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A Local Version of the Kawamata-Viehweg Vanishing Theorem

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A Local Version of the Kawamata-Viehweg Vanishing Theorem Pure and Applied Mathematics Quarterly Volume 7, Number 4 (Special Issue: In memory of Eckart Viehweg) 1477|1494, 2011 A Local Version of The Kawamata-Viehweg Vanishing Theorem J¶anosKoll¶ar To the memory of Eckart Viehweg Abstract: We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered. Keywords: vanishing theorems, Cohen-Macaulay, log canonical. The, by now classical, Kawamata{Viehweg vanishing theorem [Kaw82, Vie82] says that global cohomologies vanish for divisorial sheaves which are Q-linearly equivalent to a divisor of the form (nef and big) + ¢. In this note we prove that local cohomologies vanish for divisorial sheaves which are Q-linearly equivalent to a divisor of the form ¢. If X is a cone over a Fano variety, one can set up a perfect correspondence between the global and local versions. More generally, we study the depth of various sheaves associated to a log canonical pair (X; ¢). The ¯rst signi¯cant result in this direction, due to [Elk81], says that if (X; 0) is canonical then X has rational singularities. In particular, OX is CM. The proof has been simpli¯ed repeatedly in [Fuj85], [Kol97, Sec.11] and [KM98, 5.22]. Various generalizations for other divisorial sheaves and to the log canonical case were established in [KM98, 5.25], [Kov00], [Ale08] and [Fuj09b, Secs.4.2{3]. Received: May 26, 2010; Revised: Nov. 9, 2010. 1478 J¶anosKoll¶ar Here we prove a further generalization, which, I believe, covers all the theorems about depth mentioned above. We work with varieties over a ¯eld of characteristic 0. For the basic de¯nitions and for background material see [KM98]. De¯nition 1. Let X be normal and D1;D2 two Q-divisors. We say that D1 is locally Q-linearly equivalent to D2, denoted by D1 »Q;loc D2, if D1 ¡ D2 is Q-Cartier. The same de¯nition works if X is not normal, as long as none of the irreducible components of the Di is contained in Sing X. Note that this is indeed a local property. That is, if fXi : i 2 Ig is an open cover of X and D1jXi »Q;loc D2jXi for every i, then D1 »Q;loc D2. The following can be viewed as a local version of the Kawamata-Viehweg van- ishing theorem. Theorem 2. Let (X; ¢) be dlt, D a (not necessarily e®ective) Z-divisor and 0 0 ¢ · ¢ an e®ective Q-divisor on X such that D »Q;loc ¢ . Then OX (¡D) is CM. Here dlt is short for divisorial log terminal [KM98, 2.37] and CM for Cohen{ Macaulay. A sheaf F is CM i® all its local cohomologies vanish below the maximal i dimension; that is, i® Hx(X; F ) = 0 for i < codimX x for every point x 2 X. i Similarly, depthx F ¸ j i® Hx(X; F ) = 0 for i < j; see [Gro68, III.3.1] or [Har77, Exrcs.III.3.3{5]. Examples illustrating the necessity of the assumptions are given in (4.5{10). The proof of Theorem 2 also works in the complex analytic case. (Normally one would expect that proofs of a local statement as above automatically work for analytic spaces as well. However, many of the papers cited above use global techniques, and some basic questions are still unsettled; see, for instance, (4.3).) Weaker results hold for log canonical and semi log canonical pairs. For basic de¯nitions in the semi log canonical case (abbreviated as slc) see [K+92, Sec.12] or [Fuj09b]. Theorem 3. Let (X; ¢) be semi log canonical and x 2 X a point that is not a log canonical center of (X; ¢). Local Kawamata-Viehweg Vanishing 1479 (1) Let D be a Z-divisor such that none of the irreducible components of D is contained in Sing X. Let ¢0 · ¢ be an e®ective Q-divisor on X such 0 that D »Q;loc ¢ . Then depthx OX (¡D) ¸ minf3; codimX xg: (2) Let Z ½ X be any closed, reduced subscheme that is a union of lc centers of (X; ¢). Then depthx OX (¡Z) ¸ minf3; 1 + codimZ xg: In contrast with (2), my proof does not work in the complex analytic case; see (17). 4 (Applications and examples). (4.1) In (2) we can take ¢0 = 0. Then D can be any Q-Cartier divisor, reproving [KM98, 5.25]. The D = ¢0 · ¢ case recovers [Fuj09b, 4.13]. (4.2) The D = 0 case of (3.1) is a theorem of [Ale08, Fuj09b] which says that if (X; ¢) is lc and x 2 X is not a log canonical center then depthx OX ¸ minf3; codimX xg. This can fail if x is a log canonical center, for instance when x 2 X is a cone over an Abelian variety of dimension ¸ 2. (4.3) If (X; ¢) is slc then K + ¢ is Q-Cartier, hence ¡K » ¢. Then ¡ ¢ X X Q;loc » OX ¡(¡KX ) = !X . Thus if x 2 X is not a log canonical center then depthx !X ¸ minf3; codimX xg. (Note that while OX is CM i® !X is CM, it can happen that OX is S3 but !X is not; see [Pat10]. Thus (4.3) does not seem to be a formal consequence of (4.2).) Let f :(X; ¢) ! C be a semi log canonical morphism to a smooth curve C (cf. [KM98, 7.1]) and Xc the ¯ber over a closed point. None of the lc centers are contained in Xc, thus if x 2 Xc has codimension ¸ 2 then depthx !X=C ¸ 3. Therefore, the restriction of !X=C to Xc is S2, hence it is isomorphic to !Xc . More generally, !X=C commutes with arbitrary base change. (When the general ¯ber is klt, this follows from [Elk81]; for projective morphisms a proof is given in [KK10], but the general case has not been known earlier. As far as I know, the complex analytic case is still unproved.) (4.4) Assume that (X; ¢) is slc. For any n ¸ 1, write ¡ ¡ ¢ ¡nKX ¡ bn¢c »Q ¡n KX + ¢) + n¢ ¡ bn¢c »Q;loc n¢ ¡ bn¢c: 1480 J¶anosKoll¶ar P¡ 1 ¢ Assume now that ¢ = 1 ¡ Di with mi 2 N [ f1g. Then n¢ ¡ bn¢c = P mi ci D for some c 2 N where 0 · c < m for every i. Thus c · m ¡ 1 for i mi i i i i i i every i, that is, n¢ ¡ bn¢c · ¢. Thus, if x 2 X is not a log canonical center then ¡ ¢ depthx OX nKX + bn¢c ¸ minf3; codimX xg: In particular, if f :(X; ¢) ! C is a proper slc morphism to a smooth curve and ¡ ¢ Xc is any ¯ber then the restriction of OX nKX + bn¢c to any ¯ber is S2 and hence the natural map ¡ ¢ ¡ ¢ OX nKX + bn¢c jXc !OXc nKXc + bn¢cc is an isomorphism. This implies that the Hilbert function of the ¯bers ¡ ¡ ¢¢ Â Xc; OXc nKXc + bn¢cc is deformation invariant. (Note that, because of the rounding down, the Hilbert function is not a polynomial in the usual sense, rather a polynomial whose coef- ¯cients are periodic functions of n. The period divides the index of (X; ¢), that ¡ ¢ is, the smallest n0 2 N such that n0¢ is a Z-divisor and n0 KX + ¢ is Cartier.) (4.5) It is also worthwhile to note that while the assumptions of Theorem 2 depend only on the Q-linear equivalence class of D, being CM is not preserved by Q-linear equivalence in general. For instance, let X be a cone over an Abelian variety A of dimension ¸ 2. Let DA be a Z-divisor on A such that mDA » 0 for some m > 1 but DA 6» 0. Let DX be the cone over DA. Then DX »Q;loc 0, OX (DX ) is CM but OX is not CM. These assertions follow from the next easy characterization of CM divisorial sheaves on cones: (4.6) Claim. Let Y ½ Pn be projectively normal, H the hyperplane class on Y and D a Cartier divisor on Y . Let X ½ An+1 be the cone over Y with vertex v and DX the cone over D. Then X i ¡ ¢ i¡1¡ ¢ Hv X; OX (DX ) = H Y; OY (D + mH) for i ¸ 2: m2Z In particular, OX (DX ) is CM i® i¡ ¢ H Y; OY (D + mH) = 0 8 m 2 Z; 8 0 < i < dim Y: ¤ Local Kawamata-Viehweg Vanishing 1481 4 (4.7) Consider the quadric cone X := (x1x2 = x3x4) ½ A with vertex v = (0; 0; 0; 0). It is the cone over the quadric surface Q =» P1 £ P1 ½ P3. It contains two families of planes with typical members A := (x1 = x3 = 0) and B := (x1 = x4 = 0). By (4.6) ¡ ¢ ¡ ¢ 2 P 1 Hv X; OX (aA + bB) = m2Z H Q; OQ(a + m; b + m) ¡ ¢ ¡ ¢ P 0 1 0 1 = 0·m·jb¡aj¡2 H P ; OP1 (m) ­ H P ; OP1 (jb ¡ aj ¡ 2 ¡ m) : Thus we see that OX (aA + bB) is CM only if jb ¡ aj < 2. P 1 (4.8) As another application of (2), assume that (X; aiDi) is dlt and 1¡ n · ai · 1 for every i for some n 2 N. Then, for every m, ¡ P ¢ P ¡ P ¢ m K + D = m(1 ¡ a )D + m K + a D X i i Pi i i X i i i »Q;loc im(1 ¡ ai)Di: If 1 · m · n ¡ 1 then 0 · m(1 ¡ ai) · ai, thus by (2) and by Serre duality we conclude that [¡m] P [m+1] P !X (¡m Di) and !X (m Di) are CM for 1 · m · n ¡ 1.
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