / X ) ⊗ O X ( L ) ⊗ I ( ϕ L ) (cid:17) = , for every q > . Our proof of (1) in Theorem 4.2 depends on an induction on dim X due to a Bertiniproperty related with log terminal singularities (cf. Lemma 4.1 in [21]) and (2) is an im-mediate result of strong openness of multiplier ideal sheaves and the above local vanishingtheorem.Moreover, analogous to the argument of the case of hypersurfaces as in [39], we canestablish the following Theorem 4.4.
Let X ⊂ M be locally a complete intersection and ϕ ∈ Psh( M ) such that theslope ν x ( ϕ | X ) = for every x ∈ X. Then, the analytic adjoint ideal sheaf Ad j X ( ϕ ) = O M i ff b I ( ϕ | X ) = O X i ff X is normal and has only rational singularities i ff ( X , x ) is canonical forany x ∈ X. Here, the slope of ϕ | X at x is defined by ν x ( ϕ | X ) : = sup { γ ≥ (cid:12)(cid:12)(cid:12) ϕ | X ≤ γ log X k | f k | + O (1) } ∈ [0 , + ∞ ) , ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 13 where ( f k ) are local generators of the maximal ideal m x of O X , x (cf. [5]).5. L og canonical singularities of plurisubharmonic functions This section is devoted to investigate the local structure of log canonical singularities ofplurisubharmonic functions (see [16, 10, 11] for some global properties related to various L extension theorems).5.1. Finite generation of graded system of ideal sheaves.
In this subsection, we studythe finite generation of certain graded system of ideal sheaves from symbolic powers; see[34, 37] for the basic references. Firstly, let us recall some related concepts and notations.
Definition 5.1.
Let X be a complex manifold and Z ⊂ X an analytic set with ideal sheaf I Z ⊂ O X . Let Z = S k ∈ Λ Z k be a global irreducible decomposition of Z and m Λ : = ( m k ) k ∈ Λ be a Λ -tuple with m k ∈ N .The m Λ -th symbolic power I < m Λ > Z of I Z is the ideal sheaf of germs of holomorphicfunctions that have multiplicity ≥ m k at a general point of Z k for each k ∈ Λ , i.e., I < m Λ > Z : = { f ∈ O X (cid:12)(cid:12)(cid:12) ord x f ≥ m k for general point x of each Z k } . In particular, if m k = m for every k ∈ Λ , then I < m Λ > Z is nothing but the m -th symbolicpower I < m > Z of I Z . Remark . (1) The inequality ord x f ≥ m k means that all the partials of f of order < m k vanish at x .(2) If the inequality ord x f ≥ m holds at a general point x ∈ Z k for every k , by uppersemi-continuity, it holds at every point of Z . Remark . Let π : e X → X be a log resolution of I Z obtained by first blowing up X along Z . Since Z k is generically smooth for every irreducible component Z k of Z , there is aunique irreducible component of the exceptional divisor of Bl Z ( X ) mapping onto Z k , whichdetermines an irreducible divisor E k ⊂ e X . Then, by the definition, it follows that I < m Λ > Z = π ∗ O e X (cid:16) − X k m k E k (cid:17) , where the sum does make sense in view of that the family ( Z k ) k ∈ Λ is locally finite. Definition 5.4.
Let X be a complex manifold, I ⊂ O X an ideal sheaf and Z ⊂ X anirreducible analytic set. The order of vanishing ord Z ( I ) of I along Z is the largest integer m such that I ⊂ I < m > Z . In addition, we setord Z ( c · I ) : = c · ord Z ( I )for any real number c > Definition 5.5. (cf. [14]). Let X be a complex manifold, Z ⊂ X an irreducible analytic set.Let ϕ be a (quasi-) plurisubharmonic function on X and x ∈ X a point.The Lelong number ν x ( ϕ ) of ϕ at x is defined to be ν x ( ϕ ) : = sup { γ ≥ | ϕ ( z ) ≤ γ log | z − x | + O (1) near x } , on some coordinate neighborhood of x . We set ν x ( ϕ ) = + ∞ if ϕ ≡ −∞ .The generic Lelong number of ϕ along Z is defined as ν Z ( ϕ ) : = inf z ∈ Z ν z ( ϕ ) . Remark . (1) By the upper semi-continuity of Lelong number, it follows that ν Z ( ϕ ) = ν z ( ϕ ) for a general point z ∈ Z ; specifically, for any point outside a countable union ofproper analytic subsets of Z . In addition, for any x ∈ X , we have1 n ν x ( ϕ ) ≤ c − x ( ϕ ) ≤ ν x ( ϕ ) . (2) Let z ∈ Z be a regular point. Then, we can deduce that ν Z ( ϕ ) = max { γ ≥ | ϕ ( z ) ≤ γ log | I Z | + O (1) near z } . Definition 5.7.
Let X be a complex space (not necessarily reduced). A graded system ofideal sheaves a • = { a m } on X is a collection of coherent ideal sheaves a m ⊂ O X ( m ∈ N )such that a = O X and a i · a j ⊂ a i + j , ∀ i , j ≥ . Definition 5.8.
The
Rees algebra
Rees( a • ) of a • is the graded O X -algebraRees( a • ) : = M m ≥ a m . A graded system of ideal sheaves a • is finitely generated on X if Rees( a • ) is finitelygenerated as a graded O X -algebra in the sense that there is an integer m such that Rees( a • )is generated as an O X -algebra by its terms of degrees ≤ m .Let X be a complex manifold and a • = { a m } a graded system of ideal sheaves on X . Wedefine a Siu plurisubharmonic function ϕ a • associated to a • by (cf. [33]) ϕ a • : = log ∞ X m = ε m | a m | m = log ∞ X m = ε m ( | f m | + · · · + | f mr m | ) m on a domain Ω ⊂ X , where every term a m is an ideal sheaf with a choice of finite generators f m , ..., f mr m , and ε m approach 0 so fast as m → ∞ that the infinite series locally convergesuniformly. Lemma 5.9.
Let a • = { a m } be a finitely generated graded system of ideal sheaves on Ω ∋ oin C n . Then, there exists m > and one Siu plurisubharmonic function ϕ a • associated to a • such that ϕ a • = ϕ a • , m + O (1) near o with ϕ a • , m : = log( m P m = ε m | a m | m ) .Proof. Since a • is finitely generated on Ω , there exists m > m > m , a m = X k + k + ··· + m k m = m a k · a k · · · a k m m . Note that a ( m − m ⊂ a m ! by the definition of graded system of ideal sheaves, and then we canconclude that for some constant C > m ), | a m | ( m − ≤ C · (cid:18) | a | + | a | + · · · + | a m | m (cid:19) m ! , which implies that | a m | m ≤ C m ! · (cid:18) | a | + | a | + · · · + | a m | m (cid:19) for all m > m (shrinking Ω if necessary). Thus, from the definition of Siu plurisubhar-monic function, after shrinking ε m , it follows that ϕ a • = ϕ a • , m + O (1) on Ω . (cid:3) Lemma 5.10. (cf. [33], Theorem 2.2)
For each Siu plurisubharmonic function ϕ a • definedabove on Ω ⊂ X, it follow that I ( c · ϕ a • ) = I ( c · a • ) on Ω for all c > . At the end of this part, we prove the following result on the finite generation of certaingraded system of ideal sheaves.
Proposition 5.11. (Chenyang Xu).
Let X be a smooth complex quasi-projective varietyand A ⊂ X an algebraic subvariety ( not necessarily irreducible). Let a • = { a m } be agraded sequence of ideals on X given by a m = I < m Λ > A with m k = e k m for each k and e k thecodimension of A k in Ω . If LCT( a • ) = , then a • is finitely generated on X. ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 15
Proof.
Let A k (1 ≤ k ≤ s ) be all the irreducible components of A , and E k the divisorobtained by blowing up the generic point of A k . Write E = P ≤ k ≤ s E k and a m = µ ∗ O X ′ (cid:0) − m P ≤ k ≤ s e k E k (cid:1) , where µ : X ′ → X is a model which contains all E k .For each E k , we know ord E k ( a m ) ≥ e k m by the definition, thusord E k ( a • ) = lim m →∞ m ord E k ( a m ) ≥ e k . We also have the log discrepancy A X ( E k ) = e k . Therefore, the assumption implies that foreach k , 1 = LCT( a • ) ≤ A X ( E k )ord E k ( a • ) ≤ e k e k = , which implies that ord E k ( a • ) = e k and E k computes the log canonical threshold of a • .The rest is a standard argument using the minimal model program: for any 0 < ε < ffi ciently large m such thatLCT( 1 m a m ) > − ε, ∀ m ≥ m . Thus we know that for each E k , the log discrepancy0 < A X , − ε m a m ( E k ) : = A X ( E k ) − − ε m ord E k ( a m ) ≤ e k − (1 − ε ) e k = ε e k , where the first inequality uses that LCT( m a m ) > − ε , and the middle inequality followsfrom that 1 m ord E k ( a m ) ≥ inf m ≥ ord E k ( a m ) m = ord E k ( a • ) = e k . We can choose ε su ffi ciently small such that ε e k < k , then by Corollary 1.4.3in [6], we know that there exists a model π : e X → X such that the exceptional divisor Ex( π )is precisely E , the sum of E k , and − E is π -ample, which implies that a • is finitely generatedon X . (cid:3) Remark . As we will see, by combining (2) of Theorem 4.2 with (1) of Remark 5.16,the algebraic variety A in X with the reduced complex structure is a weakly normal complexspace.5.2. The relation between log canonical locus and symbolic powers.
This part is de-voted to the algebraic properties of log canonical locus of plurisubharmonic functions. Atfirst, by combining Demailly’s analytic approximation of plurisubharmonic functions withproperties of asymptotic multiplier ideals, we are able to establish the following result,which is important for us to study the local structure of log canonical singularities forgeneral plurisubharmonic functions.
Theorem 5.13.
Let o ∈ Ω ⊂ C n be the origin, u ∈ Psh( Ω ) with c o ( u ) = and A = N ( I ( u )) .Then, it follows thatc o ( ϕ a • ) = LCT o ( a • ) = and I ( ϕ a • ) o = I ( a • ) o = I ( u ) o , where a • = { a m } is the graded system of ideal sheaves on Ω given by a m = I < m Λ > A withm k = e k m for each k and e k the codimension of A k in Ω .Remark . As an enhanced version of Theorem 5.13, we expect that there exists aplurisubharmonic function u A with analytic singularities near o such that c o ( u A ) = c o ( u ) = I ( u A ) o = I ( u ) o , which enables us to give a positive solution to Question 1.1 by (2)of Theorem 4.2. Corollary 5.15.
With the same hypothesis as in Theorem 5.13. Then: (1)
The following statements are equivalent: ( a ) There exists a plurisubharmonic function u A with analytic singularities near o, suchthat c o ( u A ) = and I ( u A ) o = I ( u ) o . ( b ) There is m > such that c o ( m log | a m | ) = . (2) If A is locally a (scheme-theoretic) complete intersection, thenc o (log | a | ) = and I (log | a | ) o = I ( u ) o . Remark . (1) Suppose that a • is finitely generated (e.g., A is locally a complete in-tersection). Then, by Lemma 5.9 there exists m > ϕ a • = ϕ a • , m + O (1) (shrinking ε m if necessary), which implies that c o ( u A ) = I ( u A ) o = I ( u ) o by taking u A = ϕ a • , m .(2) If ( A , o ) is of dimension n − n −
1, by the above Corollary,we obtain that c o (log | a | ) = I (log | a | ) o = I ( u ) o . In particular, if n = A , o ) is a singular curve of embedding dimension 2, then ( A , o )is weakly normal by (2) of Theorem 4.2. Corollary 5.17.
Let u ∈ Psh( Ω ) such that c o ( u ) = and A = N ( I ( u )) . Then, we candeduce that (1) If dim o A = n − , then any union of ( n − -dimensional irreducible components of ( A , o ) is a semi-log-canonical hypersurface. (2) Assume that o is an isolated singularity of A. Then the multiplier ideal subschemeV ( u ) is weakly normal near o.Remark . As an immediate consequence of Corollary 5.17, we can obtain Theorem1.2 in [22].5.3.
A solution to Question 1.1 in dimension three.
In the last part, we give a positiveanswer to Question 1.1 in the three dimensional case.5.3.1.
Proof of Theorem 1.15.
By Theorem 5.13, we have c o ( ϕ a • ) = I ( ϕ a • ) o = I ( u ) o , where a • = { a m } is the graded system of ideal sheaves on Ω given by a m = I < m Λ > A with m k = e k m for each k and e k the codimension of A k in Ω . Thanks to Lemma 5.9 and (2) ofTheorem 4.2, it is su ffi ciently to show that a • is finitely generated on some neighborhoodof o .(i) When dim o A =
1, it follows that o is at most an isolated singularity of A . Then, wecan achieve the finite generation by a combination of Artin’s algebraization theorem withProposition 5.11.(ii) When ( A , o ) is of pure dimension 2, the desired result follow from the fact that c o ( I ( u )) ≥ c o ( u ) = A , o ) is algebraic (cf. [36, 41]).(iii) When dim o A = A , o ) is not of pure dimension. After shrinking Ω , we mayassume that each irreducible component of A contains the origin o . Let A = A ∪ A withdim o A k = − k ( k = , a • near o byProposition 5.11 and the algebraicity of ( A k , o ). (cid:3) Remark . One can refer to Remark A.5 in Appendix A for an alternative argument ofthe proof of Theorem 1.15.
ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 17
Proof of Corollary 1.17.
Put Z = t S α = A k α , where A k α is an irreducible componentof A through o for each α (shrinking Ω if necessary) and of codimension e k α . Let m Λ = ( e k , . . . , e k t ) and consider the plurisubharmonic function e u : = log (cid:16) | I < m Λ > Z | + e u (cid:17) , where I < m Λ > Z is the m Λ -th symbolic power of I Z . Thanks to Theorem 1.15, it is su ffi -ciently to prove that c o ( e u ) = I ( e u ) o = I Z , o , which is a consequence of Demailly’sanalytic approximation of plurisubharmonic functions and (2) of Remark 5.6 (cid:3) Proof of Theorem 1.19. ( a ) When ( A , o ) is of pure dimension 2, the desired resultimmediately follows from (ii) in the proof of Theorem 1.15. ( b ) When dim o A = A , o ) is not of pure dimension. Let( A , o ) = ( A , o ) ∪ ( A , o )with dim o ( A k , o ) = − k ( k = , V ( u ) isweakly normal near o .Therefore, in view of Lemma 2.3 in [2], it follows from the fact that ( A , o ) ∩ ( A , o ) = { o } that T o A ∩ T o A =
0, where T o A k ( k = ,
2) is the Zariski tangent space of A k at o ,which implies that both ( A , o ) and ( A , o ) are regular, and they intersect transversely at o .Thus, up to a change of variables at o , we obtain that I ( u ) o = ( z z , z z ) · O . ( c ) When dim o A =
1, i.e., ( A , o ) is a curve with embedding dimension at most 3. Then,Theorem 1.15 implies that V ( u ) is weakly normal at o . Thus, we obtain that ( A , o ) = ( N ( z , z ) , o ), ( N ( z , z z ) , o ) or ( N ( z z , z z , z z ) , o ) in some local coordinates near o ,which implies that I ( u ) o = ( z , z ) · O , ( z , z z ) · O or ( z z , z z , z z ) · O , respec-tively. ( d ) When dim o A =
0, the desired result is straightforward. (cid:3)
Remark . We refer to [23] for an alternative argument of the case ( b ) by the so-called“inversion of adjunction” ([17], Theorem 2.5).5.3.4. Proof of Corollary 1.22.
Since the statement is local, we may assume that V ( a • ) isthe multiplier ideal subscheme associated to a • on a bounded Stein domain Ω ⊂ C . Let ϕ a • be a Siu plurisubharmonic function associated to a • on Ω . Then, we can deduce fromLemma 5.10 that I ( ϕ a • ) = I ( a • ) on Ω and c x ( ϕ a • ) = LCT( a • ) = x ∈ V ( a • ).Thus, we can immediately achieve the desired result by Theorem 1.15 and Theorem 1.19. (cid:3) Remark . Similar to Corollary 1.17, we obtain that any union of irreducible compo-nents of V ( a • ) with the reduced complex structure is weakly normal as well.A ppendix A. C onstructibility of log canonical threshold of graded system of idealsheaves from symbolic powers
In this appendix, we are ready to discuss something on the constructibility of log canoni-cal thresholds from symbolic powers, which turns out to be helpful for us to study Question1.1.
Definition A.1. (cf. [42]). Let X be a complex manifold. A subset Z ⊂ X is analyticallyconstructible if Z = S λ ∈ Λ ( V λ \ W λ ), where { V λ } λ ∈ Λ is a locally finite family of irreducibleanalytic sets and W λ $ V λ is an analytic set for every λ ∈ Λ .A function f : X → [ −∞ , + ∞ ] is analytically constructible if each set f − ( t ) is con-structible for any t ∈ [ −∞ , + ∞ ], and the family { f − ( t ) } t ∈ [ −∞ , + ∞ ] is locally finite on X .Similarly, we can define a counterpart in the algebraic setting. In particular, every alge-braically constructible set or function is analytically constructible. Remark
A.2 . (1) The di ff erence, finite intersection and union of analytically constructiblesets is also an analytically constructible set.(2) Let Ω ⊂⊂ X be a relatively compact domain. Then, the range f ( Ω ) of analyticallyconstructible function f | Ω is a finite subset in [ −∞ , + ∞ ]. Remark
A.3 . (Chevalley-Remmert; cf. [42], p. 291). Let π : e X → X be a proper holomor-phic mapping of complex manifolds. Then the image of every analytically constructibleset in e X is an analytically constructible set in X . Proposition A.4.
Let X be a complex manifold of dimension three and Z ⊂ X an analyticset. Then, the log canonical threshold
LCT x ( a • ) is an analytically constructible functionon X, where a • = { a m } is the graded sequence of ideals on X given by a m = I < m Λ > Z withm k = e k m for each k and e k the codimension of Z k in X.Proof. We can achieve this by a similar argument to the proof of Theorem 1.15. Withoutloss of generality, we may assume that dim Z k > Z k of Z .(i) When dim Z =
1, i.e., Z ⊂ X is a curve. Then, the singular locus of Z is at most a dis-crete subset in X . Note that LCT x ( a • ) = + ∞ i ff x ∈ X \ Z , and LCT x ( a • ) = x ∈ Z reg . Hence, we can deduct that LCT x ( a • ) is an analytically constructible function on X .(ii) When dim Z k = k , i.e., Z is a hypersurface of X . Thus, we obtain thatthe symbolic power of I Z coincides with the ordinary power, i.e., I < m > Z = I mZ for all m ,which implies that LCT x ( a • ) = LCT x ( I Z ) for any point x ∈ X . Let π : e X → X be a logresolution of I Z , and then by a standard argument on the computation of LCT x ( I Z ) via thelog resolution π (see [17], Proposition 1.7), we conclude the constructibility of LCT x ( a • ).(iii) When dim Z = Z k = k . Let e Z α ( α = ,
2) be the union ofall α -codimensional irreducible components of Z , and consider the graded system of idealsheaves a ′• , a ′′• on X given by a ′ m = I < m > e Z and a ′′ m = I < m > e Z respectively. Thus, it followsfrom (i) and (ii) that LCT x ( a ′• ) and LCT x ( a ′′• ) are analytically constructible functions on X .On the other hand, we note that LCT x ( a • ) = LCT x ( a ′• ) for any x ∈ e Z \ e Z and LCT x ( a • ) = LCT x ( a ′′• ) for any x ∈ e Z \ e Z . Finally, combining the discreteness of analytic set e Z ∩ e Z in X and the constructibility of LCT x ( a ′• ) and LCT x ( a ′′• ), we conclude that the functionLCT x ( a • ) is constructible on X . (cid:3) Remark
A.5 . (An alternative argument on the proof of Theorem 1.15).
At first, we observethat o is at most an algebraic singularity of A . Then, there exists an a ffi ne algebraic variety b A ⊂ C n such that ( b A , o ) = ( A , o ) and we may assume that all irreducible components of b A contain the origin o . Thus, by Theorem 5.13, we haveLCT o ( b a • ) = LCT o ( a • ) = I ( ϕ b a • ) o = I ( u ) o , where b a • = { b a m } is the graded system of ideal sheaves on C n given by b a m = I < m Λ > b A .Combining the lower semi-continuity of log canonical thresholds with Proposition A.4,we are able to infer that LCT( b a • ) = U of o in C n . Thus, by Proposition 5.11, it follows that b a • is finitely generated on U . Therefore, a • is finitely generated on a neighborhood of o (in the complex topology), which impliesthat c o ( ϕ a • , m ) = m >
0, and I ( ϕ a • , m ) = I ( u ) near o by Lemma 5.9. Finally,using (2) of Theorem 4.2, we conclude that the multiplier ideal subscheme V ( u ) is weaklynormal, shrinking Ω if necessary. (cid:3) In view of Proposition A.4, it is natural to raise the following question:
Question A.1.
Let X be a complex manifold and Z ⊂ X an analytic set. Let a • = { a m } be the graded sequence of ideal sheaves on X given by a m = I < m Λ > Z , where m k = e k m for ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 19 each k and e k is the codimension of Z k in X. Is the log canonical threshold LCT x ( a • ) ananalytically constructible function on X?Remark A.6 . (1) From the argument in the proof of Proposition A.4, we can deduce thatthe answer of Question A.1 is positive if Z ⊂ X is locally a complete intersection or allsingularities of Z are isolated.(2) Analogous to Remark A.5, we are able to derive that if ( A , o ) is algebraic, then apositive answer of Question A.1 implies a positive solution to Question 1.1.Finally, by combining the argument in Remark A.5 with Lemma 5.10, we can deducethe following observation on log canonical locus of pair associated to graded system ofideals in the algebraic setting. Proposition A.7.
Let X be a smooth complex projective variety and a • = { a m } any gradedsystem of ideals on X such that LCT( a • ) = . If the answer of Question A.1 is positive,then any union of the irreducible components of multiplier ideal subscheme V ( a • ) is weaklynormal. A ppendix B. S emi - log - canonical hypersurface singularities of dimension two name symbol defining function h ∈ C [ z , z , z ]smooth A z Du Val A n z + z + z n + , n ≥ D n z + z z + z n − , n ≥ E z + z + z E z + z + z z E z + z + z simple elliptic X , z + z + z + λ z z z , λ , J , z + z + z + λ z z z , λ , T , z + z + z + λ z z z , λ + , T p , q , r z z z + z p + z q + z r , p + q + r < A ∞ z + z pinch point D ∞ z + z z degenerate cusp T , ∞ , ∞ z + z z T , q , ∞ z + z q + z z , q ≥ T ∞ , ∞ , ∞ z z z T p , ∞ , ∞ z z z + z p , p ≥ T p , q , ∞ z z z + z p + z q , q ≥ p ≥ T able
1. Semi-log-canonical hypersurface singularities of dimension two
Acknowledgements . The author would like to sincerely thank Prof. Xiangyu Zhouand Qi’an Guan for their generous support and encouragements. He is very grateful toProf. Chenyang Xu for valuable discussions and suggestions, especially for providing aproof of Proposition 5.11. He is also indebted to Prof. Jean-Pierre Demailly for helpfulcorrespondence, and Prof. William Allen Adkins for sharing his works.
ULTIPLIER IDEAL SHEAVES ON COMPLEX SPACES WITH SINGULARITIES 21 R eferences [1] W. A. Adkins, A. Andreotti, J. V. Leahy, An analogue of Oka’s theorem for weakly normal complex spaces ,Pacific J. Math. 68 (1977), 297–301.[2] W. A. Adkins, J. V. Leahy,
A topological criterion for local optimality of weakly normal complex spaces ,Math. Ann. 243 (1979), 115–123.[3] F. Ambro,
Basic properties of log canonical centers , in Classification of Algebraic Varieties, EMS Seriesof Congress Reports, Vol. 3, European Mathematical Society, Zürich, 2011, pp. 39–48.[4] M. Andreatta, A. Silva,
On weakly rational singularities in complex analytic geometry , Ann. Mat. PuraAppl. (4) 136 (1984), 65–76.[5] R. Berman, S. Boucksom, P. Eyssidieux, et al.,
Kähler-Einstein metrics and the Kähler-Ricci flow on logFano varieties , J. Reine Angew. Math. 751 (2019), 27–89.[6] C. Birkar, P. Cascini, C. Hacon, J. McKernan,
Existence of minimal models for varieties of log generaltype , J. Amer. Math. Soc. 23 (2010), 405–468.[7] A. Bravo,
A remark on strong factorizing resolutions , RACSAM 107 (2013), 53–60.[8] A. Bravo, O. Villamayor,
A strengthening of resolution of singularities in characteristic zero , Proc. Lond.Math. Soc. 86 (2003), 327–357.[9] J. Y. Cao,
Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compactKähler manifolds , Compos. Math. 150 (2014), 1869–1902.[10] J. Y. Cao, J.-P. Demailly, S. Matsumura,
A general extension theorem for cohomology classes on nonreduced analytic subspaces , Sci. China Math. 60 (2017), 949–962.[11] T. O. M. Chan,
Extension from lc centres and the extension theorem of Demailly but with estimates ,preprint, arXiv: 1811.02204.[12] B. Claudon,
Invariance for multiples of the twisted canonical bundle , Ann. Inst. Fourier (Grenoble) 57(2007), 289–300.[13] T. de Fernex, R. Docampo,
Jacobian discrepancies and rational singularities , J. Eur. Math. Soc. 16 (2014),165–199.[14] J.-P. Demailly,
Analytic Methods in Algebraic Geometry , Higher Education Press, Beijing, 2010.[15] J.-P. Demailly,
On the cohomology of pseudoe ff ective line bundles , in Complex Geometry and Dynamics,Abel Symposia, Vol. 10, Springer, Switzerland, 2015, pp. 51–99.[16] J.-P. Demailly, Extension of holomorphic functions defined on non reduced analytic subvarieties , in TheLegacy of Bernhard Riemann after One Hundred and Fifty Years, Advanced Lectures in Mathematics, Vol.35.1, Int. Press, Somerville, MA, 2016, pp. 191–222.[17] J.-P. Demailly, J. Kollár,
Semi-continuity of complex singularity exponents and Kähler-Einstein metrics onFano orbifolds , Ann. Sci. École Norm. Sup. (4) 34 (2001), 525–556.[18] L. Ein, S. Ishii, M. Musta¸t˘a,
Multiplier ideals via Mather discrepancy , in Minimal Models and ExtremalRays (Kyoto, 2011), Adv. Stud. Pure Math., Vol. 70, Math. Soc. Japan, Tokyo, 2016, pp. 9–28.[19] E. Eisenstein,
Generalizations of the restriction theorem for multiplier ideals , preprint, arXiv: 1001.2841.[20] H. Grauert, R. Remmert,
Coherent Analytic Sheaves , Grundlehren Math. Wiss., 265, Springer-Verlag,Berlin, 1984.[21] D. Greb,
Rational singularities and quotients by holomorphic group actions , Ann. Sc. Norm. Super. PisaCl. Sci. (5) 10 (2011), 413–426.[22] Q. A. Guan, Z. Q. Li,
Multiplier ideal sheaves associated with weights of log canonical threshold one ,Adv. Math. 302 (2016), 40–47.[23] Q. A. Guan, Z. Q. Li,
Multiplier ideal sheaves associated with weights of log canonical threshold one II ,preprint.[24] Q. A. Guan, Z. Q. Li,
Analytic adjoint ideal sheaves associated to plurisubharmonic functions , Ann. Sc.Norm. Super. Pisa Cl. Sci. (5) XVIII (2018), 391–395.[25] Q. A. Guan, X. Y. Zhou,
A solution of an L extension problem with an optimal estimate and applications ,Ann. of Math. (2) 181 (2015), 1139–1208.[26] Q. A. Guan, X. Y. Zhou, A proof of Demailly’s strong openness conjecture , Ann. of Math. (2) 182 (2015),605–616. See also arXiv: 1311.3781.[27] Q. A. Guan, X. Y. Zhou,
Restriction formula and subadditivity property related to multiplier ideal sheaves ,to appear in J. Reine Angew. Math., DOI: https: // doi.org / / crelle-2019-0043.[28] H. Guenancia, Toric plurisubharmonic functions and analytic adjoint ideal sheaves , Math. Z. 271 (2012),1011–1035.[29] S. Ishii,
Introduction to singularities , Springer, Tokyo, 2014.[30] M. Jonsson, M. Musta¸t˘a,
Valuations and asymptotic invariants for sequences of ideals , Ann. Inst. Fourier(Grenoble) 62 (2012), 2145–2209.[31] L. Kaup, B. Kaup,
Holomorphic Functions of Several Variables. An Introduction to the FundamentalTheory , Translated from the German by Michael Bridgland, de Gruyter Studies in Math., Vol. 3, Walter deGruyter, Berlin, 1983. [32] D. Kim,
Themes on Non-analytic Singularities of Plurisubharmonic Functions , Complex Analysis andGeometry, Springer Proceedings in Mathematics & Statistics, vol. 144, Springer, Tokyo, 2015, pp. 197–206.[33] D. Kim, H. Seo,
Jumping numbers of analytic multiplier ideals (with an appendix by Sébastien Boucksom) ,preprint, arXiv: 1908.02474v1.[34] J. Kollár,
Singularities of the Minimal Model Program , Cambridge Tracts in Mathematics, Vol. 200, Cam-bridge Univ. Press, Cambridge, 2013.[35] J. Kollár, S. Mori,
Birational geometry of algebraic varieties , Cambridge Tracts in Mathematics 134,Cambridge Univ. Press, Cambridge, 1998.[36] J. Kollár, N. Shepherd-Barron,
Threefolds and deformations of surface singularities , Invent. Math. 91(1988), 299–338.[37] R. Lazarsfeld,
Positivity in Algebraic Geometry II , Ergeb. Math. Grenzgeb. (3), 49, Springer-Verlag, Berlin,2004.[38] A. Li, I. Swanson,
Symbolic powers of radical ideals , Rocky Mountain J. Math. 36 (2006), 997–1009.[39] Z. Q. Li,
Analytic adjoint ideal sheaves associated to plurisubharmonic functions II , to appear in Ann. Sc.Norm. Super. Pisa Cl. Sci. (5).[40] Z. Q. Li, X. Y. Zhou,
Multiplier ideal sheaves on Q -Gorenstein complex spaces , preprint.[41] W. F. Liu, S. Rollenske, Two-dimensional semi-log-canonical hypersurfaces , Le Matematiche (Catania)67 (2012), 185–202.[42] S. Łojasiewicz,
Introduction to Complex Analytic Geometry , Birkhäuser, Basel, 1991.[43] S.-I. Matsumura,
Injectivity theorems with multiplier ideal sheaves for higher direct images under Kählermorphisms , preprint, arXiv: 1607.05554.[44] A. M. Nadel,
The behavior of multiplier ideal sheaves under morphisms , in Complex Analysis, Aspects ofMathematics, Vol. E 17, Vieweg & Teubner Verlag, Braunschweig, 1991, pp. 205–222.[45] A. Nobile,
Some properties of the Nash blowing-up , Pacific J. Math. 60 (1975), 297–305.[46] T. Ohsawa,
On the extension of L holomorphic functions. V: E ff ects of generalization , Nagoya Math. J.161 (2001), 1–21. Erratum: Nagoya Math. J. 163 (2001), 229.[47] T. Ohsawa, K. Takegoshi, On the extension of L holomorphic functions , Math. Z. 195 (1987), 197–204.[48] M. P˘aun, Siu’s invariance of plurigenera: a one-tower proof , J. Di ff erential Geom. 76 (2007),485–493.[49] H. H. Pham, The weighted log canonical threshold , C. R. Math. Acad. Sci. Paris (4) 352 (2014), 283–288.[50] R. Richberg,
Stetige streng pseudokonvexe Funktionen , Math. Ann. 175 (1968), 257–286.[51] Y.-T. Siu,
Invariance of plurigenera , Invent. Math. 134 (1998), 661–673.[52] Y.-T. Siu,
Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance ofsemipositively twisted plurigenera for manifolds not necessarily of general type , in Complex Geometry(Collection of papers dedicated to Hans Grauert), Springer-Verlag, Berlin, 2002, pp. 223–277.[53] Y.-T. Siu,
Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonial bundles ,in Finite or Infinite Dimensional Complex Analysis and Applications, Springer-Verlag, Boston, MA, 2004,pp.45–83.[54] Y.-T. Siu,
Multiplier ideal sheaves in complex and algebraic geometry , Sci. China Ser. A 48 (2005), 1–31.[55] S. Takagi,
Adjoint ideals along closed subvarieties of higher codimension , J. Reine Angew. Math. 641(2010), 145–162.[56] S. Takayama,
Pluricanonical systems on algebraic varieties of general type , Invent. Math. 165 (2006),551–587.[57] S. Takayama,
On the invariance and lower semi-continuity of plurigenera of algebraic varieties , J. Alge-braic Geom. 16 (2007), 1–18.[58] K. Takegoshi,
Higher direct images of canonical sheaves tensorized with semi-positive vector bundles byproper Kähler morphisms , Math. Ann. 303 (1995), 389–416.[59] B. Teissier,
Variétés polaires II. Multiplicités polaires, sections planes, et conditions de Whitney , in Alge-braic geometry (La Rábida, 1981), Lecture Notes in Math., Vol. 961, Springer, Berlin, 1982, pp. 314–491.[60] D. Varolin,
A Takayama-type extension theorem , Compos. Math. 144 (2008), 522–540.[61] J. Varouchas,
Kähler spaces and proper open morphism , Math. Ann. 283 (1989), 13–52.[62] J. Włodarczyk,
Resolution of singularities of analytic spaces , in Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry //