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Next, we obtain a restriction on the singularities of any split scheme. To formulate it, we need the following.

We refer to [AnBo—69], [Man—80] for more on this notion and on the weaker notion of semi-normality. Thus, normal varieties are weakly normal. Thus, f is an isomorphism. Every split scheme is weakly normal. This is a nonzero ideal of B contained in A. The opposite inclusion follows from Remark 1. By the above claim and Proposition 1. Thus, E contains no irreducible components of X. By the claim, X is split compatibly with E.

Further consequences of the existence of a Frobenius splitting concern the vanishing of all higher cohomology groups of line bundles, or equivalently of invertible sheaves. To establish them, we need the following preliminary result. Let L be an invertible sheaf on a scheme X.

Now, F is the identity map on points, and F is the p-th power map. The second isomorphism follows from the projection formula. Let L be an invertible sheaf on a split scheme X. Here and elsewhere, when no confusion is likely, we have abused the notation and denoted L Y by L itself. Here the isomorphism on the extreme right is only semilinear. Since Y is compatibly split, the horizontal arrows are compatibly split as well.

Frobenius Splitting: General Theory We now come to the main results of this section. By the Serre vanishing theorem [Har—77, Chap. III, Proposition 5. Combined with Lemma 1. Let X be a split projective scheme equipped with an ample invertible sheaf L. Assume that X is Cohen—Macaulay and equidimensional. III, Theorem 7. Using Lemma 1.

III, Corollary 7. Consider, for example, the case where X is a projective nonsingular irreducible curve of genus g. IV, Corollary 3. Hence, X is not split. Consequences of Frobenius splitting 17 As another class of examples, consider a hypersurface X of degree d in the projective space Pn. Then, by [Har—77, Chap.

III, Proposition 8. By [loc cit. III, Theorem 8. By the theorem on formal functions [loc cit. III, Theorem Here My is the maximal ideal of OY,y. To check this, denote by Ks the kernel of this map.

## Progress in Mathematics

On the other hand, the action 18 Chapter 1. The next relative vanishing theorem will be a key ingredient in obtaining versions of the Grauert—Riemenschneider and Kawamata—Viehweg theorems in the presence of a splitting Theorems 1. Let D, resp. E, be a closed subscheme of X, resp. We begin with reduction arguments similar to those of the proof of Lemma 1. We argue by contradiction, and assume that H i X, ID is nonzero. Choose an irreducible component of this support, with generic point y. Frobenius Splitting: General Theory Proof.

In particular, weakly normal curves are those reduced curves having only ordinary multiple points as singularities. Show that X is normal.

## A proof of Wahl's conjecture in the symplectic case

In particular, the normalization of a split scheme is split. Let I be the conductor as in the proof of Proposition 1. Recall from Remark 1. OX , cf. II, Exercise 5. III, Exercise 6.

## Frobenius Splitting Methods in Geometry and Representation Theory

Together with Lemma 1. We will recover these results in a more direct and explicit way later in this section. We begin with the simplest 1. Recall that the k[t1 ,. Frobenius Splitting: General Theory This space has another structure of k[t1 ,. OAn by [Har—77, Chap. Let Tr : k[t1 ,. We claim that Tr is a free generator of this module as a module over k[t1 ,. Set identifying An , F! Thus, Tr generates the k[t1 ,. This completes the proof of the claim.

Observe that the splitting of Example 1. We now aim at extending the results of Example 1. Here are some basic properties of its cohomology spaces. Since d is an Ap -linear Proof. By the 1 , Let X be a nonsingular variety of dimension n. We now express the trace map in local coordinates.

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Let X be a nonsingular variety of dimension n and let t1 ,. This implies our formula.

Criteria for splitting 27 Notice that the formula for Tr gives back the generator of An , F! OAn found in Example 1. More generally, we have the following.

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Let X be a nonsingular variety. We can now state an important characterization of split varieties, which follows immediately from Proposition 1. Here, t1 ,. To formulate this condition, we need the following. Let X be a nonsingular variety of dimension n and let Y1 ,. Equivalently, there exists a system of local coordinates t1 ,.

Criteria for splitting 29 1. In particular, this subscheme is reduced.

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Choose a system of local coordinates t1 ,. Let Y be the unique irreducible component of this zero scheme containing x, and let f be a local equation of Y at x.

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Since Y is nonsingular at x, we may choose a system of local coordinates t1 ,. Let t1m g t1 ,. Applying Lemma 1.