A Local Version of the Kawamata-Viehweg Vanishing

October 24, 2018 A LOCAL VERSION OF THE KAWAMATA-VIEHWEG VANISHING THEOREM JANOS´ KOLLAR´ To the memory of Eckart Viehweg 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 first significant 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 simplified 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]. Here we prove a further generalization, which, I believe, covers all the theorems about depth mentioned above. We work with varieties over a field of characteristic 0. For the basic definitions and for background material see [KM98]. Definition 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 definition 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 {Xi : i ∈ I} is an open cover of X and D1|Xi ∼Q,loc D2|Xi for every i, then D1 ∼Q,loc D2. The following can be viewed as a local version of the Kawamata-Viehweg van- arXiv:1005.4843v1 [math.AG] 26 May 2010 ishing theorem. Theorem 2. Let (X, ∆) be dlt, D a (not necessarily effective) Z-divisor and ∆′ ≤ ′ ∆ an effective 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– i Macaulay. A sheaf F is CM iff Hx(X, F ) = 0 for i < codimX x for every point i x ∈ X and depthx F ≥ j iff 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).) 1 2 JANOS´ KOLLAR´ Weaker results hold for log canonical and semi log canonical pairs. For basic definitions 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 ∈ X a point that is not a log canonical center of (X, ∆). (1) Let D be a Z-divisor such that none of the irreducible components of D is contained in Sing X. Let ∆′ ≤ ∆ be an effective Q-divisor on X such that ′ D ∼Q,loc ∆ . Then depthx OX (−D) ≥ min{3, codimX x}. (2) Let Z ⊂ X be any closed, reduced subscheme that is a union of lc centers of (X, ∆). Then depthx OX (−Z) ≥ min{3, 1 + codimZ x}. 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. Then D can be any Q-Cartier divisor, reproving [KM98, 5.25]. The D = ∆′ ≤ ∆ 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 ∈ X is not a log canonical center then depthx OX ≥ min{3, codimX x}. This can fail if x is a log canonical center, for instance when x ∈ X is a cone over an Abelian variety of dimension ≥ 2. (4.3) If (X, ∆) is slc then KX + ∆ is Q-Cartier, hence −KX ∼Q,loc ∆. Then O −(−K ) =∼ ω . Thus if x ∈ X is not a log canonical center then depth ω ≥ X X X x X min{3, codimX x}. (Note that while OX is CM iff ωX is CM, it can happen that OX is S3 but ωX is not. 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 fiber over a closed point. None of the lc centers are contained in Xc, thus if x ∈ Xc has codimension ≥ 2 then depthx ωX/C ≥ 3. Therefore, the restriction of ωX/C to Xc is S2, hence it is isomorphic to ωXc . More generally, ωX/C commutes with arbitrary base change. (When the general fiber is klt, this follows from [Elk81]; for projective morphisms a proof is given in [KK09], 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 −nK −⌊n∆⌋∼Q −n K +∆)+ n∆ −⌊n∆⌋ ∼Q n∆ −⌊n∆⌋. X X ,loc 1 N Assume now that ∆ = 1 − mi Di with mi ∈ ∪ {∞}. Then n∆ −⌊n∆⌋ = ci P Di for some ci ∈ N where 0 ≤ ci < mi for every i. Thus ci ≤ mi − 1 for Pi mi every i, that is, n∆ −⌊n∆⌋≤ ∆. Thus, if x ∈ X is not a log canonical center then depth O nK + ⌊n∆⌋ ≥ min{3, codim x}. x X X X In particular, if f : (X, ∆) → C is a proper slc morphism to a smooth curve and Xc is any fiber then the restriction of O nK + ⌊n∆⌋ to any fiber is S and hence X X 2 the natural map O nK + ⌊n∆⌋ | → O nK + ⌊n∆ ⌋ is an isomorphism. X X Xc Xc Xc c LOCALKAWAMATA-VIEHWEGVANISHING 3 This implies that the Hilbert function of the fibers χ X , O nK + ⌊n∆ ⌋ c Xc Xc c is deformation invariant. (Note that, because of the rounding down, the Hilbert function is not a polynomial, rather a polynomial whose coefficients are periodic functions of n. The period divides the index of (X, ∆), that is, the smallest n0 ∈ N such that n ∆ is a Z-divisor and n K + ∆ is Cartier.) 0 0 X (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 ⊂ Pn be projectively normal, H the hyperplane class on Y and D a Cartier divisor on Y . Let X ⊂ An+1 be the cone over Y with vertex v and DX the cone over D. Then Hi X, O (D ) = Hi−1 Y, O (D + mH) for i ≥ 2. v X X X Y m∈Z In particular, OX (DX ) is CM iff Hi Y, O (D + mH) =0 ∀ m ∈ Z, ∀ 0 <i< dim Y. Y 4 (4.7) Consider the quadric cone X := (x1x2 = x3x4) ⊂ A with vertex v = (0, 0, 0, 0). It is the cone over the quadric surface Q =∼ P1 × P1 ⊂ P3. It contains two families of planes with typical members A := (x1 = x3 = 0) and B := (x1 = x4 = 0). By (4.6) 2 1 H X, OX (aA + bB) = H Q, OQ(a + m,b + m) v Pm∈Z 0 1 0 1 = H P , OP1 (m) ⊗ H P , OP1 (|b − a|− 2 − m) . P0≤m≤|b−a|−2 Thus we see that OX (aA + bB) is CM only if |b − a| < 2. 1 (4.8) As another application of (2), assume that (X, aiDi) is dlt and 1 − ≤ P n ai ≤ 1 for every i for some n ∈ N. Then, for every m, m KX + Di = m(1 − ai)Di + m KX + aiDi Pi Pi Pi ∼Q,loc m(1 − ai)Di. Pi If 1 ≤ m ≤ n − 1 then 0 ≤ m(1 − ai) ≤ ai, thus by (2) and by Serre duality we conclude that [−m] [m+1] ω (−m Di) and ω (m Di) areCMfor1 ≤ m ≤ n − 1. (4.8.1) X P X P If, in addition, KX is Q-Cartier, then m Di ∼Q,loc m(1 − ai)Di, hence Pi Pi OX (−m Di) is CM for 1 ≤ m ≤ n − 1. (4.8.2) P Results like these are quite fragile. As an example, let X ⊂ A4 be the quadric cone with the 2 families of planes |A| and |B|. Then X, A + 1 (B + B ) and X, 9 (A + A )+ 6 (B + B + B ) 1 2 1 2 10 1 2 10 1 2 3 are both dlt, giving that OX (−A1 − B1 − B2) and OX (−A1 − A2 − B1 − B2 − B3) are CM.

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