A characteristic p proof of Wahl's vanishing theorem for rational surface singularities

Arch. Math. 73 (1999) 256 ± 261 0003-889X/99/040256-06 $ 2.70/0  Birkhäuser Verlag, Basel, 1999

Archiv der Mathematik

A characteristic p proof of Wahls vanishing theorem for rational surface singularities By NOBUO HARA Dedicated to Professor Satoshi Arima Abstract. We give a quick new proof of Wahls vanishing theorem for rational surface singularities, using prime characteristic technique. Our method also gives a criterion for F-rationality in dimension 2.

1. Introduction. In his paper [13], J. Wahl proved some vanishing theorems for resolutions of normal surface singularities, being motivated by applications to deformations of surface singularities. Among them, Theorem C and Theorem D of [13] are hard to prove, and fail in characteristic p > 0. In this note we shall give a quick new proof to one of them, namely Theorem D, using the method developped in [3]. We begin with fixing the notation to be used throughout this note. Let R ˆ oY;y be the local ring of a normal surface singularity …Y; y† over an algebraically closed field k, and let f : X ! Spec R be the minimal good resolution, i.e., the smallest resolution whose reduced exceptional divisor E ˆ f ÿ1 …y†red consists of smooth curves E1 ; . . . ; Es intersecting transversally. The weighted dual graph G of …Y; y† is the numerical data consisting of the genera of Ei s and the intersection numbers Ei
Ej. We denote by VX …ÿlog E† the oX -dual of the sheaf W1X …log E† of differential 1-forms with log poles along E. Theorem 1 (Wahl [13, Theorem D]). Let …Y; y† be a rational singularity with weighted dual graph G. If char k ˆ 0 or char k ˆ p is greater than a certain bound determined only by G, then we have the following vanishing of local cohomology with supports along E: HE1 …X; VX …ÿlog E†…E†† ˆ 0: We can slightly generalize Theorem 1. Following [7] and [12], we say that …Y; y† is a Du Bois singularity if the natural map R1 f oX ! H 1 …E; oE † is an isomorphism. A rational singularity is Du Bois. Theorem 2. Let …Y; y† be a Du Bois singularity with weighted dual graph G. Then, if char k ˆ 0 or char k ˆ p is greater than a certain bound given explicitly by G, we have HE1 …X; W1X …log E†† ˆ 0: Mathematics Subject Classification (1991): 14B05, 14J17, 13A35.

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Given the weighted dual graph G of a singularity …Y; y†, we can bound char k ˆ p > 0 for s P which the theorems hold as follows: Let D ˆ ri Ei be the minimal positive integral cycle iˆ1 with the property …†

D
Ei % min f0; 2 ÿ 2gi ÿ ti g for all i ˆ 1; . . . ; s;

where gi is the genus of Ei, and ti is the number of intersection of Ei with other Ej s. Then, if p > rG ˆ maxfr1 ; . . . ; rs g, Theorems 1 and 2 hold true. We can find the cycle D by the method of ªcomputation sequence.º The inequality p > rG also gives a sufficient condition for a rational singularity to be F-rational, although it is not best possible both for the vanishing theorems and F-rationality. To prove the theorems, we reduce things to characteristic p > 0 and employ some characteristic p method. Unfortunately, our method does not apply to the other vanishing theorem of Wahl [13, Theorem C]. However, we remark that similar technique gives a proof of the Kodaira vanishing theorem for smooth projective surfaces in characteristic 0 (cf. [6]).

2. Proof of theorems. We first show the implication Theorem 2 ) Theorem 1, and next prove Theorem 2. Finally we examine the bound p > rG given by …†. Theorem 2 ) Theorem 1: We first recall the dual form of Grothendiecks formal function theorem (e.g., [13, Proposition 2.2]), which says that, for a locally free sheaf f on X one has HE1 …X; f†  lim H 0 …Z; f oZ …Z††; ÿ!

where Z runs through all positive cycles supported in E, and the direct limit maps are injective. Therefore, HE1 …X; f† ˆ 0 if and only if H 0 …Z; f oZ …Z†† ˆ 0 for all Z > 0 with Supp…Z† 7 E. So, to deduce Theorem 1 from Theorem 2, it is sufficient to show that there is an inclusion map VX …ÿlog E†…E† oZ …Z† ,! W1X …log E† oZ …Z† for every positive cycle Z on E. Now let …Y; y† be a rational singularity. Then f : X ! Spec R is the minimal resolution, so that the canonical divisor KX of X is f -nef, i.e., ki ˆ KX
Ei ^ 0 for i ˆ 1; . . . ; s. Given a positive cycle Z on E, pick an effective Cartier divisor K 0 on Z whose image by the degree map Pic…Z† ! Z s is …k1 ; . . . ; ks †. Then we have oZ …ÿKX †  oZ …ÿK0 †, since the rationality of …Y; y† implies that the degree map is an isomorphism [1]. Thus we have an inclusion oZ …ÿKX † ,! oZ , and tensoring with W1X …log E†…Z† yields VX …ÿlog E†…E ‡ Z† oZ ,! W1X …log E†…Z† oZ , as required. Frobenius and de Rham complex (cf. [3]). Our strategy to attack Theorem 2 is entirely different from Wahls. We reduce things to the case char k ˆ p > 0, and make use of the Frobenius morphism F: X ! X, via which we can regard the push-forward of the log de Rham complex F WX …log E† as a complex of oX -modules. More generally, for an integral s P rj Ej supported in E, the usual differential map induces a complex divisor D ˆ jˆ1

F …WX …log E†…D†† of oX -modules. We denote the ith cocycle and coboundary of F …WX …log E†…D†† by ziD and biD , respectively, and simply write zi ˆ zi0, bi ˆ bi0 for

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the case D ˆ 0. Then if 0 % r1 ; . . . ; rs % p ÿ 1, we have the following exact sequences of locally free oX -modules for each i: …ai † 0 ! ziD ! F …WiX …log E†…D†† ! bi‡1 D ! 0; …bi † 0 ! biD ! ziD ! WiX …log E† ! 0: It would be needless to explain about …ai †. The second one …bi † is induced by the Cartier  operator Cÿ1 : WiX …log E† ! zi =bi (see [8]) and the natural inclusion F WX …log E† ! F …WX …log E†…D††; which is a quasi-isomorphism by the assumption 0 % rj % p ÿ 1 [3, Lemma 3.2]. P r o o f o f Th e o r e m 2 . We prove the theorem for char k ˆ p  0. Then the case char k ˆ 0 follows by a standard technique of reduction modulo p (see e.g. [4, §5]). The main point of our proof in the case char k ˆ p is the following Proposition 3. Let …Y; y† be a Du Bois singularity of char k ˆ p > 0 and assume that there is an integral divisor D on X satisfying the following condition: (y) 0 % D % …p ÿ 1†E and HE1 …X; W1X …log E†…D†† ˆ 0. Then we have HE1 …X; W1X …log E†† ˆ 0: P r o o f. The exact sequence …a1 † and the vanishing of (y) imply HE1 …X; z1D † ˆ 0, since b2D is torsion-free. Then HE1 …X; b1D † ˆ 0 by …b1 †. Applying this vanishing to the exact sequence …a0 †, which is nothing but F

0 ! oX ! F …oX …D†† ! b1D ! 0; we see that the induced map F: HE2 …X; oX † ! HE2 …X; oX …D†† is injective. Since the map F: oX ! F …oX …D†† in …a0 † factors into the usual Frobenius F: oX ! F oX and the inclusion F oX ,! F …oX …D††, the Frobenius map F: HE2 …X; oX † ! HE2 …X; oX † is also injective. Here we switch from the exact sequences …ai † and …bi † to those for D ˆ 0. We restart from …a0 † for D ˆ 0, F

0 ! oX ! F oX ! b1 ! 0: Then, the injectivity of the Frobenius map on HE2 …X; oX †, together with the GrauertRiemenschneider vanishing HE1 …X; oX † ˆ 0 (cf. [13, Theorem A]), implies HE1 …X; b1 † ˆ 0. We consider the following commutative diagram with exact rows, which is induced by the exact sequence …b1 † and the quasi-isomorphism F WX …log E† ! F …WX …log E†…D††. 0 ˆ HE1 …X; b1 †

ÿ!

HE1 …X; z1 †

ÿ! HE1 …X; W1X …log E††

# 0ˆ

HE1 …X; z1D †

jj ÿ!

HE1 …X; W1X …log E††

We now have that HE1 …X; z1 † ˆ 0. So, to prove HE1 …X; W1X …log E†† ˆ 0, it suffices to show that HE1 …X; b2 † ˆ 0, by virtue of the exact sequence …a1 † for D ˆ 0. Finally we use …b2 † for D ˆ 0, 0 ! b2 ! F W2X …log E† ! W2X …log E† ! 0: This sequence tells us that it is enough to show HE1 …X; W2X …log E†† ˆ HE1 …X; wX …E†† ˆ 0. By duality, it is equivalent to saying that R1 f oX …ÿE† ˆ 0. Looking at the exact sequence

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0 ! oX …ÿE† ! oX ! oE ! 0, we see that it is a direct consequence of our assumption that …Y; y† is a Du Bois singularity. Now we continue the proof of Theorem 2 for char k ˆ p > 0. First we sketch an outline. If s P Dˆ ri Ei is an effective divisor whose intersection number with each Ei is less than a iˆ1

certain bound, then HE1 …X; W1X …log E†…D†† ˆ 0 as we will see in Lemma 4 below. The coefficients r1 ; . . . ; rs of such a D possibly depend on the weighted dual graph G of …Y; y†, but we can choose them independently of char k ˆ p. So, if p > maxfr1 ; . . . ; rs g, then condition (y) in Proposition 3 is satisfied, and the required vanishing follows. A dual form of the Serre vanishing tells us that the vanishing HE1 …X; W1X …log E†…D†† ˆ 0 is achieved if ÿD is a large multiple of an f -ample Cartier divisor. However, we can give a little more economic choice of D (though it is not best possible yet). Lemma 4. Let gi be the genus of Ei and let ti ˆ Ei
…E ÿ Ei †. Consider an integral cycle s P Dˆ ri Ei such that iˆ1

…† D
Ei % minf0; 2 ÿ 2gi ÿ ti g for i ˆ 1; . . . ; s. Then we have HE1 …X; W1X …log E†…D†† ˆ 0: P r o o f. By [13, Proposition 2.4], it is sufficient to show that for any positive cycle Z supported in E, there exists a component Ei of Supp…Z† such that H 0 …Ei ; W1X …log E† oEi …D ‡ Z†† ˆ 0: To see this, choose an Ei such that Z
Ei < 0 (note that …Ei
Ej † is negative definite), and recall the following exact sequence [13, (1.10.2)]: 0 ! oEi ! VX …ÿlog E† oEi ! VEi …Ei ÿ E† ! 0: Tensoring with oX …KX ‡ E ‡ D ‡ Z† gives 0 ! oEi …KX ‡ E ‡ D ‡ Z† ! W1X …log E† oEi …D ‡ Z† ! oEi …D ‡ Z† ! 0: Since H 0 …Ei ; oEi …KX ‡ E ‡ D ‡ Z†† ˆ H 0 …Ei ; oEi …D ‡ Z†† ˆ 0 by our numerical assumption, we have H 0 …Ei ; W1X …log E† oEi …D ‡ Z†† ˆ 0, as desired. Given a weighted dual graph G, we can find the unique minimal effective cycle D with property …† in Lemma 4, using the algorithm called ªcomputation sequenceº (cf. [9]). s P ri Ei is such cycle for a Du Bois singularity …Y; y† and if Consequently, if D ˆ iˆ1

char k ˆ p > rG ˆ maxfr1 ; . . . ; rs g, then HE1 …X; W1X …log E†† ˆ 0. This completes the proof of Theorem 2. When …Y; y† is a rational singularity, we can make the inequality p > rG slightly better. In this case, we can accept the condition …†

D
Ei % minfdi ÿ 2; di ÿ ti g for i ˆ 1; . . . ; s,

instead of condition …†, where ÿdi ˆ E2i is the self-intersection number of Ei. To see this, we note that for any positive cycle Z supported in E, one has …KX ‡ Z†
Z % ÿ 2 (since …Y; y† is rational), so that …KX ‡ Z†
Ei < 0 for some Ei 7 Supp…Z†. Then argue as in the proof of Lemma 4.

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3. A criterion for F-rationality in dimension two. The notion of F-rational rings in characteristic p > 0 is defined in terms of tight closure ([2], [5]), and proved to characterize rational singularity ([3], [10], [11]). Namely, a ring of characteristic 0 has rational singularity if and only if its modulo p reduction is F-rational for p  0. In dimension two, F-rational rings always have rational singularities. On the other direction, the following question arises: Given the weighted dual graph G of a rational surface singularity, can we find an explicit lower bound of characteristic p for which singularities with graph G are F-rational? When the graph G is star-shaped, we can give the best possible answer to this question [4]. When the graph is not star-shaped, we do not have the effective bound yet. But we can apply our method to give the following cheap criterion: Proposition 5. Let …Y; y† be a rational singularity of char k ˆ p > 0 and D ˆ

s P iˆ1

ri Ei be the

minimal positive cycle with property …†. Then R ˆ oY;y is F-rational if p > maxfr1 ; . . . ; rs g. P r o o f. For every e ˆ 0; 1; 2; . . ., the cycle pe D satisfies property …†, so that one has HE1 …X; W1X …log E†…pe D†† ˆ 0 by Lemma 4. As we have seen in the proof of Proposition 3, this implies that the Frobenius map F: HE2 …X; oX † ! HE2 …X; oX …D†† is injective as well as F: HE2 …X; oX …peÿ1 D†† ! HE2 …X; oX …pe D†† 1 for all e ˆ 1; 2; . . . : Therefore the Q-divisor A ˆ D applies to the following result, which p comes from the argument in [3, §4]: Proposition 6 ([3]). Let char k ˆ p > 0 and let A ˆ

s P iˆ1

ai Ei be a Q-divisor on X such that

0 < ai < 1. Assume that the e-times iterated Frobenius map F e : HE2 …X; oX † ˆ HE2 …X; oX …‰AŠ†† ! HE2 …X; oX …‰pe AŠ†† is injective for all e ˆ 1; 2; . . . : Then R ˆ oY;y is F-rational if and only if …Y; y† is a rational singularity. E x a m p l e s. (i) Let …Y; y† be a rational double point of type E6. Then our method shows that if char k ˆ p ^ 7, then HE1 …X; VX …ÿlog E†…E†† ˆ HE1 …X; W1X …log E†† ˆ 0 and R ˆ oY;y is F-rational. In this case, however, the effective bound is p ^ 5 both for Wahls Vanishing Theorem D and F-rationality ([4], [13]). (ii) The criterion of F-rationality given in [4] does not apply to the case where the dual graph G is not star-shaped. Let us consider a rational singularity …Y; y† with non-star-shaped graph:

Here ªÿ3º in the circle means that the corresponding component Ei has self-intersection E2i ˆ ÿ3, while the self-intersection numbers of the other components are ÿ2. The numbers

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outside of the circles denote the coefficients of the minimal positive cycle D satisfying …†. Hence our results are applicable if p ^ 5. Actually, the above graph has no ªbad cycle for Theorem Dº in the sense of [13], so that HE1 …X; VX …ÿlog E†…E†† ˆ 0 holds for all p. Moreover, a direct computation shows HE1 …X; W1X …log E†…Z0 †† ˆ 0 for the fundamental cycle j 2, then HE1 …X; W1X …log E†† ˆ 0 and R is F-rational. Z0 . This implies that if p ˆ Q u e s t i o n . For a rational surface singularity …Y; y† of characteristic p > 0, is there any implication between the vanishing HE1 …X; W1X …log E†† ˆ 0 and the F-rationality of oY;y ? We know that the vanishing implies the F-rationality when …Y; y† has a k -action [2]. A c k n o w l e d g m e n t s. The author thanks Tomohiro Okuma for inspiring the formulation in terms of the concept of Du Bois singularities. This work was partially supported by Waseda University Grant for Special Research Project 97A-584 and 98A-140. References [1] M. ARTIN, Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math. 84, 485 ± 496 (1962). [2] R. FEDDER and K.-I. WATANABE, A characterization of F-regularity in terms of F-purity. In: ªCommutative Algebra,º MSRI Publ. 15, 227 ± 245. New York 1989. [3] N. HARA, A characterization of rational singularities in terms of injectivity of Frobenius maps. Amer. J. Math. 120, 981 ± 996 (1998). [4] N. HARA and K.-I. WATANABE, The injectivity of Frobenius acting on cohomology and local cohomology modules. Manuscripta Math. 90, 301 ± 315 (1996). [5] M. HOCHSTER and C. HUNEKE, Tight closure, invariant theory and the BriancËon-Skoda theorem. J. Amer. Math. Soc. 3, 31 ± 116 (1990). [6] C. HUNEKE and K. E. SMITH, Tight closure and the Kodaira vanishing theorem. J. Reine Angew. Math. 484, 127 ± 152 (1997). [7] S. ISHII, Du Bois singularities on a normal surface. Adv. Stud. Pure Math. 8, 153 ± 163 (1986). [8] N. KATZ, Nilpotent connections and monodromy theorem. Publ. Math. IHES 39, 175 ± 232 (1970). [9] H. B. LAUFER, On rational singularities. Amer. J. Math. 94, 597 ± 608 (1972). [10] K. E. SMITH, F-rational rings have rational singularities. Amer. J. Math. 119, 159 ± 180 (1997). [11] V. SRINIVAS and V. B. MEHTA, A characterization of rational singularities. Asian J. Math. 1, 249 ± 271 (1997). [12] J. STEENBRINK, Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math. 40 (2), 513 ± 536 (1983). [13] J. WAHL, Vanishing theorems for resolutions of surface singularities. Invent. Math. 31, 17 ± 41 (1975). Eingegangen am 3. 6. 1998 *) Anschrift des Autors: Nobuo Hara Department of Mathematics Waseda University 3-4-1 Okubo Shinjuku Tokyo 169 Japan

*) Eine überarbeitete Fassung ging am 8. 2. 1999 ein.