Continuous CM-regularity and generic vanishing: With an appendix by Atsushi Ito (Okayama University)

Debaditya Raychaudhury

doi:10.1515/advgeom-2023-0028

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Continuous CM-regularity and generic vanishing

With an appendix by Atsushi Ito (Okayama University)

Debaditya Raychaudhury

Published/Copyright: January 24, 2024

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From the journal Advances in Geometry Volume 24 Issue 1

Abstract

We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, O_X(1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV.

Here we answer the question in the affirmative for many pairs (X, O_X(1)) which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if F is continuously k-regular for some positive integer k ≤ dim X, then F is a GV_−(k−1) sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the ℚ-twisted bundles on polarized abelian varieties (X, O_X(1)), and we show that this function can be extended to a continuous function on N¹(X)_ℝ. We also provide syzygetic consequences of our results for O_ℙ(E)(1) on ℙ(ɛ) associated to a 0-regular bundle ɛ on polarized abelian varieties.

In particular, we show that O_ℙ(E)(1) satisfies the N_p property if the base-point freeness threshold of the class of O_X(1) in N¹(X) is less than 1/(p + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito.

Keywords: Vector bundles; Castelnuovo–Mumford regularity; generic vanishing; syzygies

MSC 2010: Primary 14F06; 14F17; Secondary 13D02

1 Introduction

Given a smooth projective variety X ⊆ ℙⁿ, it is well-known that the geometry of the embedding is reflected by the coherent sheaves on X with suitable positivity properties. One of the most fundamental notion of such properties that governs the complexity of a sheaf F is given by its Castelnuovo–Mumford (CM) regularity with respect to the pair (X, O_X(1)).

In this article we study a variant of CM-regularity that was introduced by Mustopa in [29] for polarized irregular varieties (X, O_X(1)). It is defined as follows: F is continuously k-regular if the cohomological support loci Vⁱ(F(k − i)) are not empty for i ≥ 1 (strictly speaking, this definition is slightly more general than that in [29] as we are not assuming global generation of O_X(1)). The structures of these cohomological support loci are of great importance in the study of the geometry of irregular varieties. An important notion in this topic is the notion of generic vanishing (GV for short) introduced by Green and Lazarsfeld in the pioneering works [9] and [10]. Fundamental contributions from Hacon [15] and Pareschi–Popa [34], [35], [36] through the derived category approach and Fourier–Mukai functors led to subsequent developments of the theory of generic vanishing.

Turning to details, a coherent sheaf F on X is said to be GV if codim Vⁱ(F) ≥ i for all i. This property is intimately related to the positivity of F; in particular, a GV sheaf on an abelian variety is nef. In [29], Mustopa asked the following question.

Question 1.1

([29, Question (∗)]). Let X be a smooth projective variety of dimension d ≥ 1 and let O_X(1) be an ample and globally generated line bundle on X. Let F be a torsion-free coherent sheaf on X. If F is continuously 1-regular for (X, O_X(1)), is F a GV sheaf?

The question above was motivated by Beauville’s construction in [5] of rank 2 Ulrich bundles on abelian surfaces (X, O_X(1)). It turns out that for these bundles ɛ, ɛ(−1) is indeed GV. It is easy to see that the answer to Question 1.1 is affirmative for polarized curves. It was shown in [29] that the answer to the question is also affirmative for

a large class of polarized surfaces that includes the case of any polarized abelian surface (loc. cit., Theorem B and Corollary C),
certain polarizations on Cartesian and symmetric products of curves (loc. cit., Propositions 3.1 and 3.2),
some scrollar embeddings of ruled threefolds over a curve (loc. cit., Proposition 4.1).

Continuous CM-regularity for semihomogeneous bundles on abelian varieties has been studied by Küronya and Mustopa [21] and later by Grieve [14]. In particular, Küronya and Mustopa [21] show the following: if ɛ is a semihomogeneous bundle on abelian variety (X, O_X(1)) of dimension g as in the set-up of Question 1.1, and if moreover c₁(ɛ) is a rational multiple of c₁(O_X(1)), then even more is true, namely ɛ(1 − g) is GV. In [14], Grieve established a description of continuous CM-regularity of semihomogeneous bundles on abelian varieties. This description was in terms of a normalized polynomial function studied in [13], and obtained via the Wedderburn decomposition of the endomorphism algebra of the abelian variety. A further study of the index and generic vanishing theory of simple semihomogeneous bundles was also carried out in [14], building on and refining [12] and [11].

Besides the notion of GV sheaves, a related notion in the generic vanishing theory is that of M-regularity of coherent sheaves. A coherent sheaf F on X is said to be M-regular if codim Vⁱ(F) > i for all i > 0. In this direction, Mustopa asked whether a continuous 0-regular torsion-free coherent sheaf on a polarized smooth projective variety (X, O_X(1)) with O_X(1) globally generated is M-regular; see [29, Remark 1.6].

Following this train of thought, it is natural to propose the more general question: given a continuously k-regular torsion-free coherent sheaf on a smooth polarized variety (X, O_X(1)) with O_X(1) globally generated and 1 ≤ k ≤ dim X, is it true that codimVⁱ(F) ≥ i − k + 1 for all i? Here we remark that the notion of generic vanishing was generalized in [36] where Pareschi–Popa define a sheaf F to be GV_−k for an integer k ≥ 0, if codim(Vⁱ(F)) ≥ i − k for all i. In view of this, here we devote ourselves to answering the question below, which is more general than [29, Question (∗) and Remark 1.6].

Question 1.2

Let X be a smooth projective variety of dimension d ≥ 1 and let O_X(1) be an ample and globally generated line bundle on X. Let F be a torsion-free coherent sheaf on X. Assume that F is continuously k-regular for (X, O_X(1)) for some integer k with 0 ≤ k ≤ d.

If 1 ≤ k ≤ d, is F a GV_−(k−1)sheaf?
If k = 0, is F an M-regular sheaf?

The following is the main result of this article that answers the above question in the affirmative for many pairs (X, O_X(1)). We also note that the result below does not require the hypothesis that the polarization O_X(1) is globally generated.

Theorem A. Let (X, H) be a polarized smooth projective variety. Assume that there exist a globally generated line bundle H₁on X and an ample line bundle H₂on Alb(X) such thatH=H1+albX∗H2,where alb_X : X → Alb(X) is the Albanese map. Let F be a torsion-free coherent sheaf on X that is continuously k-regular for (X, H) for some integer k with 0 ≤ k ≤ dim X. Then the following statements hold.

Vⁱ(F) = 0 for i ≥ k + 1.
If k ≠ 0, then codim(V^k(F)) ≥ 1.

In particular, the answer to Question 1.2 is affirmative for the pair (X, H).

Note that the above result answers Question 1.2 in the affirmative for any polarized abelian variety. Moreover, it also shows that the question has an affirmative answer in many cases that include for example the case (a): X = Y × A where Y is a regular smooth projective variety, A is an abelian variety, and O_X(1) is ample and globally generated; and the case (b): X is a projective bundle ℙ(ɛ) on an abelian variety A associated to an ample and globally generated vector bundle ɛ, and O_X(1) = T + F where T is the tautological bundle, and F is the pull-back of any ample line bundle on A. Observe also that the above theorem proves a stronger statement than what is asked for in Question 1.2 (2) when (X, H) satisfies the hypotheses of Theorem A: it shows that continuously 0-regular sheaves are in fact IT₀ (see Definition 3.3 and Corollary 5.4). We will give other variants of Theorem A in Subsection 3.3.

The proof of Theorem A is homological in nature and relies on an inductive argument, but the crucial ingredient of the proof is a relative set-up of continuous CM-regularity that we develop in this article.

Inspired by the recent development of the cohomological rank function by Jiang–Pareschi [20], we further extend the notion of continuous CM-regularity to define a real-valued regularity Q−regl_(E〈n_〉)for ℚ-twisted bundles E〈n_〉where l_is an ample class in N¹(X) on an abelian variety X and n_∈N1(X)Q.For these, we prove the following

Theorem B. Let X be an abelian variety of dimension g, and let ɛ be a vector bundle on X. Ifl_∈N1(X)is a polarization, then the functionQ−regl_(E〈−〉):N1(X)Q→Rsendingn_∈N1(X)Q to Q−regl_(E〈n_〉)can be extended to a continuous functionR−regl_(E〈−〉):N1(X)R→R.

We now mention the immediate consequences of Theorem A for continuously k-regular torsion-free coherent sheaves F on polarized abelian varieties (X, O_X(1)). It follows immediately that

(Corollary 5.1) if k = 1, then χ(F) ≥ 0 with equality if and only if V⁰(F) is a divisor,
(Corollary 5.4) if k = 1 then F is nef, and if k = 0 then it is ample.

Let us denote the (usual) CM-regularity by regOX(1)(−).In Corollary 5.6 we establish the sub-additivity of CM-regularity for polarized abelian varieties. To be more precise, for any torsion-free coherent sheaves ɛ and F on a polarized abelian variety (X, O_X(1)) such that at least one of ɛ and F is locally free, we show that the following inequality holds:

(1.1) reg O X ( 1 ) ( E ⊗ F ) ≤ reg O X ( 1 ) ( E ) + reg O X ( 1 ) ( F ) .

We remark that it was shown earlier by Totaro [40, Theorem 3.4] that (1.1) holds for arbitrary polarized smooth projective varieties when O_X(1) is very ample and satisfies a certain Koszul hypothesis, see also [8, Theorem 1.1] for the case X = ℙ^d.

We also deduce syzygetic consequences from Theorem A for projective bundles on abelian varieties. Before stating our result, we spend a few words on linear series on abelian varieties to set the context.

It was a conjecture of Lazarsfeld that on a polarized abelian variety (X, H), tH satisfies the N_p property if t ≥ p+3. Lazarsfeld’s conjecture was proven by Pareschi [32]. The result was further extended by Pareschi–Popa [35] where it was shown that tH satisfies the N_p property if t ≥ p + 2 and |H| contains no base-divisor. Further, denote the ideal sheaf at the origin of X by I₀ and define

r ( H ) := inf c ∈ Q ∣ there exists an effective Q -divisor D ≡ c H such that J ( X , D ) = J 0

where J(X, D) is the multiplier ideal. Lazarsfeld–Pareschi–Popa proved in [24] that if r(H) < 1/(p + 2) then H satisfies the N_p property. Very recently, Jiang–Pareschi defined in [20] the base-point freeness thresholdβ(h_)with 0<β(h_)≤1for a polarization h_∈N1(X)on an abelian variety X; it can be characterized as follows: β(h_)<x⟺J0〈xh_〉is IT₀ for x ∈ ℚ. It was shown by Caucci [6] that β(h_)≤r(H),and further the above results are unified to shown that H satisfies the N_p property if β(h_)<1/(p+2).The study of a more general property Nprvia base-point freeness threshold was carried out by Ito [17]. Sharp results on projective normality and higher syzygies of general polarized abelian varieties were also established by Ito [18] and [19].

The following is our result on syzygies of projective bundles associated to continuously 0-regular vector bundles on polarized abelian varieties (X, H), which is an immediate consequence of Theorem A and Theorem A.1 in the Appendix A due to Atsushi Ito.

Corollary C. Let (X, H) be a polarized abelian variety, and let ɛ be a continuously 0-regular vector bundle for (X, H). Then O_ℙ(E)(1) satisfies the N_p property ifβ(h_)<1/(p+2)whereh_is the class of H in N¹(X).

Organization of the paper. In Section 2, we recall preliminaries of (continuous) CM-regularity. In Section 3 we first discuss the theory of generic vanishing, and then we proceed to prove Theorem A; at the end of this section, we prove a few variants of Theorem A. We define the ℚ CM-regularity in Section 4 and prove Theorem B. Finally, Section 5 is devoted to the proofs of Corollaries 5.1, 5.4, 5.6 and Corollary C.

Conventions. We work over the field of complex numbers ℂ. We tacitly assume that the varieties are irregular and the morphisms are non-constant. We use the additive and multiplicative notation interchangeably for tensor products of line bundles, and the sign “≡” is used for numerical equivalence. The rest of the notation is standard in algebraic geometry.

Acknowledgements: It is my great pleasure to acknowledge my gratitude to Professor Purnaprajna Bangere, and Professor Angelo Felice Lopez for their constant encouragement and support. In particular, I am extremely grateful to Professor Angelo Felice Lopez for suggesting me to think about projective normality of Ulrich bundles, which is how this project got started. I am indebted to Professor Atsushi Ito for generously sharing his ideas on syzygies of projective bundles on abelian varieties, and for kindly agreeing to write the Appendix A of this article. I sincerely thank Professors Nathan Grieve, V. Kumar Murty, Robert Lazarsfeld, Yusuf Mustopa, and Mihnea Popa for their valuable comments and suggestions on an earlier draft. I also benefited from conversations with Jayan Mukherjee for which I thank him. I am grateful to the anonymous referee for a careful reading of the manuscript and for suggesting several improvements. During the preparation of this work, I was supported by a Simons Postdoctoral Fellowship from the Fields Institute for Research in Mathematical Sciences.

2 Continuous CM-regularity of coherent sheaves

2.1 Definition and first properties

This subsection is devoted to the definitions and basic properties of (continuous) CM-regularity. We start with the definition of a partial variant of the usual CM-regularity; cf. [22, Definition 1.8.4] and [40, Lemma 3.2].

Definition 2.1

(CM-regularity). Let X be a smooth projective variety and let H be a line bundle on X. Also, let q, k be integers with q ≥ 0. A coherent sheaf F on X is called C_q_,k for (X, H) if H^q⁺ⁱ(F((k − i)H)) = 0 for all integers i ≥ 1. When H is ample, we say that F is k-regular for (X, H) for k ∈ ℤ if it is C_0,k for (X, H).

The following important result in the study of partial regularity was proven by Totaro [40], and is well-known in the set-up of the usual (i.e., non-partial) Castelnuovo–Mumford regularity.

Lemma 2.2

([40, Lemma 3.2]). Let (X, H) be a smooth projective variety with H globally generated. Let F be a coherent sheaf on X, and q, k be integers with q ≥ 0. If F is C_q_,kfor (X, H) then it is C_q_,k+mfor (X, H) for any integer m ≥ 0.

When H is ample and globally generated, define the regularity of F as

reg H ( F ) := min { m ∈ Z ∣ F is m -regular for ( X , H ) } .

We now recall the definition of the cohomological support loci, cf. [33, Definition 1.2], that are of fundamental importance in the study of irregular varieties.

Definition 2.3

(Cohomological support loci). Let X be a smooth projective variety and let a : X → A be a morphism to an abelian variety A. Let F be a coherent sheaf on X. The i-th cohomological support locusVai(F)with respect to a for i ∈ ℕ is defined as

V a i ( F ) := ζ ∈ Pic 0 ( A ) ∣ h i F ⊗ a ∗ ζ ≠ 0 .

We simply write Vⁱ(F) for ValbXi(F)where alb_X : X → Alb(X) is the Albanese morphism of X.

As we discussed earlier, continuous CM-regularity is a slightly coarser measure of positivity than CM-regularity, and was introduced by Mustopa [29]. For our purpose, we need to generalize the definition of continuous CM-regularity, cf. [29, Definition 1.1], to a relative and partial set-up that we describe next.

Definition 2.4

(Continuous CM-regularity). Let X be a smooth projective variety and let H be a line bundle on X. Further, let a : X → A be a morphism to an abelian variety A. Let F be a coherent sheaf on X and let q, k be integers with q ≥ 0. The sheaf F is called Cq,k′afor (X, H) if Vaq+i(F((k−i)H))≠Pic0(A)for all integers i ≥ 1. When H is ample, we say that F is continuously k-regular for (X, H) if it is c0,k′albXfor (X, H).

When H is ample and globally generated, we define the continuous regularity of F as

reg H cont ( F ) := min m ∈ Z ∣ ∀ i > 0 : V i ( F ( ( m − i ) H ) ) ≠ Pic 0 ( Alb ( X ) ) ≅ Pic 0 ( X ) .

In general, we have the inequality regHcont(F)≤regH(F).However, strict inequalities are possible:

Example 2.5

Let H be an ample and globally generated line bundle on an abelian variety X of dimension g. Then g=regHcontOX<regHOX=g+1.

We will use the following fact without any further reference.

Remark 2.6

Let a : X → A be a morphism from a smooth projective variety X to an abelian variety A. Let F be a coherent sheaf and H be a line bundle on X. It is a consequence of Proposition 2.1 (respectively 2.4) that if F is C_q_,k (respectively Cq,k′afor (X, H) for integers q, k ≥ 0, then it is also C_q_+k,0 (respectively Cq+k,0′a.

We will see that in practice, it is often useful to work with relative continuous CM-regularity rather than CM-regularity. The following observation (where we use semicontinuity to see that the first three equivalent conditions imply the fourth) highlights this and shows that the former property is stable under perturbations by elements of a^∗ Pic⁰(A).

Observation 2.7

(See also [29, Lemma 1.2]). Let X be a smooth projective variety and let H be a line bundle on X. Let a : X → A be a morphism to an abelian variety A, and let F be a coherent sheaf on X. Also, let q, k be integers with q ≥ 0. The following conditions are equivalent:

F is C q , k ′ a for ( X , H ) ,
F ⊗ a ∗ ζ is C q , k ′ a for ( X , H ) for some (equivalently, for all) ζ ∈ Pic⁰(A),
F is C q , k ′ a for X , H + a ∗ ζ for some (equivalently, for all) ζ ∈ Pic⁰(A),
F ⊗ a ∗ ζ is C q , k for ( X , H ) for some (equivalently, for general) ζ ∈ Pic⁰(A).

Thus, the (partial) continuous CM-regularity is determined by the class of the line bundle H in the Néron–Severi group Pic(X)/Pic⁰(X).

2.2 Behavior of continuous CM-regularity

In this subsection, we study the behavior of continuous CM-regularity of torsion-free coherent sheaves with respect to restriction and pull-back. We introduce the property (P_a) that is crucial for us since it allows us to produce smooth sections in the appropriate linear series by Bertini’s theorem.

Definition 2.8

(Property (Pa)). We say that a polarized smooth projective variety (X, H) satisfies (P_a) where a : X → A is a morphism to an abelian variety if for all ζ ∈ Pic⁰(A), H + a^∗ζ is globally generated.

The above property is desirable for various geometric reasons aside from the one we stated before; we point out another such reason here. A useful notion for sheaves on irregular varieties is their continuous global generation property that we will define in Definition 3.6. It follows from [7, Proposition 3.1] that if H satisfies Property PalbX,then H is continuously globally generated.

We now make the following

Observation 2.9

Let (X, H) be a polarized smooth projective variety and let a : X → A be a morphism to an abelian variety. Assume that (X, H) satisfies (P_a). Then for any ζ ∈ Pic⁰(A) and any smooth and irreducible member Y ∈ |H + a^∗ζ|, the pair (Y, H |_Y ) satisfiesPaYwhere a|_Y : Y → A is the restriction of a.

It is important for us to note that continuous CM-regularity satisfies better restriction properties than ordinary CM-regularity. We highlight this by means of an example which shows that in general, if ɛ k-regular for (X, H), ɛ|_Y need not be k-regular for (Y, H|_Y ) if Y ∈ |H + ξ| where ξ is non-trivial in Pic⁰(X).

Example 2.10

Let X be an abelian surface, and let H be an ample and globally generated line bundle. Fix a line bundle 0 ≠ ξ ∈ Pic⁰(X) and observe that H + ξ is globally generated (this fact has been pointed out to me by the referee, whom I thank). Indeed, consider the isogeny φH:X→Xˆ=Pic0(X)that sends x∈X to tx∗H⊗H⊗−1where t_x : X → X is translation by x. Since φ_H is an isogeny, in particular surjective, there exists x ∈ X such that H+ξ=tX∗Hwhence the global generation of H +ξ follows. Thus, H +ξ is ample and globally generated, and consequently there exists a smooth curve Y ∈ |H + ξ|. It is evident that 2H + ξ is 0-regular for (X, H). However, we claim that (2H + ξ)|_Y is not 0-regular for (Y, H|_Y ). To see this, observe that (H + ξ)|_Y = K_Y by adjunction, whence h¹((H + ξ)|_Y ) ≠ 0 by Serre duality.

However, for the continuous variant of CM-regularity, we have

Lemma 2.11

Let (X, H) be a polarized smooth projective variety and let a : X → A be a morphism to an abelian variety A. Assume that (X, H) satisfies (P_a). Let F be a torsion-free coherent sheaf that isCq,k′afor (X, H) where q, k are integers with q ≥ 0. Then for any ζ ∈ Pic⁰(A) and any smooth and irreducible member Y ∈ |H + a^∗ζ|, there exists ζ^' ∈ Pic⁰(A) such that F ⊗ a^∗ζ^'|_Y is C_q_,kfor (Y, H|_Y ).

Proof. It is safe to assume that k = 0. Consider the following part of the long exact sequence

H q + i F ( − i H ) ⊗ a ∗ ζ ′ → H q + i F ( − i H ) Y ⊗ a ∗ ζ ′ → H q + i + 1 F − ( i + 1 ) H − a ∗ ζ + a ∗ ζ ′

obtained from twisting the restriction sequence by general ζ^' ∈ Pic⁰(A) and passing to cohomology. The statement now follows from semicontinuity. □

We now recall the definition of a strongly generating morphism as presented in the introduction of [3].

Definition 2.12

(Strongly generating morphisms). Let a : X → A be a morphism from a smooth projective variety X to an abelian variety A. We call the morphisma strongly generating if the induced map a∗:Aˆ=Pic0(A)→Pic0(X)is injective.

Inspired by [31], given a morphism a : X → A as above, we will work with the covering trick, i.e. we will consider the following base-change diagram

(2.1)

where μ : Ã → A is an isogeny of abelian varieties. It turns out that if a is strongly generating, then X ̃ is smooth and irreducible; see the proof of [2, Lemma 2.3]. This is precisely the reason why we consider strongly generating morphisms.

Most of the time throughout the article, we will only consider the case when A ̃ = A and μ = n_A is the multiplication by n isogeny for an integer n ≥ 1, and in this case we denote X ̃ by X_n, a ̃ by a_n, and μ̃ by μ_n. Moreover, in this case, if a is strongly generating then so is a_n.

We show that continuous CM-regularity behaves well with respect to the above covering trick.

Lemma 2.13

Let X be a smooth projective variety and let H be a line bundle on X. Also, let a : X → A be a morphism to an abelian variety A that is strongly generating. Let F be a coherent sheaf on X that isC0,k′afor (X, H ). Let μ : Ã→ A be an isogeny and consider the base-change diagram (2.1). Then μ̃^∗F isC0,k′a˜for (X̃ , H ̃ := μ̃^∗H).^∗

Proof. Without loss of generality, we may assume k = 0. For ζ ∈ Â , we have by the projection formula

H i μ ˜ ∗ F − i H + a ∗ ζ = H i F − i H + a ∗ ζ ⊕ ⨁ k = 1 d − 1 H i F − i H + a ∗ ζ + a ∗ ζ k

where d:=deg(μ),μ∗OA˜=OA⊕⨁k=1d−1ζkand ζ_k ∈ Â for 1 ≤ k ≤ d − 1. The conclusion now follows from semicontinuity and the commutativity of the diagram (2.1). □

3 Continuous CM-regularity and generic vanishing

3.1 The theory of generic vanishing

Throughout this subsection, for an abelian variety A, we denote by Â the dual abelian variety identified with Pic⁰(A).

3.1.1

Let X be a smooth projective variety and let alb_X : X → A := Alb(X) be the Albanese map. Consider Â = Pic⁰(A) ≅ Pic⁰(X) and let P be a Poincaré line bundle on A × Â . Let P := (alb_X × id_A_̂ )^∗P. Let D(X) and D(Â ) be the bounded derived categories of Coh(X) and Coh(Â ). In this situation, we have the following two Fourier–Mukai transform functors

R Φ P : D ( X ) → D ( A ˆ ) , R Φ P ( − ) := R p A ˆ ∗ p X ∗ ( − ) ⊗ P , R Ψ P : D ( A ˆ ) → D ( X ) , R Ψ P ( − ) := R p X ∗ p A ˆ ∗ ( − ) ⊗ P .

A reference for the following definitions can be found for example in [37, Proposition/Definition 2.1 and Proposition/Definition 2.7].

Definition 3.1

(Generic vanishing and M-regularity). Let X be a smooth projective variety and let F be a coherent sheaf on X.

F is called generic vanishing, abbreviated as GV, if codim(Vⁱ(F)) ≥ i for all integers i > 0. More generally, for an integer k ≥ 0, F is called GV_−k if codim(Vⁱ(F)) ≥ i − k for all integers i.
F is called Mukai regular, abbreviated as M-regular, if codim(Vⁱ(F)) > i for all i > 0.

Evidently, GV = GV₀. The following fundamental theorem is due to Hacon [15] and Pareschi and Popa [37].

Theorem 3.2

([36, Theorem 3.7 and Corollary 3.11]). Let X be a smooth projective variety with dim Alb(X) = g. Let F be a coherent sheaf on X, and let k ≥ 0 be an integer. Then the following are equivalent:

F is GV_−k,
codim Supp(RⁱΦ_PF) ≥ i − k for all integers i,
for any sufficiently positive ample line bundle L on Â , H ⁱ(F ⊗ RΨ_P_[g]L^⊗−1) = 0 for all integers i > k.

3.1.2

Let X be an abelian variety of dimension g. We recall the notions of M–regularity and Index Theorems with prescribed indices; cf. [34, the end of Section 1 on p. 5].

Definition 3.3

(IT sheaves). Let X be an abelian variety and let F be a coherent sheaf on X. The sheaf F is said to satisfy the Index Theorem with index k for some k ∈ ℤ, abbreviated as IT_k, if Vⁱ(F) = 0 for all i ≠ k.

It is clear that on an abelian variety X, a coherent sheaf F satisfies IT₀ ⇒ F is M-regular ⇒ F is GV. Also, note that an ample line bundle on an abelian variety X satisfies IT₀. In the abelian case, we denote the Fourier–Mukai transform functors by

R S ˆ : D ( X ) → D ( X ˆ ) , R S : D ( X ˆ ) → D ( X ) .

A fundamental result of Mukai, see [27, Theorem 2.2], shows that RŜ : D(X) → D(X̂ ) is an equivalence of derived categories, and we have the following inversion formulae:

(3.1) R S ˆ ∘ R S = ( − 1 ) X ˆ ∗ [ − g ] and R S ∘ R S ˆ = ( − 1 ) X ∗ [ − g ]

where (−1)_X_̂ and (−1)_X are multiplications by (−1) on X̂ and X, respectively.

It turns out that if F on X is IT_k for some k ∈ ℤ, then RŜF = R^kŜF[−k] and R^kŜF is a locally free sheaf. In particular, if L is an ample line bundle on X̂ , then RSL^⊗−1 = R^gSL^⊗−1[−g].

3.1.3

A result of Pareschi–Popa on preservation of vanishing says that on an abelian variety, a tensor product of a GV sheaf F and an IT₀ sheaf G is IT₀ if one of F and G is locally free; see [37, Proposition 3.1]. A variation of their proof yields the following statement, which is more suited for our purposes.

Proposition 3.4

Let X be an abelian variety of dimension g. Let F and G be coherent sheaves on X with one of them locally free. Assume that G is IT₀and m ≥ 0 is an integer. If one of the following holds:

either Vⁱ(F) = 0 for all integers i ≥ m + 1, or
F is GV_−m,

then Vⁱ(F ⊗ G) = 0 for all integers i ≥ m + 1.

Proof. The proof is identical to that of [37, Proposition 3.1]. Let ζ ∈ Pic⁰(X); we aim to show that the group Hⁱ(F ⊗ G ⊗ ζ) = 0 for all i ≥ m + 1. As G ⊗ ζ is also IT₀, we have that RŜ(G ⊗ ζ) = R⁰Ŝ(G ⊗ ζ) =: N_ζ is locally free. By Mukai’s inversion formulae (3.1), we have G⊗ζ=RS(−1)Xˆ∗Nζ[g].Thus, we deduce that H ⁱ(X, F ⊗ G ⊗ ζ) is isomorphic to

(3.2) H i X , F ⊗ R S ( − 1 ) X ˆ ∗ N ζ [ g ] ≅ H i X ˆ , R S ˆ F ⊗ _ ( − 1 ) X ˆ ∗ N ζ [ g ] ≅ H g + i X ˆ , R S ˆ F ⊗ _ ( − 1 ) X ˆ ∗ N ζ

where the first isomorphism in (3.2) is obtained by an exchange formula of Pareschi–Popa; see [36, Lemma 2.1]. To this end, consider the following spectral sequence

E 2 j k := H j X ˆ , R k S ˆ F ⊗ ( − 1 ) X ˆ ∗ N ζ ⟹ H j + k X ˆ , R S ˆ F ⊗ _ ( − 1 ) X ˆ ∗ N ζ .

Observe that the E2jkterm above vanishes if j+k ≥ g+m+1. Indeed, if (1) holds, then R^kŜF = 0 if k ≥ m +1, and if (2) holds, then dim Supp(R^kŜF) ≤ g −k+m. Thus, E∞jk=0 if j+k≥g+m+1,whence Hj+kXˆ,RSˆF⊗(−1)Xˆ∗Nζ=0in the same range. □

We include a consequence of the above proposition which we will not use anywhere in the sequel.

Corollary 3.5

Let F and G be coherent sheaves on an abelian variety X with one of them locally free. If F is GV_−kand G is GV for some integer k ≥ 0, then F ⊗ G is GV_−k.

Proof. This is identical to [37, Theorem 3.2]. Let L be a sufficiently positive ample line bundle on X̂ . Since G is GV, G ⊗ R^gSL^⊗−1 is IT₀ by Theorem 3.2. Thus, by Proposition 3.4, Hⁱ(F ⊗ G ⊗ R^gSL^⊗−1) = 0 for i ≥ k + 1 whence the assertion follows from Theorem 3.2. □

3.1.4

One of the most essential and useful tools in the study of irregular varieties is the notion of continuous global generation that we define next; cf. [37, Definition 5.2].

Definition 3.6

(Continuous global generation). Let X be an irregular variety. A sheaf F on X is continuously globally generated if for any non-empty open subset U ⊂ Pic⁰(X), the following sum of evaluation maps is surjective:

⨁ ζ ∈ U H 0 ( F ⊗ ζ ) ⊗ ζ ∗ → F .

It was shown by Pareschi–Popa [37, Corollary 5.3] that an M-regular sheaf is continuously globally generated. Another result of them asserts that if F is a continuously globally generated sheaf and L is a continuously globally generated line bundle on X, then F ⊗ L is globally generated; see [34, Proposition 2.12]. Thus, if H₁ and H₂ are two ample line bundles on an abelian variety X, then H₁ + H₂ is globally generated. It has been pointed out by the referee that alternatively, one could also use [4, Theorem 1.1] to get the above statement.

3.2 Proof of Theorem A

We now prove Theorem A stated in the introduction.

Proposition 3.7

Let (X, H) be a polarized smooth projective variety and let a : X → A be a morphism to an abelian variety A. Assume that (X, H) satisfies (P_a) and let F be a torsion-free coherent sheaf on X that isCq−1,0′afor (X, H) for some positive integer q ≤ dim X. ThenVai(F)=∅for all integers i ≥ q.

Proof. The proof is based on induction on dim X =: n and we divide the proof into two steps. The following first step verifies the base case which is when n = 1:

Step 1. Here we prove the statement for i = n. By hypothesis, we know that Hⁿ(F(−(n−q+1)H)⊗a^∗ζ) = 0 for some ζ ∈ Pic⁰(A). Since (X, H) satisfies (P_a), for any given ζ^' ∈ Pic⁰(A) we can choose a smooth and irreducible member Y ∈ (n − q + 1)H + a^∗(ζ^' − ζ). Consider the restriction exact sequence

0 → F ( − ( n − q + 1 ) H ) ⊗ a ∗ ζ → F ⊗ a ∗ ζ ′ → F ⊗ a ∗ ζ ′ Y → 0.

Passing to cohomology, we obtain the desired vanishing Hⁿ(F ⊗ a^∗ζ^') = 0.

Step 2. Now we prove the statement by induction and thanks to the previous step, we assume n ≥ 2. Also, because of Step 1, we assume that 1 ≤ q ≤ i ≤ n − 1. Let ζ ∈ Pic⁰(A). We want to show that Hⁱ(F ⊗ a^∗ζ) = 0. Our proof is inspired by the proof of [26, Lemma 3.3]. By Observation 2.7 there exists ζ^' ∈ Pic⁰(A) such that F ⊗ a^∗ζ^' is C_q_−1,0 for (X, H). Observe that by Lemma 2.2, Hⁱ(F ⊗ a^∗ζ^' ⊗ (−jH)) = 0 for all integers j ≤ i − q + 1. To this end, consider the restriction exact sequence

(3.3) 0 → F ⊗ a ∗ ζ → F ⊗ H + a ∗ ζ ′ → F ⊗ H + a ∗ ζ ′ Y → 0

where Y ∈ |H +a^∗ζ^' −a^∗ζ| is a smooth and irreducible member (which exists by Bertini thanks to (P_a)). Passing to the cohomology of (3.3), we deduce that it is enough to prove that the restriction map

H i − 1 F ⊗ H + a ∗ ζ ′ → H i − 1 F ⊗ H + a ∗ ζ ′ Y

surjects since H ⁱ(F ⊗ (H + a^∗ζ^')) = 0. On the other hand, choose a general smooth and irreducible member Z ∈ |H+a^∗ζ −a^∗ζ^'| such that F|_Z is torsion-free (such a section exists thanks again to the fact that (X, H) satisfies property (P_a)). Then we know that Hⁱ⁻¹(F ⊗ (H + a^∗ζ^')) → Hⁱ⁻¹(F ⊗ (H + a^∗ζ^')|_Y_+Z) surjects by Lemma 2.2 as Hⁱ(F ⊗ a^∗ζ^'(−H)) = 0. Consequently, it is enough to show that the map

H i − 1 F ⊗ H + a ∗ ζ ′ Y + Z → H i − 1 F ⊗ H + a ∗ ζ ′ Y

surjects. Consider the following commutative diagram with exact rows and exact left column

which by the snake lemma yields the following short exact sequence:

0 → O Z ( − Y ) → O Y + Z → O Y → 0.

Twisting the above by F ⊗ (H + a^∗ζ^') and passing to cohomology, we deduce that it is enough to prove the following vanishing: H_i(F ⊗ (H + a^∗ζ^') ⊗ (−(H + a^∗ζ^' − a^∗ζ))|_Z) = Hⁱ(F ⊗ a^∗ζ|_Z) = 0. But F|_Z is torsion-free and Cq−1,0′aZfor (Z, H|_Z) by Lemma 2.11, and (Z, H|_Z) satisfies (P_a^|_Z ) by Observation 2.9. Thus we are done by the induction hypothesis. □

Theorem 3.8

Let (X, H) be a polarized smooth projective variety and let a : X → A be a morphism to an abelian variety A. Assume that (X, H) satisfies (P_a). Let F be a torsion-free coherent sheaf on X that isCq−1,0′afor (X, H) for some positive integer q ≤ dim X. ThenVaq+i(F(−tH))=∅for all integers i, t with 0 ≤ t ≤ i ≤ dim X − q.

Proof. Clearly the statement holds by Proposition 3.7 if t = 0, in particular it holds if dim X =: n = 1. Consider the restriction sequence

0 → F − t H + a ∗ ζ → F − ( t − 1 ) H + a ∗ ζ → F − ( t − 1 ) H + a ∗ ζ Y → 0

where Y ∈ |H| is a general smooth and irreducible member such that F|_Y is torsion-free. The cohomology sequence of the above, and an easy induction on n and t finishes the proof (thanks to Lemma 2.11 and Observation 2.9). □

Corollary 3.9

Let (X, H) be a polarized smooth projective variety and let a : X → A be a morphism to an abelian variety A. Assume that one of the following two conditions holds:

(X, H) satisfies property (P_a), or
the morphism a is strongly generating, and there exists an isogeny μ : Ã→ A such that (X̃ , μ̃^∗H) satisfies (P_a_̃ ), where X̃ , a ̃ and μ̃ are as in (2.1).

Let F be a torsion-free coherent sheaf on X that isC0,k′afor some integer k with 0 ≤ k ≤ dim X. Then

V a i ( F ) = ∅ for all integers i ≥ k + 1,
if k ≠ 0, then codimVak(F)≥1.

Proof. The assertion follows immediately from Theorem 3.8 if (1) holds. Now assume that (2) holds. The assertion (B) is obvious, so it is enough to show (A), that is Vⁱ(F) = 0 for i ≥ k + 1. By Lemma 2.13, we know that μ̃^∗F is C0,k′a˜for (X̃ , μ̃^∗H). Also, μ̃^∗F is torsion-free and coherent. Since (X̃ , μ̃^∗H) satisfies (P_a_̃ ), using Theorem 3.8 we conclude that Va˜iμ˜∗F=∅for i ≥ k + 1. It follows from the projection formula that μ∗Vai(F)⊆Va˜iμ˜∗Ffor all i where μ^∗ : Pic⁰(A) → Pic⁰(A) is the induced map, whence the assertion follows. □

We are now ready to provide the

Proof of Theorem A. This is an immediate consequence of Corollary 3.9 (2). Indeed, set a := alb_X, A := Alb(X), and let n_A : A → A be the multiplication by n isogeny. Then a is strongly generating. Observe that μn∗H=μn∗H1+an∗nA∗H2,and consequently Xn,μn∗Hsatisfies (P_an ) for n ≥ 2.

3.3 A few variants

In this subsection, we prove several variants of Theorem A.

3.3.1

We use the following corollary to show that the moduli space of Gieseker-stable sheaves on abelian surfaces answers Question 1.2 in the affirmative.

Corollary 3.10

Let X be a smooth projective variety, and let alb_X : X → A := Alb(X) be the Albanese map of X.

Assume that there is an isogeny μ : Ã→ A such that the following two conditions hold:

X ×_Alb(X)A ̃ ≅ Y × A ̃ for a regular smooth projective variety Y, and
the induced map X ×_Alb(X)A ̃ → A ̃ is the projection pr_A_̃under the identification in (1).

Let O_X(1) is an ample and globally generated line bundle on X. Let F be a torsion-free coherent sheaf on X that is continuously k-regular for (X, O_X(1)) where 0 ≤ k ≤ dim X. Then

Vⁱ(F) = 0 for i ≥ k + 1.
If k ≠ 0, then codim(V^k(F)) ≥ 1.

In particular, Question 1.2 has an affirmative answer for X.

Proof. Since (B) is obvious, we only need to show (A). We have the following base-change diagram

Set H = O_X(1). Note that since Y is regular, ã is also the Albanese map of X̃ . It is easy to verify that it is enough to show that Vⁱ(μ̃^∗(F)) = 0 for i ≥ k + 1. To this end, we apply Lemma 2.13 to deduce that μ̃^∗(F) is a torsion-free coherent sheaf that is continuously k-regular. Now, μ̃^∗(H) = H₁ ⊠ H₂ where H₁ and H₂ are ample and globally generated line bundles on Y and A ̃ respectively. Consequently the assertion follows from Theorem A. □

Example 3.11

Let A be an abelian surface and let v ∈ H^ev(A, ℤ) be a primitive Mukai vector satisfying v > 0 and 〈v, v〉 ≥ 6 (see [41] for details). Let L be a very general ample divisor on A, and let M_L(v) be the moduli space of Gieseker-stable sheaves on A with respect to L with Mukai vector v. By the results of loc. cit., we know that M_L(v) is a smooth projective variety, and Alb(M_L(v)) = A × Â . Moreover, if we set n(v)=12〈v,v〉and a:=albML(v),then we have the base-change diagram (see loc. cit. (4.10), (4.11))

where K _L(v) is a regular smooth projective variety (it is a hyperkähler manifold deformation equivalent to a generalized Kummer variety), and a_n_(v) = pr_A_×Â is the projection. Thus Question 1.2 has an affirmative answer for X = M _L(v) by the previous corollary.

We have the following example when the polarization is a sufficiently positive adjoint linear series:

Example 3.12

Let X be a smooth projective variety and let H ≡ K_X + sL where s ≥ dim X + 1 and L is an ample and globally generated line bundle on X. By Corollary 3.9 (1), the answer to Question 1.2 is affirmative for the pair (X, H) whenever H is ample.

3.3.2

The following result is a variant of Theorem A where we assume that H₁ = K _X + Q for a nef line bundle Q, but require that alb_X : X → Alb(X) is finite onto its image.

Theorem 3.13

Let (X, H) be a polarized smooth projective variety. Assume that the Albanese map alb_X : X → Alb(X) is finite onto its image. Further assume that there exist a nef line bundle Q on X and an ample line bundle H₂on Alb(X) such thatH=KX+Q+albX∗H2.Let F be a torsion-free coherent sheaf on X that is continuously k-regular for (X, H) for some integer k with 0 ≤ k ≤ dim X. Then the following statements hold.

Vⁱ(F) = 0 for i ≥ k + 1.
If k ≠ 0, then codim(V^k(F)) ≥ 1.

In particular, the answer to Question 1.2 is affirmative for the pair (X, H).

Proof. This is also an immediate consequence of Corollary 3.9 (2). Indeed, set a := alb_X and A := Alb(X), and note that μn∗H≡KXn+μn∗(Q)+an∗n2H2satisfies Panfor n≫ 0. □

3.3.3

Let (X, H) be a polarized smooth projective variety, and let F be a torsion-free coherent sheaf on X that is continuously k-regular as in the statements of Theorems A, 3.13 and Corollary 3.9. Then in particular we have shown that F is GV_−(k−1) whence by Theorem 3.2H ⁱ(F ⊗ RΨ_P_[g]L^⊗−1) = 0 for all i ≥ k and for any sufficiently positive ample line bundle L on Â . We now show that in some cases the vanishing in fact holds for any ample line bundle L on Â . First we need the following result.

Corollary 3.14

Let a : X → A be a morphism from a smooth projective variety X to an abelian variety A that is finite onto its image. Let H be an ample line bundle on X that satisfies one of the following:

(X, H) satisfies property (P_a); or
a is strongly generating, and moreover H := H₁ + a^∗H₂where H₁is a line bundle on X and H₂is an ample line bundle on A satisfying one of the following:
1. H ₁ is globally generated; or
2. H ₁ = K _X + Q for a nef line bundle Q.

Let F be a torsion-free coherent sheaf on X that isC0,k′afor (X, H) for some positive integer k ≤ dim X. Then Hⁱ(F ⊗ a^∗L) = 0 for any ample line bundle L on A and for any integer i ≥ k.

Proof. It follows from the proofs of Theorems A, 3.13 and Corollary 3.9 that F is GV_−(k−1) with respect to a (this means that codimVai(F)≥i−k+1for all i). Consequently, the vanishing Hⁱ(F ⊗ a_∗L) = 0 for i ≥ k and any ample line bundle L on A follows from the projection formula and Proposition 3.4. □

The above vanishing gives the following

Corollary 3.15

Let X be a smooth projective variety and let a := alb_X : X → A := Alb(X) be its Albanese morphism. Assume that a is finite onto its image and let H be an ample line bundle on X such that (X, H) satisfies (P_a). Let F be a torsion-free coherent sheaf on X that isC0,k′afor (X, H) for some positive integer k ≤ dim X. Then H_i(F ⊗ RΨ_P_[g]L^⊗−1) = 0 for all i ≥ k and for any ample line bundle L on Â where g = q(X).

Proof. Fix an ample line bundle L on Â . We recall from the proof of [36, Theorem B] that RΨP[g]L⊗−1=albX∗R0SL∗.Denoting translations by an element y ∈ Â by t_y : Â → Â , we obtain an isogeny φ_L : Â → A by sending y ∈ Â to ty∗L⊗L⊗−1.To this end, consider the following base-change diagram:

(3.4)

On the other hand, [27, Proposition 3.11] shows that φL∗R0SL∗=H0(L)⊗L.Consequently, we deduce using the projection formula that there is an injection

H i F ⊗ R Ψ P [ g ] L ⊗ − 1 = H i F ⊗ alb X ∗ R 0 S L ∗ ↪ H i φ ^ ∗ F ⊗ alb X ∗ R 0 S L ∗ = H 0 ( L ) ⊗ H i φ ^ ∗ F ⊗ a ^ ∗ L .

But φ̂^∗F is a torsion-free coherent sheaf that is C0,k′aˆfor (X̂ , φ̂^∗H) by Lemma 2.13. Observe that φ̂^∗H satisfies (P_a_̂ ), as the induced map Pic⁰(A) → Pic⁰(Â ) is also an isogeny. Consequently, H ⁱ((φ̂^∗F) ⊗ â^∗L) = 0 for all integers i ≥ k by Corollary 3.14, and the assertion follows. □

4 Continuous ℚ CM-regularity on abelian varieties

Throughout this section, X is an abelian variety of dimension g and ɛ is a vector bundle on X.

4.1 Continuous ℚ CM-regularity for vector bundles

In view of the recent development of the cohomological rank function by Jiang–Pareschi [20] that was motivated by the continuous rank function introduced and studied in [1] and [3], it is natural to extend the notion of continuous CM-regularity to an ℝ-valued regularity function on ℚ-twisted vector bundles.

Definition 4.1

(Cohomological rank function). Define hgen i(E)for i ∈ ℕ as the dimension of Hⁱ(ɛ ⊗ ζ) for general ζ ∈ Pic⁰(X). Given a polarization l_∈N1(X)and x = a/b ∈ ℚ with a, b ∈ ℤ and b > 0, following [20, Definition 2.1] we define the cohomologicalrank functionhEi(xl_)=b−2ghgen ibX∗E⊗L⊗abwhere L is a line bundle representing l.

We note that if L is an ample line bundle, then ɛ being continuously k-regular for (X, L) for k ∈ ℤ is equivalent to the condition hEi((k−i)l_)=0for all integers i ≥ 1.

Let l_∈N1(X)be a polarization. Assume, for some y = a/b ∈ ℚ with a, b ∈ ℤ and b > 0, that hEi((y−i)l_)=0for all integers i ≥ 1. This means that hgen ibX∗E⊗L⊗(a−ib)b=0for all integers i ≥ 1. This is equivalent to the condition that bX∗E⊗L⊗abis continuously 0-regular for X,L⊗b2.We claim that hεiah+cd−il_=0for all integers i ≥ 1 and c, d > 0. To see this, we need to show that hgen i(bd)X∗E⊗L⊗(ad+bc−ibd)bd=0for i ≥ 1, which is equivalent to (bd)X∗E⊗L⊗(ad+bc)bdbeing continuously 0-regular for X,L⊗(bd)2.But the given condition implies that (bd)X∗E⊗L⊗abd2is continuously 0-regular for X,L⊗(bd)2,and consequently the required continuous regularity follows by Corollary 3.14. Thus we define the following

Q − reg l ( E ) := inf y ∈ Q ∣ h E i ( ( y − i ) l _ ) = 0 for all integers i ≥ 1 .

Example 4.2

(Continuous ℚ CM-regularity for Verlinde bundles). We compute an example of Q−regl_(ε)where X := J(C) is the Jacobian of a smooth projective curve C of genus g≥1,l_=sθ_is the class of sΘ where Θ is a symmetric theta-divisor on X, s ≥ 2 is an integer, and ɛ := 𝔼_r_,k is a Verlinde bundle. We recall the definition of these bundles. For a pair of positive integers (r, k), let U_C(r, 0) be the moduli space of semistable bundles of rank r and degree 0 on C, and let det : U_r := U_C(r, 0) → X̂ = X be the determinant map. The Verlinde bundle associated to (r, k) is by definition Er,k:=det∗OUr(kΘ˜)where Θ̃ is the generalized theta-characteristic of C. For details about these bundles, we refer to [39] and [30].

Küronya and Mustopa showed in [21, Proposition 3.2] that regOX(sΘ)contEr,k=g−krs if 2łgcd(r,k).We claim that Q−regsθ_Er,k=g−krs if 2łgcd(r,k).

We show this by following their proof. First of all, by the proof of [21, Proposition 3.2] one can write 𝔼_r_,k = ⨁𝕎_a_,b ⊗ ζ_i for ζ_i ∈ X̂ where a = r/gcd(r, k), b = k/gcd(r, k) and 𝕎_a_,b is semihomogeneous, i.e. for any x ∈ X there exists ξ ∈ X̂ such that tx∗Wa,b=Wa,b⊗ξ.We also know that rank(𝕎_a_,b) = a^g and det(𝕎_a_,b) = O_X(a^g⁻¹bΘ). Observe that we have the following equality:

Q − reg s θ _ E r , k = inf y ∈ Q ∣ ∀ i ≥ 1 : h E r , k i ( ( y − i ) s θ _ ) = 0 = inf y ∈ Q ∣ ∀ i ≥ 1 : h W a , b i ( ( y − i ) s θ _ ) = 0 = inf r ′ s ′ ∈ Q s ′ > 0 and s ′ X ∗ W a , b ⊗ ( s Θ ) ⊗ r ′ s ′ is continuously 0-regular for X , s ′ 2 s Θ .

Now s′X∗ Wa,b⊗(sΘ)⊗r′s′is semihomogeneous and c1s′X∗Wa,b⊗(sΘ)⊗r′s′∈N1(X)is ℚ-proportional to s′2sθ_. Thus, by [21, Proposition 2.8],Q−regsθEr,k≤r′s′if and only if s′X∗Wa⋅b⊗Θ⊗r′s′s−gs′2sΘis nef, which in turn holds if and only if ag−1s′2b+ar′s′s−gs′2s≥0by [21, Proposition 2.7]. A simple computation shows that ag−1s′2b+ar′s′s−gs′2s≥0⟺r′s′≥g−krs.

4.2 Continuous ℚCM-regularity for ℚ-twisted bundles

Given a class n_∈Pic(X)/Pic0(X)=:N1(X),following [23, Chapter 6] we define the ℚ-twisted bundle E〈xn_〉 for x∈Qas the equivalence class of pairs (E,xn_)with respect to the equivalence relation generated by declaring E⊗M⊗e,yn_∼(E,em_+xn_)where M ∈ Pic(X), m_is its class in N¹(X), e ∈ ℤ and y ∈ ℚ. Now, given a polarization l_∈N1(X),we define the continuous ℚ CM-regularity of E〈xn_〉as follows:

Q − reg l _ ( ε 〈 x n _ 〉 ) := Q − reg b 2 l _ b X ∗ ε ⊗ N ⊗ a b

where x = a /b, a, b ∈ ℤ and b > 0. It is a formal verification that this quantity is well-defined. We only show that it does not depend on the representation of the ℚ-twist. The fact that it does not depend on its representation as a ℚ-twisted bundle under the equivalence described above can also be easily checked.

Let abn1_=cdn2_∈N1(X)Qwith a, b, c, d ∈ ℤ, b, d > 0. Set n_′:=adn1=bcn2∈N1(X).Then we have

h b X ∗ E ⊗ N 1 ⊗ a b i ( r / s − i ) b 2 l _ = 0 ∀ i ≥ 1 ⟺ h gen i ( b s ) X ∗ E ⊗ N 1 ⊗ s 2 a b ⊗ L ⊗ ( r − i s ) b 2 s = 0 ∀ i ≥ 1 ⟺ h g e n i ( b s d ) X ∗ E ⊗ N 1 ⊗ d 2 s 2 a b ⊗ L ⊗ ( r − i s ) b 2 d 2 s = 0 ∀ i ≥ 1 ⟺ h g e n i ( b s d ) X ∗ E ⊗ N ′ ⊗ b d s 2 ⊗ L ⊗ ( r − i s ) b 2 d 2 s = 0 ∀ i ≥ 1.

Similar computations as above show that hdX∗E⊗N2⊗cdi(r/s−i)d2l_=0for all i ≥ 1 is equivalent to

h gen i ( s d ) X ∗ E ⊗ N 2 ⊗ s 2 c d ⊗ L ⊗ ( r − i s ) d 2 s = 0 ∀ i ≥ 1 ⟺ h gen i ( b s d ) X ∗ E ⊗ N 2 ⊗ b 2 s 2 c d ⊗ L ⊗ ( r − i s ) b 2 d 2 s = 0 ∀ i ≥ 1

which is equivalent to the condition hgen i(bsd)X∗E⊗N′⊗bds2⊗L⊗(r−is)b2d2s=0for all i ≥ 1.

4.3 Generic vanishing for ℚ-twisted bundles

We mention an immediate corollary of Theorem A. In [20], Jiang and Pareschi extended the definitions of GV, M-regular and IT₀ sheaves to the ℚ-twisted cases. In particular, according to their definitions, for an ample class l_∈N1(X),a ℚ-twisted vector bundle E〈xl_〉 for x=abwith b > 0 is GV, M-regular, or IT₀ if so is bX∗E⊗L⊗ab.

Corollary 4.3

Let X be an abelian variety. Let l _ , l ′ _ ∈ N 1 ( X ) be ample classes and let ɛ be a vector bundle on X. IfQ−regl_Exl_′<1thenExl′_is IT₀, and ifQ−regl_Exl_′=1thenExl_′is GV.

Proof. Let X=rswith s > 0. First, if Q−regl_εxl_′<1, then we can find a, b ∈ ℤ with b > 0 such that we have Q−regl_εxl_′<ab<1.This means that hsX∗E⊗L′⊗abiab−is2l_=0for all i ≥ 1. But this means that the bundle(bs)X∗E⊗L′⊗b2rs⊗L⊗abs2is continuously 0-regular for X,L⊗b2s2Consequently, by Theorem A we deduce that (bs)X∗E⊗L⊗b2rs⊗abs2−b2s2Lis GV, whence by preservation of vanishing we conclude that (bs)X∗E⊗L′⊗b2rsis IT₀. Thus sX∗E⊗L′⊗rsis IT₀, and the conclusion follows. Finally, if Q−reglExl′_=1,then similarly we see that sX∗E⊗L′⊗rsis GV, and the conclusion follows. □

4.4 Proof of Theorem B

The proof of Theorem B is based on the following two results.

Lemma 4.4

Let X be an abelian variety of dimension g and let ɛ be a vector bundle on X. Let a, b ∈ ℤ with b > 0, and setx=a/b∈Q. If n_,l_∈N1(X)with l ample, and δ = c/d ∈ ℚ with c, d ∈ ℤ, d > 0, then

Q − reg l _ ( ε 〈 x n _ + δ l _ 〉 ) = Q − reg l _ ( ε 〈 x n _ 〉 ) − δ .

Proof. Let r, s ∈ ℤ with s > 0. It is enough to show the following equivalence:

(4.1) h ( b d ) X ∗ E ⊗ N ⊗ a b d 2 ⊗ L ⊗ b 2 c d i ( r / s − i ) b 2 d 2 l _ = 0 ∀ i ≥ 1 ⟺ h b X ∗ E ⊗ N ⊗ a b i ( r / s + c / d − i ) b 2 l _ = 0 ∀ i ≥ 1.

But the left hand side of (4.1) is equivalent to the following condition:

(4.2) h g e n i ( b d s ) X ∗ E ⊗ N ⊗ a b d 2 s 2 ⊗ L ⊗ b 2 c d s 2 ⊗ L ⊗ ( r − i s ) b 2 d 2 s = 0 ∀ i ≥ 1.

And the right hand side of (4.1) is equivalent to hgen i(bds)X∗E⊗N⊗abd2s2⊗L⊗(rd+cs−ids)b2ds=0for all i ≥ 1, which under simplification boils down to (4.2). The proof is now complete. □

Proposition 4.5

Let X be an abelian variety of dimension g and let ɛ be a vector bundle on X. Letl_,l_′,n_∈N1(X)with l, l^'ample. Further, let x = a/b, y = c/d ∈ ℚ with a, b, c, d ∈ ℤ, b, c, d > 0. Then

Q − reg l ε x n _ + y l ′ _ ≤ Q − reg l ( ε 〈 x n _ 〉 ) .

Proof. Let β = β₁/β₂ ∈ ℚ with β₁, β₂ ∈ ℤ and β₂ > 0. It is enough to show that if Q−regl(E〈xn_〉)<βthen Q−regl__Exn_+yl′_≤β.As before, Q−regl(ε〈xn_〉)<βimplies that hbX∗E⊗N⊗abi(β−i)b2l_=0for all i ≥ 1, which is equivalent to bβ2X∗E⊗N⊗abβ22⊗L⊗β1β2b2being continuously 0-regular for X,L⊗bβ22.This in turn is equivalent to G:=bdβ2X∗E⊗N⊗abd2β22⊗L⊗β1β2b2d2being continuously 0-regular for X,L⊗bdβ22.

We aim to show that Q−reglExn_+yl′_≤βwhich is equivalent to showing the vanishing condition h(bd)X∗E⊗N⊗abd2⊗L′⊗b2cd(β−i)b2d2l_=0for all i ≥ 1, which in turn is equivalent to showing that G⊗L′⊗b2cdβ22is continuously 0-regular for X,L⊗bdβ22.But this follows from Corollary 3.14. □

Proof of Theorem B. We adapt an argument of Ito in the proof of [16, Proposition 2.9]. First we show that Q−regl(ε〈−〉):N1(X)Q→Ris continuous. Let {ξ_i}_i_∈ℕ be a sequence in N¹(X)_ℚ converging to ξ ∈ N¹(X)_ℚ. For any rational number δ > 0 there exists N₀ ∈ ℕ such that for all i≥N0,ξi−ξ+δl_ and ξ+δl_−ξiare both ample. Thus, by Proposition 4.5 we deduce that for all i ≥ N₀ we have the following inequality:

Q − reg l _ ( ε 〈 ξ + δ l _ 〉 ) ≤ Q − reg l _ ε ξ i ≤ Q − reg l _ ( ε 〈 ξ − δ l _ 〉 ) .

Now by Lemma 4.4, we deduce that Q−regl_Eξi−Q−regl_(E〈ξ〉)≤δfor all i ≥ N₀, and that proves the claim. To finish the proof, we need to show that given a sequence {ξ_i}_i_∈ℕ in N¹(X)_ℚ converging to ξ ∈ N¹(X)_ℝ, the sequence Q−regl_εξii∈Nconverges to a real number. As before, for any rational δ > 0 there exists N₀ ∈ ℕ such that for j,k≥N0,ξj−ξ+δl_and ξ + δl − ξ_k are both ample. Using Proposition 4.5, we deduce that Q−reglEξj+δl_≤Q−reglEξk−δl_which by Lemma 4.4 implies that for all j, k ≥ N₀ we have the inequality Q−regl_εξj−δ≤Q−regl_Eξk+δ.Thus we obtain

l i m ‾ j → ∞ Q − reg l _ ε ξ j − δ ≤ l i m _ k → ∞ Q − reg l _ ε ξ k + δ .

Now, by letting δ → 0, we see that Q−regl_Eξii∈Nconverges to a real number. □

5 Applications for sheaves on abelian varieties

5.1 Three immediate consequences

We list three consequences of our generic vanishing theorems.

Corollary 5.1

Let (X, H) be a polarized smooth projective variety. Assume that there exist a globally generated line bundle H₁on X and an ample line bundle H₂on Alb(X) such thatH=H1+albX∗H2,where alb_X : X → Alb(X) is the Albanese map. Let F be a torsion-free coherent sheaf on X that is continuously 1-regular for (X, H). Then χ(F) ≥ 0 with equality if and only if either V⁰(F) = 0 or any component V⁰(F) is of codimension 1.

Proof. We know that F is GV from Theorem A, and moreover Vⁱ(F) = 0 for i ≥ 2. It follows from the generic vanishing theory that χ(F) ≥ 0 with equality if and only if codim(V⁰(F)) ≥ 1. Now, by [36, Proposition 3.15] and [33, Lemma 1.8], we know that a codimension q component of V⁰(F) is also a component of V^q(F), whence the conclusion follows. □

Remark 5.2

For a coherent sheaf F on a smooth projective variety X of dimension d, one defines RΔF := RHom(F, K_X). If F is GV, then RΦ_P(RΔF) is a sheaf concentrated in degree d, i.e. RΦ_P(RΔF) = R^dΦ_P(RΔF)[−d]. The generic vanishing theory tells us that for a GV sheaf F, χ(F) is the rank ofRΔFˆand the support ofRΔFˆis −V⁰(F) where RΔFˆ=RdΦP(RΔF).Thus, if X, H, F are as in Corollary 5.1, then χ(F) ≥ 0 with equality if and only if any (non-empty) component of the support ofRΔFˆis of codimension one.

It is worth mentioning that for a GV sheaf F on an abelian variety, V⁰(F) ≠ 0 by [33, Lemma 1.12].

Remark 5.3

The conclusion of Corollary 5.1 also applies in the set-up of Corollaries 3.9, 3.10 and Theorem 3.13, all with k = 1.

Corollary 5.4

Let (X, H) be a polarized abelian variety and let F be torsion-free coherent sheaf on X. Assume that F is continuously k-regular for (X, H) for some k ∈ ℕ. Then the following statements hold.

If 1 ≤ k ≤ dim X, then F is a GV_−(k−1)sheaf. In particular, if k = 1, then F is nef.
If k = 0, then F is an IT₀sheaf, in particular F is ample.

Proof. This follows immediately from Theorem A combined with the facts that GV sheaves on abelian varieties are nef, see [37, Theorem 4.1], and IT₀ sheaves are ample, see [7, Corollary 3.2]. □

Remark 5.5

Let (X, H) be a polarized abelian variety and let ɛ be a vector bundle of rank r that is continuously 0-regular for (X, H). Then for any subvariety Z ⊆ X of dimension k, we have

c 1 ( E ) k ⋅ Z = c 1 ( E ( − H ) ) + r H k ⋅ Z ≥ r k H k Z

as ɛ(−H) is nef by the above corollary. In particular, c₁(ɛ)ⁿ ≥ rⁿHⁿ where n := dim X, and for any x ∈ X we have the Seshadri constant ϵ(det(ɛ), x) ≥ rϵ(H, x). This refines the bound of [25, Theorem 7.2] for abelian varieties.

Corollary 5.6

Let (X, H) be a polarized abelian variety. Let F₁and F₂be torsion-free coherent sheaves on X with one of them locally free. Assume that F₁is continuously k₁-regular and F₂is continuously k₂-regular for (X, H). ThenF1⊗F2 is C0,k1+k2for (X, H).

Proof. Without loss of generality, we may assume that k₁ = k₂ = 0. For any positive integer i ≤ dim X we have Hⁱ(F₁ ⊗ F₂(−iH)) = Hⁱ((F₁(−iH)) ⊗ F₂). By Theorem A, we know that F₁(−iH) is GV_−(i−1) and F₂(−H) is GV. By the preservation of vanishing, F₂ is IT₀, whence by Proposition 3.4 we obtain the required vanishing Hⁱ(F₁ ⊗ F₂(−iH)) = 0. □

Remark 5.7

We remark that the proof of the above corollary also shows the following: let (X, H) be a polarized abelian variety and let F₁, F₂ be torsion-free coherent sheaves on X with one of them locally free. If F₁ is continuously 0-regular for (X, H) and F₂ is IT₀, then F₁ ⊗ F₂ is 0-regular for (X, H). This can also be thought of as a generalization of a result of Murty and Sastry, see [28, Proposition 5.4.1].

5.2 Syzygies of tautological bundles of zero-regular bundles

We start with some background on syzygies. Let X be a smooth projective variety and let L be a very ample line bundle on X. Consider the embedding X ⊆ ℙ^r given by the complete linear series |L| where r = h⁰(L) − 1. One has a minimal graded free resolution of R(X,L):=⨁q≥0H0(qL)as an S := Sym^∙(H⁰(L)) module as follows:

0 → E r + 1 = ⨁ j S − a r + 1 , j → E r = ⨁ j S − a r , j → ⋯ → E 1 = ⨁ j S − a 1 , j → E 0 = ⨁ j S − a 0 , j → R ( X , L ) → 0.

For a reference for the following definition, see for example [22, Definition 1.8.50].

Definition 5.8

(Projective normality and the Np property). Suppose we are in the situation as above.

The embedding given by the complete linear series |L| is called projectively normal if E₀ = S.
We say that L satisfies the N_p property if the embedding by |L| is projectively normal, and a_ij = i + 1 for all i with 1 ≤ i ≤ p.

In practice, to calculate the syzygies of a projective variety one needs to calculate cohomology groups involving the syzygy bundles that we define next; see for example [38, Section 3].

Definition 5.9

(Syzygy bundle). Let X be a smooth projective variety and let ɛ be a globally generated vector bundle on X. The syzygy bundle M_E is the kernel of the map H⁰(ɛ) ⊗ O_X → ɛ, i.e. we have the exact sequence

(5.1) 0 → M E → H 0 ( E ) ⊗ O X → E → 0.

The following proposition of Park will be used in the proof of Theorem A.1 and we include it here.

Proposition 5.10

([38, Proposition 3.2]). Let X be a smooth projective variety and let ɛ be an ample and globally generated bundle on X. Then O_ℙ(E)(1) on ℙ(ɛ) satisfies the N_p property ifHk⋀iME⊗E⊗j=0for 0 ≤ i ≤ p + 1 and j, k ≥ 1.

We now prove Corollary C that is obtained by an immediate application of our result combined with the theorem of Ito below (Theorem A.1).

Proof of Corollary C. Note that for any x ∈ ℚ with x < 1 we have Q−regh(E〈−Xh_〉)<1by Lemma 4.4, whence ε〈−Xh_〉is IT₀ by Corollary 4.3. Since β(h_)<1/(p+2),we have (p+2)β(h_)<1.Thus for 0 < ϵ ≪ 1 we have x=1−ϵ≥(p+2)β(h_),whence the assertion follows immediately by Theorem A.1. □

A On Syzygies of projective bundles on abelian varieties Appendix by Atsushi Ito

In this appendix, we follow the notation in the previous sections. In particular, we work over the field ℂ of complex numbers.

Let (X, L) be a polarized abelian variety and let l_∈N1(X)be the class of L. In [20], Jiang and Pareschi define the basepoint-freeness threshold β(l) ∈ (0, 1]. This invariant is quite useful to study syzygies of polarized abelian varieties, since [20] and Caucci [6] show that L satisfies the N_p property if β(l_)<1/(p+2)

The purpose of this appendix is to prove the following theorem:

Theorem A.1

Let (X, L) be a polarized abelian variety and let E be a vector bundle on X. Let p ≥ 0 be an integer. Assume that there exists a rational numberrx≥(p+2)β(l_)such thatE〈−xl_〉is M-regular. Then O_ℙ(E₎(1) satisfies the N_p property.

Remark A.2

In the case where E=L,L〈−Xl_〉is M-regular if and only if x < 1; cf. [17, Example 2.1]. Hence the existence of a rational number x as in Theorem A.1 is equivalent to β(l) < 1/(p + 2) in this case.

To prove this theorem, we use the following lemma.

Lemma A.3

Let (X, L) be a polarized abelian variety and let E be a vector bundle on X. Assume that there exists a rational numberx≥β(l_)such thatE〈−xl_〉is M-regular. Then

E is IT ₀ and globally generated.
Let ME be the syzygy bundle of E defined by (5.1). For a rational number y > 0, ME 〈yl〉 is IT₀if1x+1y≤1β(l_).

Proof. (1) Since E〈−xl_〉is M-regular and x≥β(l_)>0,the bundle E is IT₀. The global generation of E follows from [17, Theorem 1.2 (1)].

(2) Write y = a/b with integers a, b > 0. Then

M E 〈 y l _ 〉 = M E a b l _ is I T 0 ⟺ b X ∗ M E ⊗ L ⊗ a b is IT 0 ⟺ b X ∗ b X ∗ M E ⊗ L ⊗ a b = M E ⊗ b X ∗ L ⊗ a b is I T 0 ,

where the first equivalence follows from the definition and the second one holds since the property IT₀ is preserved by the pushforward by isogenies; cf. [17, (2.4)].

Consider the exact sequence

0 → M E ⊗ b X ∗ L ⊗ a b → H 0 ( E ) ⊗ b X ∗ L ⊗ a b → E ⊗ b X ∗ L ⊗ a b → 0

obtained by tensoring b_X_∗(L^⊗ab) with (5.1). Since L^⊗ab is IT₀, so is the pushforward b_X_∗(L^⊗ab). Since E is IT₀ as well by (1), both H⁰(E )⊗ b_X_∗(L^⊗ab) and E ⊗ b_X_∗(L^⊗ab) are IT₀. Thus Hⁱ(ME ⊗ b_X_∗(L^⊗ab) ⊗ ζ) = 0 for any i ≥ 2 and ζ ∈ Pic⁰(X), and hence ME ⊗ b_X_∗(L^⊗ab) is IT₀ if and only if

H 1 M E ⊗ b X ∗ L ⊗ a b ⊗ ζ = 0

for any ζ ∈ Pic⁰(X), which is equivalent to the surjectivity of

(A.1) H 0 ( E ) ⊗ H 0 b X ∗ L ⊗ a b ⊗ ζ → H 0 E ⊗ b X ∗ L ⊗ a b ⊗ ζ .

In conclusion, (2) follows from the surjectivity of (A.1) for any ζ ∈ Pic⁰(X). Set F = b_X_∗(L^⊗ab) ⊗ ζ . Recall that y = a/b. Since

1 x + b a = 1 x + 1 y ≤ 1 β ( l _ )

by assumption, it suffices to show that

φ l _ ∗ Φ ( E ) ⊗ φ l _ ∗ Φ ( − 1 ) X ∗ F 1 x + b a l _

is M-regular by [17, Proposition 4.4], where φl_:X→Xˆ=Pic0(X)is the isogeny induced by the polarization l_,Φ=RSˆ:D(X)→D(Xˆ)is the Fourier–Mukai functor associated to the Poincaré line bundle on X × X̂ , and (−1)_X is the multiplication map by (−1) on X. We note that Φ(E ) and Φ(−1)X∗Fare locally free sheaves since E and (−1)X∗Fare IT₀.

The rest is to check the M-regularity of φl_∗Φ(E)⊗φl_∗Φ(−1)X∗F1x+bal_,which can be shown by modifying the argument in the proof of [17, Proposition 4.4 (1)] as follows: The pullback of a ℚ-twisted sheaf G 〈tl〉 by the multiplication map n_X on X for n ≥ 1 is defined as

n X ∗ ( G 〈 t l _ 〉 ) := n X ∗ G t n X ∗ l _ = n X ∗ G n 2 t l _ .

Since M-regularity is preserved by such pullbacks, cf. [17, (2.2)], it suffices to show the M-regularity of

(A.2) a X ∗ φ l _ ∗ Φ ( E ) ⊗ φ l _ ∗ Φ ( − 1 ) X ∗ F 1 x + b a l _ = a X ∗ φ l _ ∗ Φ ( E ) ⊗ φ l _ ∗ Φ ( − 1 ) X ∗ F a 2 x + a b l _ .

Recall that F=bX∗L⊗ab⊗ζ=bX∗L⊗ab⊗bX∗ζ.We take ζ^' ∈ Pic⁰(X) such that ζ′⊗ab=bX∗ζand set L′=(−1)X∗L⊗ζ′.Then we have

( − 1 ) X ∗ F = ( − 1 ) X ∗ b X ∗ L ⊗ ζ ′ ⊗ a b = b X ∗ ( − 1 ) X ∗ L ⊗ ζ ′ ⊗ a b = b X ∗ L ′ ⊗ a b

and hence

a X ∗ φ l _ ∗ Φ ( − 1 ) X ∗ F = a X ∗ φ l _ ∗ Φ b X ∗ L ′ ⊗ a b = a X ∗ φ l _ ∗ b X ˆ ∗ Φ L ′ ⊗ a b = φ a b l _ ∗ Φ L ′ ⊗ a b = H 0 L ′ ⊗ a b ⊗ L ′ ⊗ − a b .

In fact, the second equality follows from [27, (3.4)]. The third one follows from bXˆ∘φl_∘aX=φabl_.The last one follows from [27, Proposition 3.11 (1)] since the numerical class of L^'⊗ab is abl by L′=(−1)X∗L⊗ζ′≡L.Hence it holds that

a X ∗ φ l _ ∗ Φ ( E ) ⊗ φ l _ ∗ Φ ( − 1 ) X ∗ F a 2 X + a b l _ = a X ∗ φ l _ ∗ Φ ( E ) ⊗ H 0 L ′ ⊗ a b ⊗ L ′ ⊗ − a b a 2 X + a b l _ = a X ∗ φ l _ ∗ Φ ( E ) ⊗ H 0 L ′ ⊗ a b a 2 X l _ = H 0 L ′ ⊗ a b ⊗ a X ∗ φ l _ ∗ Φ ( E ) 1 X l _ ,

where the second equality follows from L^' ≡ L. Since E〈−Xl_〉is M-regular, φl_∗Φ(E)1xl_is M-regular as well by [17, Proposition 4.1]. Thus so is the pullback aX∗φl_∗Φ(E)1xl_and hence we obtain the M-regularity of (A.2), which implies (2). □

Proof of Theorem A.1. Since x≥(p+2)β(l_)≥β(l_),the bundle E is globally generated by Lemma A.3 (1).

To prove the N_p property for O_ℙ(E₎(1), it suffices to show that HkME⊗i⊗E⊗j=0for 0 ≤ i ≤ p + 1, j, k ≥ 1 by Proposition 5.10 since we work over ℂ, whose characteristic is zero. Hence this theorem holds if ME⊗i⊗E⊗jis IT₀ for 0 ≤ i ≤ p + 1, j ≥ 1.

If i = 0, then ME⊗i⊗E⊗j=E⊗jis IT₀ for j ≥ 1 since so is E by Lemma A.3 (1). If i≥1,ME⊗i⊗E⊗jis written as

(A.3) M E ⊗ i ⊗ E ⊗ j = M E x i l _ ⊗ i ⊗ E 〈 − x l _ 〉 ⊗ E ⊗ j − 1 ,

i.e. as a ℚ-twisted sheaf. For 1≤i≤p+1,MExil_is IT₀ by Lemma A.3 (2) since it holds that

1 x + i x ≤ p + 2 x ≤ 1 β ( l )

by assumption. Furthermore, E〈−Xl_〉is M-regular by assumption and E is IT₀. Thus their tensor product (A.3) is IT₀ by [6, Proposition 3.4]. □

rcdeba@gmail.com

Funding statement: A.I. was supported by JSPS KAKENHI Grant Numbers 17K14162, 21K03201.

Communicated by: I. Coskun

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Received: 2023-10-27

Revised: 2023-07-24

Published Online: 2024-01-24

Published in Print: 2024-01-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/advgeom-2023-0028

Keywords for this article

Vector bundles; Castelnuovo–Mumford regularity; generic vanishing; syzygies

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