Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

We study the property of \emph{continuous Castelnuovo-Mumford regularity}, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in \cite{Kuronya:Mustopa:2020} by K\"{u}ronya and Mustopa. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety's endomorphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that we established in \cite{Grieve:R-R:abVars}. In a complementary direction, we explain how these topics pertain to the \emph{Index} and \emph{Generic Vanishing Theory} conditions for simple semihomogeneous vector bundles. In doing so, we refine results from \cite{Gulbrandsen:2008}, \cite{Grieve-cup-prod-ab-var} and \cite{Mum:Quad:Eqns}.


Introduction
Recall, that a coherent sheaf F on a projective variety X is m-regular with respect to a globally generated ample line bundle O X (1), if H i (X, F (m − i)) = 0, for all i > 0.
The concept of m-regularity was formulated by Mumford [17,Lecture 14]. It remains a fundamental cohomological invariant. For example, if F is m-regular and k > m, then H 0 (X, F (k)) is spanned by the image of the natural map H 0 (X, F (k − 1)) ⊗ H 0 (X, O X (1)) → H 0 (X, F (k)).
In particular, if F is m-regular and k m, then is a globally generated O X -module. The Castelnuovo-Mumford regularity of F , with respect to O X (1), is defined to be the smallest integer m for which F is m-regular. Effective criteria for m-regularity is a foundational problem within algebraic geometry. A starting point is the following theorem of Mumford [17, p. 101].
Theorem 1.1 ([17, p. 101]). For all nonnegative integers n, there exists a polynomial F n (x 0 , . . . , x n ) so that for all coherent sheaves of ideals I on P n , if a 0 , . . . , a n ∈ Z are defined by the condition that χ(I(m)) = n i=0 (−1)h i (P n , I(m)) = n i=0 a i m i , then I is F n (a 0 , . . . , a n )-regular with respect to the tautological line bundle O P n (1).
Computationally effective methods for calculation of the Castelnuovo-Mumford regularity, reg O P n (1) (F ), for coherent sheaves F on projective n-space P n , follow from work of Bayer and Stillman [3,Theorem 1.10]. We refer to [14,Section 1.8] for more details about Castelnuovo-Mumford regularity.
Turning to the context of Abelian varieties, and more generally irregular varieties, it was noted by Green and Lazarsfeld, in [5], building on work of Mumford, [17,Lecture 14], Mukai, [15] and [16], among others, that measures of cohomological positivity and criterion for global generation of coherent sheaves, with respect to ample line bundles, is achieved via the Generic Vanishing Theory.
A systematic development of that viewpoint, from the perspective of the Fourier-Mukai transform was initiated by Hacon [11]. It was developed further in a series of articles by Pareschi and Popa (including [20], [21] and [22]). As one example, the property of Mukai regularity, see Section 6, for sheaves on a given Abelian variety, was introduced in [20]. A main result is the M-regularity criterion [20, p. 285]. More recently, motivated by work of Barja, Pardini and Stoppino, [2], Küronya and Mustopa,in [13], formulated a concept of continuous Castelnuovo-Mumford regularity, denoted by reg cont (F , O X (1)), for coherent sheaves F on a given polarized irregular variety (X, O X (1)). Briefly, this is defined to be the smallest integer m for which the cohomological support loci V i (X, F (m − i)), for all i > 0, are proper Zariski closed subsets of Pic 0 (X), the identity component of the Picard group.
For the case of Abelian varieties, Küronya and Mustopa's main result, [13,Theorem A], implies that continuous Castelnuovo-Mumford regularity is a numerical property for semihomogeneous vector bundles. It builds on [6].
which has the property that for each semihomogeneous vector bundle E over A. (Here, we identify O(1) and det(E) with their classes in N 1 R (A) the real Néron-Severi space of A.) (ii) If E is a semihomogeneous vector bundle on A with the property that the class of det(E) is a rational multiple of the class of O(1), then reg cont (E, O(1)) is equal to the smallest integer m for which E(m − g) is a generic vanishing sheaf.
Our purpose here is to build on, and refine, these results from [13]. For instance, note that our formulation of Theorem 1.3, does not require global generation on the polarizing line bundle. Nor does it ask that the algebraically closed base field be of characteristic zero. Moreover, Theorem 1.6, below, makes explicit the manner in which the function (1.1) depends on both the Wedderburn decomposition and the isogney class of the given Abelian variety. A key point is [6,Corollary 4.2], which builds upon the index theorem of Mumford [18,Chapter 16] and [19,Appendix].
Indeed, in the present article, we apply the main result of [6] to show how Albert's Theorem and the Poincaré Reducibility Theorem, for a given Abelian variety, are reflected in these cohomological properties for semihomogeneous vector bundles. Our results here allow for an explicit determination of the property of continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles. It complements the numerical description from [13]. It is expressed in terms of certain normalized polynomials that are determined by the reduced norms of the Wedderburn components of the endomorphism algebra.
As some additional results, which are of an independent interest, we build on the works [6], [7] and [8], for example, which have origins in Mumford's index theorem for line bundles on Abelian varieties [18, p. 156]. In this regard, our main results are Theorem 6.2 and Corollary 1.5. Together, they improve upon [19,Appendix Theorem 2] and the main results from [8]. Indeed, they establish more general versions of those results which apply to simple semihomogeneous vector bundles. They also refine [ Recall, that the classical formulation of these results were given by Mumford [18], with subsequent refinements by Kempf and Ramanujam [19]. We describe precisely, in Section 4, the normalized polynomial (1.3) which arises in the statement of Theorem 1.4. As indicated there, this normalized polynomial, (1.3), reflects the structure of the Wedderburn decomposition of the Abelian variety.
In this article, if D ∈ NS Q (A) is a rational divisor class on a g-dimensional Abelian variety A, then we define its index i(D) to be the number of positive roots, counted with multiplicities, of any, and in fact all, of its Hilbert polynomials Here, λ is an ample divisor class on A and We refer to [18, p. 156] and [19,Appendix Theorem 2] for more details about the fact that the index i(D), as defined here, is well-defined (i.e., independent of the choice of polarization λ). In particular, to establish that the index of rational divisor classes is well-defined, one first reduces to the case of integral divisor classes.
Note that our concept of index here, differs, in general, from that which is defined in [ Moreover, recall that the index, as defined here, arises in the Riemann-Roch Theorem. This is the content of Theorem 1.4 which we deduce from [8].
is an Abelian variety with A i simple and pairwise nonisogenous Abelian varieties. Fix an ample divisor class λ on A. Put g := dim A. Suppose that f : B → A is an isogeny from a given Abelian variety B. Let pNrd λ (·) : End 0 λ (A) → Q so that the following two assertions hold true.
, under the homomorphism that is induced by f , then (ii) The index i(D) of D is equal to the number of positive roots, counted with multiplicity, of the polynomial In this article, we define the index i(E), for a semihomogeneous vector bundle E on A, to be the number of positive roots, counted with multiplicity, of any, and, by Lemma 5.1, all of its Hilbert polynomials. Again, this concept of index differs, in general, from the concept of index that is defined in [6, Section 2.1]. Theorem 1.4 has the following consequence for the index of simple semihomogeneous vector bundles.
Corollary 1.5. In the setting of Theorem 1.4, suppose that , for E a simple semihomogeneous vector bundle on B. Then, i(E), the index of E, is equal to the number of positive roots counted with multiplicity of the polynomial As an additional application of Theorem 1.4, here we use it to build on [13, Theorem A]. In doing so, we establish the following result.
is an Abelian variety with A i simple and pairwise nonisogenous Abelian varieties. Fix an ample divisor class λ on A. Let f : B → A be an isogeny and let pNrd λ (·) : End 0 λ (A) → Q be the normalized polynomial function that is given by Theorem 1.4 (see Equation (1.3)). Fix a rational divisor class, in NS Q (B), corresponding to or for which the polynomial Finally, let E be a semihomogeneous vector bundle on B and assume that the class is identified with γ. Then reg cont (E, f * λ), the continuous Castelnuovo-Mumford regularity of E with respect to the polarization f * λ, is equal to m.
We prove Theorem 1.6 in Section 7. It is established by first reducing to the case of simple semihomogeneous vector bundles. In Sections 5 and 6, we explain how these matters relate to the Index and Generic Vanishing Theorems for simple semihomogeneous vector bundles.
In doing so, we complement related results of Gulbrandsen, [10, Propositions 5.1 and 6.3], and Kempf, [19,Appendix]. In Section 2, we establish our notation for Abelian varieties and, in Section 4, we fix our notation for endomorphism algebras and the Riemann-Roch Theorem.
Our results here, together with those of our earlier work [8], indicate, in a precise way the manner in which the Wedderburn decomposition of a given Abelian variety is reflected in cohomological and global generation properties of (higher rank) semihomogeneous vector bundles. This picture expands upon earlier work of Mumford [18].
As one direction for future investigation, it remains an interesting problem to have an explicit knowledge of the normalized polynomials pNrd λ (·), which arise in the statement of Theorem 1.6, for a wide class of Abelian varieties. For that it is necessary to have a detailed understanding of the endomorphism algebra together with the Rosati involution.
To help place matters into perspective, note that already it is an interesting question to determine which integers can be realized as Picard numbers of Abelian varieties. We refer to the article [12], as one more recent work in that direction.

Notation for Abelian varieties
In what follows, we work over a fixed algebraically closed base field k. If A is an Abelian variety, thenÂ denotes the dual Abelian variety and P denotes the normalized Poincaré line bundle on A ×Â. If x ∈ A, then τ x : A → A denotes translation by x in the group law.

Ifx
∈Â, then Px denotes the translation invariant line bundle on A that is determined byx.
Multiplication in the group law is denoted as The projections of A × A onto the first and second factors, respectively, are denoted by p 1 and p 2 . On the other hand, the projections of A ×Â onto the first and second factors, respectively, are denoted by p A and pÂ.
The endomorphism algebra of A is denoted by End(A). We also put When no confusion is likely, at times we use the same notation to denote the class, in NS(A), of a divisor D on A. By a rational divisor class, we mean an element of NS Q (A).
Moreover, by the Euler characteristic of a rational divisor class We refer to [18] for more details about Abelian varieties. Much of the most basic theory is summarized in [8, Section 2].

Albert algebras
Recall, that an Albert algebra consists of a division algebra ∆, of finite dimension over Q, together with an involution ′ : ∆ → ∆, which we denote as α → α ′ , and which is positive in the sense that If A is a simple Abelian variety, then each ample divisor class λ induces a positive involution r λ on the division algebra End 0 (A). Especially, for each fixed polarization, the pair (End 0 (A), r λ ) is an Albert algebra. We recall the definition of the involution r λ in Section 4.
We refer to [8,Theorem 5.1], and the references therein, for a statement and more detailed discussion of Albert's theorem. Several examples of Albert's theorem, as it pertains to endomorphism algebras and Rosati involutions for isotypic Abelian varieties and products thereof, are given in [8, Sections 6 and 7].

Notation for endomorphism algebras
For later use, we fix notation for endomorphism algebras of Abelian varieties. We follow the conventions of [8] closely. First of all, consider the case of an Abelian variety A of the form A := A r 1 1 × · · · × A r k k , where each of the A i are simple and pairwise nonisogeneous Abelian varieties.
In terms of the endomorphism algebra R := End 0 (A), we may write R i := M r i (∆ i ). Let Z i be the centre of ∆ i and let be the reduced norm. Put Fix an ample divisor λ on A. Recall, that the Rosati involution Within the present context, as noted in [8,Corollary 3.7], the function is the square of a rational valued homogeneous polynomial function of degree g on End 0 λ (A). It is normalized so as to take value 1 on id A = 1 R and is denoted by pNrd λ (·).
In particular, if D is a divisor on A, then, as in [8, Theorem 4.1 (a)], the Riemann-Roch Theorem may be expressed as More generally, let

Semihomogeneous vector bundles and the index theorem
The concept of semihomogeneous vector bundle over a given Abelian variety A is due to Mukai [15], building on earlier work of Mumford [18] and Atiyah [1]. Specifically, a vector bundle E over A is called semihomogeneous if for all x ∈ A, The simple semihomogeneous vector bundles over A were characterized by Mukai [15, Theorem 5.8]. In particular, for a simple vector bundle E on A, the following four conditions are equivalent We refer to [7] for the theory of theta groups that are associated to simple semihomogeneous vector bundles and to [9], building on work of Brion [4], for the main results about homogeneous Severi-Brauer varieties.
For a simple vector bundle E over A, let K(E) be the maximal subscheme of A over which m * (E) and p * A (E) are isomorphic [15,Definition 3.8]. Moreover, recall that if E is a rank r semihomogeneous vector bundle over A, then Further, if E is a simple semihomogeneous vector bundle, then • dim K(E) = dim K(det(E)); and • if χ(E) = 0, then ord(K(E)) = χ(E) 2 ; see [15,Corollary 7.9].
In what follows, we say that a vector bundle E on A is nondegenerate if its Euler characteristic In Definition 5.2 below, we use Hilbert polynomials to define a concept of index for simple semihomogeneous vector bundles. The following lemma implies that this definition is indeed well-defined. We include a proof for completeness.   Moreover, is nontrivial for all x ∈ A. Hence, by [ For (ii), if L| K 0 (L) is trivial, then, as noted in [19, Appendix Theorem 1 (ii)], there exists a nondegenerate line bundle M on A/K 0 (L) and a pointx ∈Â, which has the two properties that Moreover, it is known, see [19, p. 100], that i(L) = i(M).
In particular, the above considerations imply that To begin with, there is a natural isomorphism Then, since f : A → B is flat and since the dual morphismf :B →Â is finite, it follows that for all integers ℓ, ). Since M is nondegenerate, it has a unique nonzero cohomology group H i(M) (B, M). The claim then follows.
Finally, we conclude this section by mentioning an observation from [13,Section 4], which pertains to the nature of semihomogeneous vector bundles, and their continuous Castelnouvo-Mumford regularity, on products of nonisogenous Abelian varieties with Picard number one.

Example 5.5. Consider the case that
is a product of nonisogenous Abelian varieties, each of which have Picard number equal to one. Then, since A 1 and A 2 are not isogeneous and each have Picard number one, it follows that A has Picard number two (see for instance [8,Proposition 6.2]). Moreover, the Néron-Severi group NS Q (A) is generated by the classes of p * A i λ i , for i = 1, 2, where λ 1 and λ 2 are ample line bundles on A 1 and A 2 , respectively. Now, suppose that E is a rank r semihomogeneous vector bundle on A with restrictions E 1 and E 2 to A 1 × {0} and {0} × A 2 , respectively. Then since E is a semihomogeneous vector bundle on A, it follows easily that E 1 ⊠ E 2 is a semihomogeneous vector bundle on A.

Generic vanishing theory
In this section, we recall the most basic concepts from the Generic Vanishing Theory. We mostly follow [20,Section 2]. Let F be a coherent sheaf over a g-dimensional Abelian variety A. On the other hand, if R i pÂ * (p * A F ⊗ P) = 0 for at most one i ∈ {0, . . . , g}, then F is called a Weak Index Theorem sheaf. If R j pÂ * (p * A F ⊗ P) = 0 for all but one j, then, here, the weak index of F is defined to be the unique integer . . , g}, then F is called an Index Theorem sheaf. Recall, that if an Index Theorem sheaf F is nonzero, then the cohomology and flat base change theorem, [18, p. 51], implies that it is a Weak Index Theorem sheaf.
The following example illustrates Proposition 5.4 and Theorem 6.2, in addition to the concepts of Weak Index Theorem and Index Theorem sheaves. It also provides an illustration of [19, Appendix Theorem 1 (i)].
Example 6.1. Let E be an elliptic curve and consider its self-product The preceding discussion, as it applies, in particular, to simple semihomogeneous vector bundles is summarized in Theorem 6.2 below. Its statement includes a key aspect to the proof of [13, Theorem A], for example, namely the fact that the weak index of a simple semihomogeneous vector bundle is the same as that of its determinant line bundle. Theorem 6.2 also allows Corollary 1.5 to be deduced from Theorem 1.4. (See Section 7.) Theorem 6.2. Let E be a simple semihomogeneous vector bundle on a g-dimensional Abelian variety A. The following assertions hold true.  Indeed, in light of Lemma 5.1, the proof of assertion (i) is complete upon verification that dim K(E) is the multiplicity of 0 as a root of χ(E(N)).
For that, let r be the rank of E. Then, observe, from [15,Corollary 7.9], that dim K(E) = dim K(det(E)).
Thus, the fact that dim K(E) is the multiplicity of 0, as a root of χ(E(N)), follows from the relation that see [15,Proposition 6.12], combined with the fact that dim K(det(E) ⊗r ) is the multiplicity of 0, as a root of the polynomial As explained in the discussion that proceeds [ and so E is a Weak Index Theorem sheaf and both E and L have the same weak index.
Finally, with respect to the dual isogeny,f eachŷ ∈B is a pullback for somex ∈Â. Thus, by the projection formula The above discussion implies that L is an Index Theorem sheaf if and only if E is an Index Theorem sheaf. But, if E is nondegenerate, then so is det(E). Thus Thus, L is nondegenerate too. Thus L is an Index Theorem sheaf. It follows that E is an Index Theorem sheaf too. To finish the proof of Theorem 1.6, it thus suffices to treat the case that E is a simple semihomogeneous vector bundle.
To that end, let E be a simple semihomogeneous vector bundle on B with the property that the class det(E) rank(E) ∈ NS Q (B) is identified with γ.
By Theorem 1.3 (i), reg cont (E, f * λ) = ρ η (γ) , where ρ η (γ) is the minimum integer m ∈ Z for which either the class Let α ∈ End 0 λ (A) be the image of γ ∈ End 0 f * λ (B) under the natural homomorphism that is induced by f . Then, by Theorem 1.4, these above conditions on the class (7.2) translate into the assertion that either Indeed, that the class (7.2) is degenerate means that it has zero Euler characteristic Similarly, the condition that the class (7.2) fails to have index i, for all i ∈ {1, . . . , g}, means that the polynomial (7.4) fails to have i positive roots (counted with multiplicities) for all i ∈ {1, . . . , g}.