We also provide algorithms for determining all such degrees in practice. The coherent cohomology of toroidal compactifications of locally symmetric varieties such as Shimura varieties, with coefficients in the so-called canonical and subcanonical extensions of automorphic vector bundles, has played important roles in the study of arithmetic properties of automorphic representations see [ 17 ] for an overview.
A fundamental question in such a study is to know in which degrees the cohomology groups are nonzero, or to rule out unnecessary complication by showing that all but some explicitly predictable degrees must be zero—this is the question of vanishing that we would like to address in this article. When the locally symmetric varieties in question are compact, and when the coherent cohomology in question contributes to the Hodge graded pieces of the de Rham cohomology of automorphic local systems, the cohomology classes can be represented by harmonic forms which are directly related to automorphic forms, and there are rather general vanishing results due to Faltings [ 10 ] and Vogan and Zuckerman [ 45 ].
One of the most useful results is that, when the weight of the local system in question is regular, the corresponding de Rham cohomology is concentrated in the middle degree, and there is a similar result for the coherent cohomology contributing to the Hodge graded pieces of such de Rham cohomology already in the compact case, there are coherent cohomology of automorphic bundles which might not contribute to any de Rham cohomology.
However, when the locally symmetric varieties in question are not necessarily compact, our understanding is much less complete. Unfortunately, the techniques in [ 12 ] have not yet been generalized to also cover the case of coherent cohomology of canonical or subcanonical extensions of automorphic bundles of noncohomological weights , in the sense that the corresponding cohomology groups do not contribute to the Hodge graded pieces of the de Rham cohomology of any automorphic local system the representations of such noncohomological weights are characterized by having dual representations with irregular Harish—Chandra parameters.
To the best of our knowledge, it is still not known whether the coherent cohomology classes of such noncohomological weights are always represented by Eisenstein series and their residues. In this regard, the study in [ 32 ] of coherent cohomology of toroidal compactifications of PEL-type Shimura varieties in mixed characteristics provides nontrivial and new vanishing results for the coherent cohomology even in characteristic zero.
In fact, the results such as [ 32 , Theorems 8. On the other hand, since the methods in [ 32 ] require the existence of good mixed characteristics models not only for the Shimura varieties and their toroidal compactifications as in [ 27 ] , but also for the geometric families of abelian schemes and their toroidal compactifications as in [ 26 ] involved in the method, they have serious limitations.
While we can imagine that the methods work very similarly for abelian-type Shimura varieties, we do not know how to extend them to more general cases. Note that there are Shimura varieties unrelated to exceptional groups which can still fail to be of abelian type—there are many such Shimura varieties, as explained in [ 35 ], associated with even orthogonal groups. Also, although we still know very little about Shimura varieties associated with exceptional groups, the theory feels incomplete and unsatisfactory if we cannot say anything about them. Fortunately, the recent work [ 44 ] allows us to extend the methods in [ 32 ] to arbitrary locally symmetric varieties considered in, e.
The key point is to replace the vanishing theorems in the first three sections of [ 32 ] which were based on techniques in positive characteristics developed in [ 8 , 9 , 20 , 22 ], and [ 37 ] with a rather general vanishing theorem for mixed Hodge modules in [ 44 ] which, however, is based on complex-analytic techniques in [ 38 ], which have no useful counterparts in positive characteristics yet. While it might seem unsurprising that new vanishing theorems for automorphic cohomology are available once some new vanishing theorem for mixed Hodge modules as in [ 44 ] is known, we have been quite happily surprised by what and how much we could readily deduce from the latter, thanks to some pleasant facts in the combinatorics of root systems.
Landau Institute for Theoretical Physics
Moreover, we have also obtained new vanishing results for coherent automorphic cohomology of low weights not contributing to the Hodge graded pieces of the de Rham cohomology of local systems of regular weights , and we have found efficient algorithms for determining the degrees of vanishing in practice, in all possible Hermitian cases.
Here is an outline of the article. In Sect.
We classify all such positive parallel weights, and give concrete descriptions of them in all cases. To help the reader understand our results, we also include some illustrative examples of low ranks.
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This article is written for people who would like to understand and use our vanishing results, and our judgement is that many of them will be number theorists or algebraic geometers rather than experienced representation theorists some of the choices of conventions and notations might not be so natural for representation theorists, but they are made because of historical or practical reasons related to the geometric constructions or their number-theoretic applications. Hence, while our arguments concerning roots and weights might be rather elementary and naive, we will still spell out most of the details, for the sake of clarity and readability.
But we do not consider such efforts as merely expository—they are helpful for presenting our algorithms for determining the degrees of vanishing in all circumstances. See, e. Such a construction is functorial, exact, and compatible with tensor products and duals. We shall abusively denote the associated sheaves of sections by the same symbols. This terminology is not ideal, but suffices in many naturally occurring cases such as representations of orthogonal groups. We extend these notions and notations to the corresponding coroots in the obvious ways. Therefore, by Lemma 2. This is an immediate consequence of Proposition 2.
Corollary 2. Since 3. Combine Lemmas 3. This can be easily checked in all cases by explicit calculations cf. Nevertheless, we shall present a more conceptual argument, which we learned from Zhiwei Yun. We learned from Xinwen Zhu that the assertion in Theorem 3. For our main results to be stated in Sect.
For this purpose, by Definition 3. Thanks to the classification of Hermitian symmetric domains see, e. The proof of Theorem 4. Theorem 4. Our proof of Theorem 4. In any case, the rather geometric proofs of Theorem 4.
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To the best of our knowledge, Theorem 4. It seems plausible that the methods there using geometry in good mixed characteristics can be extended to cover all abelian-type cases, although they have not been carried out yet as far as we know. By the same method as in [ 29 ], by 2 of Theorem 4. While Theorem 4.
The analogous statements are true if we replace all varieties and sheaves with their complex analytifications with sections represented by holomorphic functions. In particular, even in PEL-type cases, Theorem 4. It also follows from the more powerful results of [ 45 ], which also work for non-Hermitian locally symmetric spaces.
Nevertheless, our proof of Theorem 4.
To better understand Theorem 4. We will see similar phenomena in Examples 5. In this section, we record some explicit algorithms for determining the degrees of vanishing in Theorem 4. We shall adopt the notation system as in Sect. The methods for writing down such dual weights will be explained in Sect. Definition 2.
Suppose we are in the context of Sect. Let us continue with the setting of Sect. Such factors can be harmless omitted even in general cf. Even so, the coherent cohomology of automorphic bundles with noncohomological weights namely, not contributing to the de Rham cohomology might still be nonzero, although it is a subtle question which is unsolved in general whether this is indeed the case.
See also Example 5. The author would like to thank Zhiwei Yun and Xinwen Zhu for their helps on understanding the conceptual meaning behind Theorem 3. He would also like to thank Michael Harris for helpful discussions on the cohomology of Shimura varieties, and to thank Junecue Suh for sending him the preprint [ 44 ]. DMS, and by an Alfred P. Sloan Research Fellowship. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of these organizations.
Skip to main content Skip to sections. Advertisement Hide. Download PDF. Research in the Mathematical Sciences December , Cite as. Vanishing theorems for coherent automorphic cohomology. Open Access. First Online: 10 November Lemma 2. Proposition 2. Proof This is an immediate consequence of Proposition 2.
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