aa r X i v : . [ m a t h . AG ] A p r ON ISOLATED LOG CANONICAL CENTERS
CHIH-CHI CHOU
Abstract.
In this paper, we show that the depth of an isolated log canonicalcenter is determined by the cohomology of the -1 discrepancy diviors over it.A similar result also holds for normal isolated Du Bois singularities. Introduction
Singularities play a significant role in the minimal model program (mmp).Among the different types of singularities, Kawamata log terminal (klt) and logcanonical (lc) are of particular importance. Many fundamental theorems are firstproved in the klt case, then extended to the lc case. And it is expected that lcshould be the largest class of singularities for which one can run mmp .One of the major differences between klt and lc is that klt singularities are ratio-nal singularities, lc singularities are Du Bois [10] but in general not rational. So it isinteresting and important to know how far lc is from being rational. Since rationalimplies Cohen-Macaulay, we can also ask if the variety X is Cohen-Macaulay atsome given point p . Or more precisely, we can calculate depth p ( O X ).There are some known results regarding this direction. For example, Fujinoshows that given a lc pair ( X, ∆) of dimension at least three, then depth p ( O X ) ≥ min { , codim p X } if ¯ p is not a lc center ( Theorem 4.21 in [4]), which is first provedby Alexeev assuming that p is a closed point and X is projective (Lemma 3.2 in[1]). Koll´ar and Kov´acs generalized this result in [12] and [16], respectively, butstill under the assumption that ¯ p is not a lc center. (See also [2] for result aboutclosed points.)In this paper, we investigate a case when ¯ p is a lc center. Assume that p is anisolated lc center, after localization we assume p is a closed point. It turns outthat there is a delicate relation between depth p ( O X ) and the cohomology group ofthe exceptional divisors over p . More precisely, given a log canonical pair ( X, ∆)and an isolated lc center p ∈ X which is a closed point, we take a log resolution f : Y → X such that K Y = f ∗ ( K X + ∆) + A − B − E. Here
A, B are effective and ⌊ B ⌋ = 0, E is the reduced divisor such that f ( E ) = p .Then we have the following, Theorem 1.1. (=Corollary 3.2) For any integer ≤ t ≤ n , we have depth p O X ≥ t if and only if H i − ( E, O E ) = 0 , ∀ < i < t. (Note that by assumption X is normal,so we know depth p O X is at least two.) : 14B05, 14F17 This result generalizes Proposition 4.7 in [5], which gives a necessary and suf-ficient condition for an index one isolated log canonical singularity to be Cohen-Macaulay.We prove this theorem by showing that the local cohomology H ip ( O X ) is theMatlis dual of H n − i ( E, K E ). The same method applies to isolated Du Bois singu-larities, (see section 3.2). In the Du Bois case, E denotes the reduced exceptionaldivisors.The most crucial ingredient of the proof is Kov´acs vanishing theorem, which saysthat R i f ∗ O Y ( − E ) = 0 , ∀ i > . With this theorem, we see that f ∗ O Y ( − E ) is quasiisomorphic to R f ∗ O Y ( − E ). By this quasi isomorphism and Grothendieck duality,we are able to see the relation between the local cohomology of X and cohomologyof O E . Because of the significant role of Kov´acs’s theorem in this paper, we givea quick proof of it in the last section. This proof, based on Fujino’s idea, onlyuses Grothendieck duality and Kawamata-Viehweg vanishing theorem instead ofthe notion of Du Bois pair in Kov´acs’s original paper. Acknowledgements.
I would like to thank professor Osamu Fujino for discussionsand answering many questions by emails. Moreover, this project is inspired byhis paper [5]. I would also like to thank professors Lawrence Ein, S´andor Kov´acsand Mihnea Popa for many useful discussions. I am also grateful to the referee forcareful reading and many useful comments.2. preliminaries
Given a pair ( X, ∆), where X is a normal variety and ∆ is a Q − linear combina-tion of Weil divisors so that K X + ∆ is Q -Cartier. Take a log resolution f : Y → X ,such that the exceptional locus and the strict transform f − ∗ ∆ are simple normalcrossing divisors. We say the pair ( X, ∆) is log canonical if K Y = f ∗ ( K X + ∆) + A − B − E, where A, B are effective, ⌊ B ⌋ = 0 and E is reduced. We say ( X, ∆) is log terminalif E is empty.In this paper we consider log canonical pair, ( X, ∆). A sub variety W ⊂ X iscalled log canonical center, if there is a log resolution as above, and some component E ′ ⊂ E such that f ( E ′ ) = W .We recall Kov´acs vanishing theorem. Theorem 2.1. (Theorem 1.2 in [17] ) Let ( X, ∆) be a log canonical pair and let f : Y → X be a proper birational morphism from a smooth variety Y such thatEx ( f ) ∪ Supp f − ∗ ∆ is a simple normal crossing divisor on Y . If we write K Y = f ∗ ( K X + ∆) + X i a i E i and put E = P a i = − E i , then R i f ∗ O Y ( − E ) = 0 for every i > . This theorem is first proved by notion of Du Bois pair under the assumption that X is Q -factorial. The proof is then simplified in [6] without assuming Q -factorial.Now we recall the duality theorems which will be used in this paper. First werecall Grothendieck duality theorem (III.11.1, VII.3.4 in [7]). Let f : Y → X N ISOLATED LOG CANONICAL CENTERS 3 be a proper morphism between finite dimensional noetherian schemes. Supposethat both X and Y admit dualizing complexes, for example when they are quasi-projective varieties. Then for any F • ∈ D − qcoh ( Y ), we have Rf ∗ R H om Y ( F • , ω • Y ) ∼ = R H om X ( Rf ∗ F • , ω • X )Here ω • X is dualizing complex. Let n be the dimension of X and assume that X is normal, then h − n ( ω • X ) := ω X = O X ( K X ), the extension of regular n -forms onsmooth locus. In this paper we only consider normal varieties, so we will use ω X and K X interchangeably. If X is Cohen-Macaulay, then h i ( ω • X ) = 0 , if i = − n , and h − n ( ω • X ) = ω X . Or equivalently, ω • X = ω X [ n ] . Now we recall local duality (V.6.2 in [7]). Suppose that (
R, p ) is a local ring. Aninjective hull I of the residue field k = R/p is a an injective R module I such thatfor any non-zero submodule N ⊂ I we have N ∩ k = 0. (See [3] Proposition 3.2.2.for more discussion.) Matlis duality says that the functor Hom ( · , I ) is a faithfulexact functor on the category of Noetherian R modules. Theorem 2.2. (Local duality ) Let ( R, p ) be a local ring and F • ∈ D + coh ( R ) . Then R Γ p ( F • ) → R Hom ( R Hom ( F • , ω • R ) , I ) is an isomorphism. In particular, if we take i -th cohomology on both hand sides, we haveH ip ( F • ) ∼ = Hom (H − i ( R Hom ( F • , ω • R )) , I )The − i comes from switching the cohomology functor H i ( · ) and Hom ( · , I ).3. Main Results
Depth of LC center.
Given a log canonical pair ( X, ∆), and an isolated lccenter p ∈ X which is a closed point. Without loss of generality, we assume X is anaffine space and p is the only closed point. By definition, we have a log resolution f : Y → X such that K Y = f ∗ ( K X + ∆) + A − B − E. Here
A, B are effective and ⌊ B ⌋ = 0, E is the reduced exceptional divisor such that f ( E ) = p . Theorem 3.1.
For < i < n , H ip ( X, O X ) is dual to H n − i ( E, K E ) by Matlisduality. For i = n , H np ( X, O X ) is dual to f ∗ O Y ( K Y + E ) .Proof. Push forward the following exact sequence on Y ,0 → K Y → K Y ( E ) → K E → . By Grauert-Riemenschneider vanishing, we have R n − i f ∗ O Y ( K Y + E ) ∼ = H n − i ( E, K E )for i < n . So to prove the statement, it suffices to prove the duality betweenH ip ( X, O X ) and R n − i f ∗ O Y ( K Y + E ) ∼ = H n − i ( E, K E ). To this end, we consider thequasi isomorphism f ∗ O Y ( − E ) ∼ = quasi R f ∗ O Y ( − E ) implied by Kov´acs vanishingtheorem. Apply Grothendieck duailty, we have R Hom ( f ∗ O Y ( − E ) , ω • X ) ∼ = quasi R Hom ( R f ∗ O Y ( − E ) , ω • X ) ∼ = quasi R f ∗ ω • Y ( E )Take − i th cohomology, we have(3.1) Ext − i ( f ∗ O Y ( − E ) , ω • X ) ∼ = R n − i f ∗ O Y ( K Y + E ) CHIH-CHI CHOU
By Matlis duality, the left hand side is isomorphic to Hom(H ip ( f ∗ O Y ( − E )) , I ),where I is injective hull of k .To prove the statement, we claim that H ip ( f ∗ O Y ( − E )) ∼ = H ip ( O X ) for i > → f ∗ O Y ( − E ) → O X → O p → , and the fact that H i ( O p ) = 0 iff i > . (cid:3) Corollary 3.2.
For any integer ≤ t ≤ n , we have depth p O X ≥ t if and only ifH i − ( E, O E ) = 0 , ∀ < i < t. (Note that by assumption X is normal, so we knowdepth p O X is at least two.)Proof. In the proof of Theorem 3.1, we showed H ip ( f ∗ O Y ( − E )) ∼ = H ip ( O X ) for i > ≤ t ≤ n , we havedepth p O X ≥ t ⇔ H ip ( X, O X ) = 0 , ∀ i < t ⇔ H ip ( X, f ∗ O Y ( − E )) = 0 , ∀ < i < t ⇔ H n − i ( E, K E ) = 0 , ∀ < i < t (Matlis duality and Equation (3.1)) ⇔ H i − ( E, O E ) = 0 , ∀ < i < t. (Serre Duality) (cid:3) Remark . The cohomology group H i ( E, O E ) is independent of resolution, be-cause H i ( E, O E ) ∼ = R i f ∗ O Y by Kov´acs vanishing theorem. And that R i f ∗ O Y iswell known to be independent of resolution. Corollary 3.4. (Proposition 4.7 [5] ) Given a closed isolated lc center p of a pair ( X, ∆) , then X is Cohen-Macauley at p if and only if H i ( E, O E ) = 0 , ∀ < i < n − . Normal isolated Du Bois singularity.
The notion of Du Bois singularitiesis a generalization of the notion of rational singularities. For a proper schemeof finite type X there exists a complex Ω • X , which is an analogue of De Rhamcomplex. Roughly speaking , X is said to have Du Bois singularities if the naturalmap O X → Ω X is a quasi isomorphism. We refer the reader to [15] and the referencethere for more discussions.In this subsection we consider the case where ( X, p ) is a normal isolated Du Boissingularity of dimension n , and f : Y → X is a log resolution such that f is anisomorphism outside of p . We claim that the idea in the previous subsection canbe applied to this case. The crucial fact we need is the following, Theorem 3.5. (Theorem 6.1 in [15] ) Take a log resolution f : Y → X as above,and let E be the reduced preimage of p . Then ( X, p ) is a normal Du Bois singularityif and only if the natural map R i f ∗ O Y → R i f ∗ O E is an isomorphism for all i > . This theorem implies that R i f ∗ O Y ( − E ) = o, ∀ i > . That is f ∗ O Y ( − E ) ∼ = quasi R f ∗ O Y ( − E )Then exactly the same proof as in previous section yields N ISOLATED LOG CANONICAL CENTERS 5
Theorem 3.6.
Given ( X, p ) is a normal isolated Du Bois singularity of dimension n . For < i < n , H ip ( X, O X ) is dual to H n − i ( E, K E ) by Matlis duality. For i = n ,H np ( X, O X ) is dual to f ∗ O Y ( K Y + E ) . In particular, f ∗ O Y ( K Y + E ) ∼ = K X . Then the corollaries in the previous section also hold.
Remark . The last statement has been proved in [8] (the Claim in Theorem 2.3).4.
Kov´acs vanishing theorem
In this section we follow Fujino’s idea to give a simple proof of Kov´acs vanishingtheorem. First we prove a similar result for dlt pair which was proved by the notionof rational pair in [13]. One of the equivalent definitions of dlt singularities is thatthere is a log resolution (Szab´o resolution [19]) f : Y → X such that the dicrepancy a ( E ; X, ∆) > − E on Y (Theorem 2.44 in [11]). Theorem 4.1.
Let ( X, ∆ X ) be a dlt pair and let f : Y → X be a Szab´o resolution.Then we can write K Y + ∆ Y = f ∗ ( K X + ∆ X ) + A − B, where A, B are effective exceptional divisors, ⌊ B ⌋ = 0 and ∆ Y is the strict transformof ∆ X . Then for any reduced subset ∆ ′ ⊆ ∆ Y , we have R i f ∗ O Y ( − ∆ ′ ) = 0 for every i > .Proof. Write K Y − f ∗ ( K X + ∆ X ) + ∆ Y = A − B, Then ⌈ A ⌉ = K Y − f ∗ ( K X + ∆ X ) + ∆ Y + B + ⌈ A ⌉ − A, which is f-exceptional and effective. Consider the following diagram of complexes, f ∗ O Y ( − ∆ ′ ) α / / β ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ R f ∗ O Y ( − ∆ ′ ) γ (cid:15) (cid:15) R f ∗ O Y ( ⌈ A ⌉ − ∆ ′ )Note that ⌈ A ⌉ − ∆ ′ = K Y − f ∗ ( K X + ∆ X ) + strict transform + δ, where δ is some effective simple normal crossing divisors such that ⌊ δ ⌋ = 0. Soby Reid-Fukuda type vanishing R i f ∗ O Y ( ⌈ A ⌉ − ∆ ′ ) = 0 for i >
0. On the otherhand, Since ⌈ A ⌉ is exceptional and ∆ ′ is strict transform, so f ∗ O Y ( ⌈ A ⌉ − ∆ ′ ) = f ∗ O Y ( − ∆ ′ ). (Lemma 12 in [12]). So β is a quasi isomorphism.Dualize this diagram we have R Hom ( f ∗ O Y ( − ∆ ′ ) , ω • X ) R Hom ( Rf ∗ O Y ( − ∆ ′ ) , ω • X ) α ∗ o o R Hom ( Rf ∗ O Y ( ⌈ A ⌉ − ∆ ′ ) , ω • X ) O O β ∗ k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ γ ∗ O O Apply Grothendieck duality we get the following composition,
CHIH-CHI CHOU R f ∗ ω • Y (∆ ′ − ⌈ A ⌉ ) γ ∗ / / β ∗ R f ∗ ω • Y (∆ ′ ) α ∗ / / R H om ( f ∗ O Y ( − ∆ ′ ) , ω • X )By Reid-Fukuda type vanishing, the complex R f ∗ ω • Y (∆ ′ ) has vanishing highercohomology. Note that β ∗ is a quasi isomorphism, so it is in fact a composition ofsheaf morphisms as following, f ∗ ω Y (∆ ′ − ⌈ A ⌉ ) γ ∗ / / f ∗ ω Y (∆ ′ ) α ∗ / / Hom ( f ∗ O Y ( − ∆ ′ ) , ω X )Since ⌈ A ⌉ is effective, γ ∗ is injective. Because f ∗ ω Y (∆ ′ ) is a rank one sheaf andthe composition α ∗ ◦ γ ∗ is an isomorphism, γ ∗ is in fact isomorphism. This impliesthat α ∗ is a quasi isomorphism, so α : f ∗ O Y ( − ∆ ′ ) → R f ∗ O Y ( − ∆ ′ ) is also a quasiisomorphism. That is, R i f ∗ O Y ( − ∆ ′ ) = 0 , ∀ i > (cid:3) With theorem 4.1, we can prove Kov´acs vanishing theorem following Fujino’sidea [6]. Consider the following maps, Y h / / f Z g / / X where g : ( Z, ∆ Z ) → ( X, ∆) is a dlt modification such that K Z + ∆ Z = g ∗ ( K X +∆). And h : Y → Z is a Szab´o resolution such that K Y = h ∗ ( K Z +∆ Z )+ A − B − ∆ Y ,where ∆ Y = h − ∗ ∆ Z . We claim that R i f ∗ O Y ( −⌊ ∆ Y ⌋ ) = 0 , ∀ i >
0. By theorem 4.1, R i h ∗ O Y ( −⌊ ∆ Y ⌋ ) =0 , ∀ i >
0. Also note that h ∗ O Y ( ⌈ A ⌉ − ⌊ ∆ Y ⌋ ) = h ∗ O Y ( −⌊ ∆ Y ⌋ ) = O Z ( −⌊ ∆ Z ⌋ ).(lemma 12 in [12]). So by Leray spectral sequence, R i f ∗ O Y ( −⌊ ∆ Y ⌋ ) = R i g ∗ O Z ( −⌊ ∆ Z ⌋ ).The latter is zero for i > f : Y → X is not a log resolution. To fix the problem, we can blow upcenters with simple normal crossing with Supp(∆ Y + A + B ). Say the blow up is π : W → Y . There are two cases can happen; If we blow up klt locus, it is a Szab´oresolution and then the divisor with − W = π − ∗ (∆ Y ). Then R i π ∗ O W ( − ∆ W ) = 0 by theorem 4.1. If we blow up centers inside non-klt locus,then the divisor with − W = π − ∗ (∆ Y ) + F , where F isthe exceptional divisor produced by blow up. Then R i π ∗ O W ( − ∆ W ) = 0 by directcalculation. In any case we showed that the higher direct image is not changed bythese two kinds of blowing up. So we can conclude Kov´acs vanishing theorem. References [1] V. Alexeev,
Limits of stable pairs , Pure and Applied Math Quaterly, 4 (2008), no. 3, 1-15.[2] V. Alexeev and C. D. Hacon,
Non-rational centers of Log Canonical singularities .arXiv:1109.4164v2. To appear in J. Algebra.[3] W. Bruns and J. Herzog,
Cohen-Macaulay rings . Cambridge studies in advanced mathematics39.[4] O. Fujino
Introduction to the log minimal model program for log canonical pairs .arXiv:0907.1506v1, 2009.[5]
On isolated log canonical singularities with index one . J. Math. Sci. Univ. Tokyo18 (2011), 299–323.[6]
A remark on Kov´acs’s vanishing theorem . Kyoto Journal of Mathematics, Vol.52, No. 4 (2012), 831–834.[7] R. Hartshorne, residues and duality.
Lecture notes of a seminar on the work of A.Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notesin mathematics, No. 20, Spriner-Verlag, Berlin, 1966.
N ISOLATED LOG CANONICAL CENTERS 7 [8] S. Ishii,
On isolated Gorenstein singularities , Math. Ann. 270 (1985), no.4, 541-554.[9] Y. Kawamata, K. Matsuda and K. Matsuki,
Introduction to the Minimal Model Program,
Algebraic Geometry, Sendai, T. Oda ed., Kinokuniya, Adv. Stud. Pure Math. 10 (1987),283-360.[10] J. Koll´ar, S. Kov´acs,
Log canonical singularities are Du Bois , J of Amer. Math. Soc. 23(2010), no. 3, 791-813.[11] J. Koll´ar, S. Mori,
Birational geometry of algebraic varieties , Cambridge University Press,1998.[12] J. Koll´ar,
A local version of the Kawamata-Viehweg vanishing theorem , Pure Appl. Math.Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1477-1494.[13] ,
Singularities of the minimal model program , Cambridge Tracts in Mathematics,vol. 200, Cambridge University Press, Cambridge, 2013, with the collaboration of S´andor J.Kov´acs. (to appear.)[14] S. J. Kov´acs, K. Schwede, and K.E.Smith,
The canonical sheaf of Du Bois singularities ,Advances in Mathematics 224 (2010), no. 4, 1618- 1640.[15] S. J. Kov´acs, K. Schwede,
Hodge theory meets the minimal model program: a survey of logcanonical and Du Bois singularities , Topology of Stratified spaces, MSRI publications, Vol58, 2011.[16] S. Kov´acs,
Irrational Centers . Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memoryof Eckart Viehweg, 1495-1515.[17] ,
Du Bois pairs and vanishing theorems , Kyoto J. Math. 51 (2011), no. 1, 4769.[18] R. Lazarsfeld,
Positivity in algebraic geometry.
I, Ergebnisse der Mathematik und ihrer Gren-zgebiete, 3. Folge, Vol. 48, Springer-Verlag, Berlin, 2004.[19] E. Szab´o,
Divisorial log terminal singularities .J. Math. Sci. Tokyo, 1:631-639, 1995..J. Math. Sci. Tokyo, 1:631-639, 1995.