June 03, 2026
Let \((A, L)\) be a polarized abelian variety of dimension \(g \geq 1\) over an algebraically closed field of characteristic zero. We prove that every nontrivial line bundle \(P\) in the connected component \(\operatorname{Pic}^0(A)\) of the Picard variety is arithmetically Cohen–Macaulay (ACM) with respect to \(L\). For \(g \geq 2\) and any fixed nontrivial \(P \in \operatorname{Pic}^0(A)\), we construct by induction an infinite sequence of indecomposable ACM vector bundles \(E_r\) of every rank \(r \geq 1\). In addition, this paper studies classification questions for ACM line bundles and shows that, for abelian varieties of dimension at least two, the category of ACM bundles is of wild representation type. This paper settles the existence problem for nontrivial ACM bundles on polarized abelian varieties and supply large explicit families of indecomposable examples
An important theme in algebraic geometry is to understand a projective variety through the vector bundles it carries. Among these, two closely related classes have received a great deal of attention: arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles. If \((X,L)\) is a polarized projective variety, a vector bundle \(E\) is ACM with respect to \(L\) if \[H^i(X,E\otimes L^{\otimes m})=0 \quad\text{for all } m\in \mathbb{Z} \text{ and } 0<i<\dim X.\] Ulrich bundles are the ACM bundles with the strongest possible cohomological behavior, and they often serve as the most rigid and useful objects in this circle of ideas.
These bundles matter for several reasons. In commutative algebra, ACM bundles correspond to maximal Cohen–Macaulay modules over homogeneous coordinate rings, so they reflect subtle properties of singularities, resolutions, and syzygies; see [1]. In algebraic geometry, Ulrich bundles became especially prominent through the work of Eisenbud, Schreyer, and Weyman, who showed that they give determinantal and Pfaffian expressions for Chow forms; see [2]. Beauville’s papers and survey give a clear picture of the importance of Ulrich bundles and of the main existence problems around them; see [3], [4].
Two broad questions organize much of the subject. The first is the existence problem: given a polarized projective variety, does it carry an ACM bundle or an Ulrich bundle? The second is the classification problem: if such bundles exist, how many indecomposable ones are there, and how complicated are their families? A useful language for the second question comes from representation theory. Following the now standard viewpoint, one studies the ACM representation type of a polarized variety, which may be finite, tame, or wild depending on the complexity of its indecomposable ACM bundles; see, for example, [5]. This perspective has proved effective in measuring how complicated the geometry of ACM bundles really is.
Much of the existing literature concerns varieties whose geometry already favors cohomological vanishing, such as projective spaces, quadrics, Grassmannians, and several classes of Fano-type varieties. Abelian varieties lie in a very different direction. If \(A\) is an abelian variety of dimension \(g\ge 2\), then \[H^i(A,\mathcal{O}_A)\neq 0 \quad\text{for every }0\le i\le g,\] so the structure sheaf itself is not ACM. This makes abelian varieties a delicate testing ground for the theory. In dimension two, Beauville constructed rank-two Ulrich bundles on abelian surfaces; see [3]. Beyond such cases, however, the existence and structure of ACM bundles on polarized abelian varieties have remained much less transparent.
The starting point of this paper is a simple observation with strong consequences. Let \((A,L)\) be a polarized abelian variety over an algebraically closed field of characteristic zero, and let \(P\in \operatorname{Pic}^0(A)\) be nontrivial. Since \(P\) is numerically trivial, a classical theorem of Mumford implies that \[H^i(A,P)=0 \quad\text{for all }i;\] see [6]. Combining this with the vanishing of higher cohomology for ample line bundles on abelian varieties, Serre duality, and the triviality of the canonical bundle, we prove that every nontrivial element of \(\operatorname{Pic}^0(A)\) is ACM with respect to every ample polarization.
In fact, we determine the full cohomology table of \(P\otimes L^{\otimes m}\) for all integers \(m\). As consequences, we show that \(\mathcal{O}_A\) is the unique non-ACM element of \(\operatorname{Pic}^0(A)\) when \(g\ge 2\), that \(\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\) gives a natural \(g\)-dimensional family of rank-one ACM line bundles, and that none of these line bundles is Ulrich, except in dimension one after one positive twist. We also show that the ACM complexity of \((A,L)\) is equal to one.
The rank-one case is only the beginning. Fix a nontrivial line bundle \(P\in \operatorname{Pic}^0(A)\). For every \(g\ge 2\) and every rank \(r\ge 1\), we construct an indecomposable ACM vector bundle \(E_r\) of rank \(r\) by iterated non-split extensions starting from \(P\). The main difficulty is to prove that suitable nontrivial extension classes exist at every step and that the resulting bundles remain indecomposable. To do this, we use Mukai’s Fourier–Mukai transform [7], [8] and relate the extension problem to finite-length modules over a regular local ring.
This viewpoint also leads to a representation-theoretic consequence. Using the Fourier–Mukai correspondence, we identify unipotent bundles on an abelian variety with finite-length modules over a regular complete local ring. From this, we deduce that for abelian varieties of dimension at least two, the category of ACM bundles is of wild representation type. Thus polarized abelian varieties support not only explicit ACM bundles in every rank, but also a maximally complicated ACM theory from the point of view of classification.
We also study classification questions for ACM line bundles. Under the assumption that the Picard number is one, we characterize exactly which line bundles are ACM with respect to a fixed polarization. For abelian surfaces, we give a more refined criterion in terms of intersection theory and elliptic quotient structures.
Taken together, our results show that polarized abelian varieties carry abundant ACM bundles despite the fact that the structure sheaf itself fails to be ACM in dimension at least two. They provide explicit families of indecomposable ACM bundles in every rank, clarify the behavior of numerically trivial line bundles with respect to a polarization, and place abelian varieties firmly on the wild side of the ACM representation-type spectrum.
Throughout, \(k\) is an algebraically closed field. Unless otherwise specified, we assume \(\operatorname{char}k=0\). All varieties are integral and projective over \(k\).
Definition 1. An abelian variety over \(k\) is a complete connected algebraic group over \(k\). If \(A\) is an abelian variety, its dual abelian variety is
\[\widehat A=\operatorname{Pic}^0(A).\] A polarization on \(A\) is the numerical class of an ample line bundle. In this article, by a polarized abelian variety we mean a pair \((A,L)\), where \(L\) is an ample line bundle on \(A\).
For \(a\in A(k)\), let \(t_a:A\to A\) denote translation by \(a\).
Definition 2. The subgroup \(\operatorname{Pic}^0(A)\subseteq \operatorname{Pic}(A)\) consists of line bundles algebraically equivalent to \(\mathcal{O}_A\). Equivalently, it is the connected component of the identity of the Picard scheme. Every element of \(\operatorname{Pic}^0(A)\) is numerically trivial.
Lemma 1. If \(P\in \operatorname{Pic}^0(A)\), then \(c_1(P)=0\) in \(\mathrm{NS}(A)\). Consequently,
\[c_1(P\otimes M)=c_1(M)\] for every line bundle \(M\) on \(A\).
Proof. By definition, \(\operatorname{Pic}^0(A)\) is the subgroup of line bundles algebraically equivalent to zero. Algebraic equivalence implies numerical equivalence. Therefore the class of \(P\) in the Néron–Severi group
\[\mathrm{NS}(A)_{\mathbb{Q}}=\operatorname{Pic}(A)/\operatorname{Pic}^0(A)\] is zero. Hence \(c_1(P)=0\) in \(\mathrm{NS}(A)_{\mathbb{Q}}\). The formula for \(c_1(P\otimes M)\) follows from additivity of the first Chern class. ◻
Lemma 2. Let \(L\) be ample on \(A\), and let \(P\in \operatorname{Pic}^0(A)\). Then \(L\otimes P\) is ample.
Proof. By Lemma 1, \(L\otimes P\) is numerically equivalent to \(L\). Ampleness of line bundles is invariant under numerical equivalence on a projective variety by Kleiman’s criterion. Since \(L\) is ample, \(L\otimes P\) is ample. ◻
Lemma 3. Let \(A\) be an abelian variety of dimension \(g\). Then
\[\omega_A\simeq \mathcal{O}_A.\]
Proof. The tangent bundle \(T_A\) is trivial. Indeed, every tangent vector at the identity element \(0\in A\) extends uniquely to a translation-invariant vector field on \(A\). Hence
\[T_A\simeq T_{A,0}\otimes_k \mathcal{O}_A.\] Taking duals and determinants gives
\[\omega_A=\det\Omega_A^1\simeq \det(T_{A,0}^{\vee})\otimes_k \mathcal{O}_A\simeq \mathcal{O}_A.\] ◻
Corollary 1 (Serre duality on an abelian variety). For every coherent sheaf \(\mathcal{F}\) on \(A\), there is a functorial duality
\[H^i(A,\mathcal{F})^\vee \simeq \operatorname{Ext}^{g-i}_A(\mathcal{F},\mathcal{O}_A).\] In particular, if \(M\) is a line bundle on \(A\), then
\[H^i(A,M)^\vee\simeq H^{g-i}(A,M^{-1}).\]
Proof. This is ordinary Serre duality:
\[H^i(A,\mathcal{F})^\vee\simeq \operatorname{Ext}^{g-i}_A(\mathcal{F},\omega_A).\] By Lemma 3, \(\omega_A\simeq \mathcal{O}_A\). If \(\mathcal{F}=M\) is a line bundle, then
\[\mathcal{H}om(M,\mathcal{O}_A)\simeq M^{-1},\] and higher sheaf Ext groups vanish because \(M\) is locally free. ◻
Theorem 1 (Riemann–Roch). Let \(A\) be an abelian variety of dimension \(g\), and let \(M\) be a line bundle on \(A\). Then
\[\chi(A,M)=\frac{c_1(M)^g}{g!}.\] In particular, if \(L\) is ample and \(P\in\operatorname{Pic}^0(A)\), then
\[\chi(A,P\otimes L^{\otimes m})=m^g\frac{(L^g)}{g!}\] for every \(m\in\mathbb{Z}\).
Proof. By the Hirzebruch–Riemann–Roch theorem,
\[\chi(A,M)=\int_A \operatorname{ch}(M)\operatorname{td}(T_A).\] The tangent bundle \(T_A\) is trivial, hence \(\operatorname{td}(T_A)=1\). Also
\[\operatorname{ch}(M)=\exp(c_1(M)).\] The degree \(2g\) component of \(\exp(c_1(M))\) is \(c_1(M)^g/g!\). Thus
\[\chi(A,M)=\frac{c_1(M)^g}{g!}.\] If \(M=P\otimes L^{\otimes m}\) with \(P\in\operatorname{Pic}^0(A)\), then \(c_1(P)=0\), so
\[c_1(P\otimes L^{\otimes m})=m c_1(L).\] Therefore
\[\chi(A,P\otimes L^{\otimes m}) = \frac{(m c_1(L))^g}{g!} = m^g\frac{(L^g)}{g!}.\] ◻
Corollary 2. If \(P\in\operatorname{Pic}^0(A)\), then
\[\chi(A,P)=0.\]
Proof. Apply Theorem 1 with \(m=0\), or directly use \(c_1(P)=0\). ◻
Definition 3. Let \(X\) be a projective variety of dimension \(n\), and let \(H\) be an ample line bundle on \(X\). A coherent sheaf \(\mathcal{F}\) on \(X\) is called arithmetically Cohen–Macaulay, or ACM, with respect to \(H\), if
\[H^i(X,\mathcal{F}\otimes H^{\otimes m})=0\] for every \(m\in\mathbb{Z}\) and every \(0<i<n\).
Remark 2. If \(n=1\), the ACM condition is vacuous, since there is no integer \(i\) satisfying \(0<i<1\).
Definition 4. Let \(X\) be a projective variety of dimension \(n\), and let \(H\) be an ample line bundle. A vector bundle \(E\) on \(X\) is called Ulrich with respect to \(H\) if
\[H^i(X,E\otimes H^{-j})=0\] for all \(i\) and all \(1\leq j\leq n\).
Remark 3. When \(H\) is very ample, this cohomological definition is equivalent to the usual definition that the graded module associated to \(E\) has a linear resolution over the homogeneous coordinate ring. It is also equivalent to saying that \(E\) is ACM and has the maximal possible number of global sections:
The key input is the following classical theorem.
Theorem 4 (Mumford’s vanishing theorem). Let \(A\) be an abelian variety of dimension \(g\), and let
\[P\in \operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}.\] Then
\[H^i(A,P)=0\] for every \(0\leq i\leq g\).
We give a Fourier–Mukai proof. Let
\[\mathcal{P}\] be the normalized Poincaré line bundle on \(A\times \widehat A\). Let
\[p_A:A\times \widehat A\to A,\qquad p_{\widehat A}:A\times \widehat A\to \widehat A\] be the projections. The Fourier–Mukai functor is
\[\Phi_{\mathcal{P}}:\mathbf{D}^b(\operatorname{Coh}A)\to \mathbf{D}^b(\operatorname{Coh}\widehat A),\] defined by
\[\Phi_{\mathcal{P}}(\mathcal{F}) = Rp_{\widehat A,*}(p_A^*\mathcal{F}\otimes \mathcal{P}).\]
We use the following standard facts due to Mukai [7]; see also [9].
Theorem 5 (Mukai). The functor \(\Phi_{\mathcal{P}}\) is an equivalence of triangulated categories. Moreover,
\[\Phi_{\mathcal{P}}(\mathcal{O}_A)\simeq k(0_{\widehat A})[-g],\] where \(k(0_{\widehat A})\) denotes the skyscraper sheaf at the origin of \(\widehat A\).
Lemma 4. Let \(P_\alpha\in\operatorname{Pic}^0(A)\) correspond to a point \(\alpha\in \widehat A(k)\). Then
\[\Phi_{\mathcal{P}}(P_\alpha)\simeq k(-\alpha)[-g],\] up to the conventional sign determined by the normalization of the Poincaré bundle.
Proof. Tensoring by \(P_\alpha\) on \(A\) corresponds under the Fourier–Mukai transform to translation by \(-\alpha\) on \(\widehat A\). More precisely,
\[\Phi_{\mathcal{P}}(P_\alpha\otimes \mathcal{F}) \simeq t_{-\alpha}^*\Phi_{\mathcal{P}}(\mathcal{F}).\] Applying this with \(\mathcal{F}=\mathcal{O}_A\) and using Theorem 5, we get
\[\Phi_{\mathcal{P}}(P_\alpha) \simeq t_{-\alpha}^*k(0_{\widehat A})[-g] \simeq k(-\alpha)[-g].\] ◻
Proof of Theorem 4. Let \(P=P_\alpha\), where \(\alpha\in\widehat A(k)\). Since \(P\neq \mathcal{O}_A\), we have \(\alpha\neq 0_{\widehat A}\).
By definition of the Fourier–Mukai transform, the derived fiber of \(\Phi_{\mathcal{P}}(P)\) at a point \(\beta\in\widehat A\) is
\[\Phi_{\mathcal{P}}(P)\otimes^{\mathbf{L}} k(\beta) \simeq R\Gamma(A,P\otimes P_\beta).\] Taking \(\beta=0_{\widehat A}\), we obtain
\[\Phi_{\mathcal{P}}(P)\otimes^{\mathbf{L}} k(0_{\widehat A}) \simeq R\Gamma(A,P).\] But by Lemma 4,
\[\Phi_{\mathcal{P}}(P)\simeq k(-\alpha)[-g].\] Since \(\alpha\neq 0\), the skyscraper sheaf \(k(-\alpha)\) has zero derived fiber at \(0_{\widehat A}\). Therefore
\[R\Gamma(A,P)=0.\] Hence
\[H^i(A,P)=0\] for all \(i\). ◻
Remark 6. Theorem 4 is also a consequence of the index theorem for line bundles on abelian varieties. In Mumford’s terminology, a line bundle \(M\) on an abelian variety has cohomology concentrated in one degree when it is nondegenerate. A nontrivial element of \(\operatorname{Pic}^0(A)\) has Euler characteristic zero and, by the Fourier–Mukai computation above, no cohomology at all. See [6] and [10].
We now prove the ACM vanishing.
Proposition 7 (Positive twists). Let \(A\) be an abelian variety, let \(L\) be ample, and let \(P\in\operatorname{Pic}^0(A)\). If \(m>0\), then
\[H^i(A,P\otimes L^{\otimes m})=0\] for all \(i>0\).
Proof. By Lemma 2, \(P\otimes L^{\otimes m}\) is ample. Since \(\operatorname{char}k=0\), Kodaira vanishing gives
\[H^i(A,\omega_A\otimes P\otimes L^{\otimes m})=0\] for \(i>0\). By Lemma 3, \(\omega_A\simeq\mathcal{O}_A\). Therefore
\[H^i(A,P\otimes L^{\otimes m})=0\] for every \(i>0\). ◻
Proposition 8 (Negative twists). Let \(A\) be an abelian variety of dimension \(g\), let \(L\) be ample, and let \(P\in\operatorname{Pic}^0(A)\). If \(m<0\), then
\[H^i(A,P\otimes L^{\otimes m})=0\] for all \(i<g\).
Proof. By Serre duality,
\[H^i(A,P\otimes L^{\otimes m})^\vee \simeq H^{g-i}(A,(P\otimes L^{\otimes m})^{-1}).\] Now
\[(P\otimes L^{\otimes m})^{-1} \simeq P^{-1}\otimes L^{\otimes (-m)}.\] Since \(m<0\), we have \(-m>0\). Hence Proposition 7 applied to \(P^{-1}\in\operatorname{Pic}^0(A)\) gives
\[H^{g-i}(A,P^{-1}\otimes L^{\otimes (-m)})=0\] whenever \(g-i>0\), i.e. whenever \(i<g\). Therefore
\[H^i(A,P\otimes L^{\otimes m})=0\] for all \(i<g\). ◻
Proposition 9 (Untwisted case). Let \(P\in\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\). Then
\[H^i(A,P)=0\] for all \(i\).
Proof. This is Theorem 4. ◻
Theorem 10. Let \((A,L)\) be a polarized abelian variety of dimension \(g\geq 1\) over an algebraically closed field \(k\) of characteristic zero. If \[P\in \operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\},\] then \(P\) is ACM with respect to \(L\).
Proof. Let \(m\in\mathbb{Z}\). We consider three cases.
If \(m>0\), then by Proposition 7,
\[H^i(A,P\otimes L^{\otimes m})=0\] for every \(i>0\), hence certainly for \(0<i<g\).
If \(m<0\), then by Proposition 8,
\[H^i(A,P\otimes L^{\otimes m})=0\] for every \(i<g\), hence for \(0<i<g\).
If \(m=0\), then Proposition 9 gives
\[H^i(A,P)=0\] for all \(i\).
Therefore \(P\) is ACM with respect to \(L\). ◻
The preceding argument gives more than the ACM property.
Theorem 11 (Cohomology table). Let \((A,L)\) be a polarized abelian variety of dimension \(g\), and let
\[P\in\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}.\] Let
\[h=\chi(A,L)=\frac{(L^g)}{g!}.\] Then:
If \(m>0\), then
\[H^i(A,P\otimes L^{\otimes m})=0\] for \(i>0\), and
\[h^0(A,P\otimes L^{\otimes m})=m^g h.\]
If \(m=0\), then
\[H^i(A,P)=0\] for every \(i\).
If \(m<0\), then
\[H^i(A,P\otimes L^{\otimes m})=0\] for \(i<g\), and
\[h^g(A,P\otimes L^{\otimes m})=(-m)^g h.\]
Proof. The vanishings for \(m>0\), \(m=0\), and \(m<0\) are respectively Proposition 7, Theorem 4, and Proposition 8.
It remains to compute the dimensions of the nonzero groups. By Theorem 1,
\[\chi(A,P\otimes L^{\otimes m})=m^g h.\]
If \(m>0\), then all higher cohomology vanishes, so
\[h^0(A,P\otimes L^{\otimes m}) = \chi(A,P\otimes L^{\otimes m}) = m^g h.\]
If \(m<0\), then all cohomology groups except possibly \(H^g\) vanish. Therefore
\[\chi(A,P\otimes L^{\otimes m}) = (-1)^g h^g(A,P\otimes L^{\otimes m}).\] On the other hand,
\[\chi(A,P\otimes L^{\otimes m})=m^g h=(-1)^g(-m)^g h.\] Thus
\[h^g(A,P\otimes L^{\otimes m})=(-m)^g h.\] ◻
Corollary 3. For \(P\in\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\), the only possible nonzero cohomology groups of \(P\otimes L^{\otimes m}\) are:
\[H^0(A,P\otimes L^{\otimes m})\quad (m>0)\] and
\[H^g(A,P\otimes L^{\otimes m})\quad (m<0).\] For \(m=0\), all cohomology groups vanish.
Corollary 4. If \(g\geq 2\), then \(\mathcal{O}_A\) is not ACM with respect to any polarization \(L\).
Proof. Since \(\dim H^1(A,\mathcal{O}_A)=g\), we have
\[h^1(A,\mathcal{O}_A)=g>0.\] If \(g\geq 2\), then \(1\) lies in the intermediate range \(0<1<g\). Thus
\[H^1(A,\mathcal{O}_A\otimes L^{\otimes 0})=H^1(A,\mathcal{O}_A)\neq 0,\] so \(\mathcal{O}_A\) is not ACM. ◻
Corollary 5. Let \(g\geq 2\). For \(P\in\operatorname{Pic}^0(A)\), the following are equivalent:
\(P\) is ACM with respect to \(L\);
\(P\neq \mathcal{O}_A\).
Thus \(\mathcal{O}_A\) is the unique non-ACM element of \(\operatorname{Pic}^0(A)\).
Proof. If \(P\neq \mathcal{O}_A\), then \(P\) is ACM by Theorem 10. If \(P=\mathcal{O}_A\), then \(P\) is not ACM by Corollary 4. ◻
Corollary 6. The punctured dual abelian variety
\[\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\] parametrizes a \(g\)-dimensional family of rank-one ACM line bundles on \((A,L)\).
Proof. The dual abelian variety \(\widehat A=\operatorname{Pic}^0(A)\) has dimension \(g\). Removing one closed point does not change the dimension. By Theorem 10, every remaining point corresponds to an ACM line bundle. ◻
Corollary 7. Every \(P\in\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\) is a slope-stable ACM line bundle with respect to polarization \(L\).
Proof. Every line bundle is slope-stable: a line bundle has rank one, and there is no proper nonzero subsheaf of smaller rank that can destabilize it in the sense of slope stability. The ACM property follows from Theorem 10. ◻
Remark 12. In particular, the construction above shows that there exist slope-stable ACM bundles on every polarized abelian variety \((A,L)\) of dimension \(g\geq 2\). More precisely, the punctured dual abelian variety \(\mathop{\mathrm{Pic^0}}(A) \setminus \{\mathcal{O}_A\}\) parametrizes a \(g\)-dimensional family of rank-one slope-stable ACM line bundles with respect to any polarization \(L\).
We now analyze the Ulrich condition for the line bundles above.
Theorem 13. Let \((A,L)\) be a polarized abelian variety of dimension \(g\geq 1\), and let
\[P\in\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}.\] Then \(P\) is not Ulrich with respect to \(L\).
Proof. If \(P\) were Ulrich, then by definition
\[H^i(A,P\otimes L^{-1})=0\] for all \(i\). But by Theorem 11, with \(m=-1\),
\[h^g(A,P\otimes L^{-1})=\chi(A,L)>0.\] Thus \(H^g(A,P\otimes L^{-1})\neq 0\). Hence \(P\) is not Ulrich. ◻
Theorem 14. Let \(P\in\operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\), and let \(a\in\mathbb{Z}\). Consider
\[M_a=P\otimes L^{\otimes a}.\] Then:
If \(g=1\), then \(M_a\) is Ulrich if and only if \(a=1\).
If \(g\geq 2\), then \(M_a\) is never Ulrich.
Proof. The Ulrich condition for \(M_a\) is
\[H^i(A,M_a\otimes L^{-j})=0\] for every \(i\) and every \(1\leq j\leq g\). Since
\[M_a\otimes L^{-j}=P\otimes L^{\otimes(a-j)},\] Theorem 11 shows that this sheaf is acyclic if and only if
\[a-j=0.\] Indeed, if \(a-j>0\), then \(H^0\neq0\); if \(a-j<0\), then \(H^g\neq0\); and if \(a-j=0\), then the sheaf is \(P\), which is acyclic.
When \(g=1\), the only required value is \(j=1\), so the condition is \(a-1=0\), hence \(a=1\).
When \(g\geq 2\), the Ulrich condition would require
\[a-j=0\] simultaneously for all \(j=1,\dots,g\), which is impossible. Therefore no \(M_a\) is Ulrich. ◻
Definition 5. Let \((A, L)\) be a polarized projective variety over a field \(k\). The ACM complexity of the polarized variety \((A, L)\), denoted \(c_{\text{ACM}}(A, L)\), is defined as the minimal rank of an Arithmetically Cohen-Macaulay (ACM) vector bundle on \(A\) with respect to the polarization \(L\). That is, \[c_{\text{ACM}}(A, L) = \min \{ \text{rank}(\mathcal{E}) \mid \mathcal{E} \text{ is an ACM vector bundle on } (A, L) \}.\] If no such vector bundle exists, we set \(c_{\text{ACM}}(A, L) = \infty\).
Corollary 8. Let \((A, L)\) be a polarized abelian variety of dimension \(g \ge 1\) over an algebraically closed field \(k\) of characteristic zero. Then the ACM complexity of \((A, L)\) is exactly \(1\).
Proof. By Definition 5, it suffices to produce an Arithmetically Cohen-Macaulay vector bundle on \((A, L)\) of rank \(1\). Let \(\text{Pic}^0(A)\) denote the dual abelian variety parametrizing numerically trivial line bundles on \(A\).
Choose any non-trivial line bundle \(P \in \text{Pic}^0(A) \setminus \{\mathcal{O}_A\}\). By Theorem 10, \(P\) is ACM with respect to \(L\). Because \(P\) is a line bundle, its rank is \(1\). Since the rank of any non-zero vector bundle is a strictly positive integer, the minimal possible rank is achieved, yielding \(c_{\text{ACM}}(A, L) = 1\). ◻
Fix a polarized abelian variety \((A,L)\) of dimension \(g\geq 2\) over an algebraically closed field \(k\) of characteristic zero, and fix a nontrivial numerically trivial line bundle \[P\in\mathop{\mathrm{Pic^0}}(A)\setminus\{\mathcal{O}_A\}.\]
Lemma 5 (Extension groups). \[\operatorname{Ext}^1_A(P,P)\simeq H^1(A,\mathcal{O}_A)\simeq k^g.\] In particular, \(\dim_k\operatorname{Ext}^1_A(P,P)=g\geq 2\), so nontrivial self-extensions of \(P\) exist.
Proof. Since \(P\) is a line bundle, \(\operatorname{Hom}(P,P)\simeq P^{-1}\otimes P\simeq\mathcal{O}_A\). The local-to-global spectral sequence degenerates at the \(E_2\)-page because \(P\) is locally free, yielding \[\operatorname{Ext}^1_A(P,P)\simeq H^1(A,\operatorname{Hom}(P,P))\simeq H^1(A,\mathcal{O}_A).\] The classical fact \(H^1(A,\mathcal{O}_A)\simeq T_0\widehat{A}\simeq k^g\) completes the proof. ◻
Any nonzero class \(\xi\in\operatorname{Ext}^1_A(P,P)\) corresponds to a non-split short exact sequence \[\label{eq:ext} 0\longrightarrow P\xrightarrow{\iota}E\xrightarrow{\pi}P\longrightarrow 0.\tag{1}\]
Lemma 6 (Rank and first Chern class). The middle term \(E\) in 1 is a vector bundle of rank \(2\) with \(c_1(E)=0\) in \(\mathrm{NS}(A)\).
Proof. Rank is additive: \(\mathrm{rk}(E)=\mathrm{rk}(P)+\mathrm{rk}(P)=2\). By additivity of the first Chern class and the fact that \(c_1(P)=0\) (Lemma 2.3), \[c_1(E)=c_1(P)+c_1(P)=0.\] ◻
Proposition 15 (Indecomposability of \(E\)). The bundle \(E\) is indecomposable.
Proof. Suppose for contradiction that \(E\simeq F_1\oplus F_2\) for nonzero locally free sheaves \(F_1,F_2\).
Step 1. Since \(\mathrm{rk}(F_1)+\mathrm{rk}(F_2)=\mathrm{rk}(E)=2\) and both ranks are positive integers, we must have \(\mathrm{rk}(F_1)=\mathrm{rk}(F_2)=1\), so \(F_1,F_2\) are line bundles.
Step 2: Both summands have degree zero with respect to \(L\). We first establish that \(E\) is semistable of slope zero with respect to \(L\). Since \(P \in \operatorname{Pic}^0(A)\), we have \(\mu_L(P) = 0\), and a line bundle is always slope-stable. Suppose for contradiction that \(E\) admits a destabilizing subbundle \(F \hookrightarrow E\) with \(\mu_L(F) > 0\). Composing with the surjection \(\pi \colon E \to P\) yields a morphism \(F \to P\).
If this morphism is nonzero, then since \(P\) is stable of slope \(0\) and \(\mu_L(F) > 0 = \mu_L(P)\), we obtain a contradiction: a nonzero morphism between semistable bundles requires the slope of the source to be at most the slope of the target.
If this morphism is zero, then \(F \subseteq \ker(\pi) = \operatorname{im}(\iota) \cong P\). Since \(P\) is a line bundle, \(F = P\) and \(\mu_L(F) = 0\), contradicting \(\mu_L(F) > 0\).
In both cases we have a contradiction. Therefore \(E\) is semistable of slope \(\mu_L(E) = 0\).
Now, since \(F_1\) and \(F_2\) are subbundles of the semistable bundle \(E\), semistability requires \[\mu_L(F_1) \leq 0 \quad \text{and} \quad \mu_L(F_2) \leq 0.\] On the other hand, slope is additive under direct sums weighted by rank: \[\mu_L(E) \;=\; \frac{\operatorname{rk}(F_1)\cdot \mu_L(F_1) + \operatorname{rk}(F_2)\cdot \mu_L(F_2)}{\operatorname{rk}(F_1) + \operatorname{rk}(F_2)} = 0.\] Since both \(\mu_L(F_i) \leq 0\) and their weighted average is \(0\), we conclude \[\mu_L(F_1) = \mu_L(F_2) = 0,\] that is, \(F_1\) and \(F_2\) are degree-zero line bundles with respect to \(L\).
Step 3. Consider the decomposition \[\operatorname{Hom}(E,P)=\operatorname{Hom}(F_1\oplus F_2,P)\simeq\operatorname{Hom}(F_1,P)\oplus\operatorname{Hom}(F_2,P).\] Since the surjection \(\pi:E\to P\) is a non-zero element in \(\operatorname{Hom}(E,P)\). Thus exactly both summands can’t be zero. Without loss of generality assume \(\pi|_{F_1}\neq 0\).
Now \(\pi|_{F_1}:F_1\to P\) is a nonzero morphism of degree zero line bundles. Hence \(\pi|_{F_1}:F_1\xrightarrow{\sim}P\) is an isomorphism.
Step 4. Define the section \[s:P\xrightarrow{(\pi|_{F_1})^{-1}}F_1\xhookrightarrow{\iota_{F_1}}F_1\oplus F_2=E,\] where \(\iota_{F_1}\) is the inclusion of the first summand. Explicitly, \(s(p)=(\phi^{-1}(p),0)\) with \(\phi=\pi|_{F_1}\). Then \[(\pi\circ s)(p)=\pi(\phi^{-1}(p),0)=\pi|_{F_1}(\phi^{-1}(p))+\pi|_{F_2}(0)=\phi(\phi^{-1}(p))=p.\] Thus \(\pi\circ s=\mathrm{id}_P\), so \(s\) is a right splitting of 1 . This implies that the extension class \(\xi=0\) in \(\operatorname{Ext}^1_A(P,P)\), contradicting the choice of a non-split extension.
Therefore the assumption \(E\simeq F_1\oplus F_2\) is false, and \(E\) is indecomposable. ◻
Lemma 7 (ACM: positive twists). For all \(m\geq 1\) and all \(i>0\), \[H^i(A,E\otimes L^{\otimes m})=0.\] In particular, the ACM condition holds for \(m>0\).
Proof. Tensor 1 with \(L^{\otimes m}\): \[0\longrightarrow P\otimes L^{\otimes m}\longrightarrow E\otimes L^{\otimes m}\longrightarrow P\otimes L^{\otimes m}\longrightarrow 0.\] The associated long exact sequence contains, for each \(i\geq 1\), \[H^{i}(A,P\otimes L^{\otimes m})\longrightarrow H^i(A,E\otimes L^{\otimes m})\longrightarrow H^i(A,P\otimes L^{\otimes m}).\] By Lemma 2.4, \(P\otimes L^{\otimes m}\) is ample for \(m\geq 1\). Kodaira–Nakano vanishing Theorem gives \[H^i(A,P\otimes L^{\otimes m})=0\quad\text{for all }i>0.\] Thus both flanking terms vanish, so \(H^i(A,E\otimes L^{\otimes m})=0\) for all \(i>0\). ◻
Lemma 8 (ACM: untwisted case). For all \(0\leq i\leq g\), \[H^i(A,E)=0.\]
Proof. Apply the long exact sequence of 1 at \(m=0\): \[H^{i}(A,P)\longrightarrow H^i(A,E)\longrightarrow H^i(A,P).\] Mumford’s vanishing theorem (Theorem 4) says \(H^i(A,P)=0\) for all \(0\leq i\leq g\). Hence \(H^i(A,E)=0\) for all \(0\leq i\leq g\). ◻
Lemma 9 (ACM: negative twists). For all \(m\leq -1\) and all \(0<i<g\), \[H^i(A,E\otimes L^{\otimes m})=0.\]
Proof. By Serre duality (Corollary 1), \[H^i(A,E\otimes L^{\otimes m})^\vee\simeq H^{g-i}(A,E^\vee\otimes L^{-m}).\] Set \(j=g-i\) (so \(0<j<g\)). Since \(m\leq -1\), we have \(-m\geq 1\). The dual of 1 is the non-split extension \[0\longrightarrow P^{-1}\longrightarrow E^\vee\longrightarrow P^{-1}\longrightarrow 0\] (with \(P^{-1}\in\mathop{\mathrm{Pic^0}}(A)\setminus\{\mathcal{O}_A\}\)). Tensoring with \(L^{-m}\) (ample) and repeating the positive-twist argument shows \[H^j(A,E^\vee\otimes L^{-m})=0\quad\text{for all }j>0.\] Thus \(H^i(A,E\otimes L^{\otimes m})^\vee=0\), so the desired vanishing holds. ◻
Theorem 16 (Existence of rank-2 indecomposable ACM bundles). Let \((A,L)\) be a polarized abelian variety of dimension \(g\geq 2\). For every \(P\in\mathop{\mathrm{Pic^0}}(A)\setminus\{\mathcal{O}_A\}\) and every nonzero class \(\xi\in\operatorname{Ext}^1_A(P,P)\simeq k^g\), the middle term \(E\) of the corresponding non-split extension 1 is an indecomposable rank-\(2\) vector bundle with \(c_1(E)=0\) that is arithmetically Cohen–Macaulay with respect to \(L\).
Proof. Rank and \(c_1\) follow from the second lemma, indecomposability from the proposition above, and the ACM property from the three vanishing lemmas (positive, untwisted, and negative twists). ◻
Corollary 9 (Moduli dimension). For each fixed \(P\in\mathop{\mathrm{Pic^0}}(A)\setminus\{\mathcal{O}_A\}\), the projective space \(\mathbb{P}(\operatorname{Ext}^1_A(P,P))\simeq\mathbb{P}^{g-1}\) parametrizes a \((g-1)\)-dimensional family of pairwise non-isomorphic indecomposable rank-\(2\) ACM bundles with \(c_1=0\). Varying \(P\) over the \(g\)-dimensional variety \(\mathop{\mathrm{Pic^0}}(A)\setminus\{\mathcal{O}_A\}\) yields a \((2g-1)\)-dimensional family.
Proof. Scalar multiples of a fixed class \(\xi\) yield isomorphic extensions (since \(\mathrm{Aut}(P)\simeq k^\times\)). The total dimension is \(\dim\mathop{\mathrm{Pic^0}}(A)+\dim\mathbb{P}^{g-1}=g+(g-1)=2g-1\). ◻
In this subsection we construct indecomposable Arithmetically Cohen-Macaulay (ACM) bundles of every rank \(r \ge 2\) by induction, using iterated non-split extensions.
Definition 6 (Iterated Extension Sequence). Define a sequence of vector bundles \(E_r\) (for \(r \ge 1\)) inductively as follows:
Base case: \(E_1 := P\).
Inductive step: For \(r \ge 2\), given \(E_{r-1}\), choose a nonzero extension class \(\xi_r \in \operatorname{Ext}^1_A(P, E_{r-1}) \setminus \{0\}\) (whose existence is guaranteed by Lemma 12 below). Let \(E_r\) be the middle term of the corresponding non-split short exact sequence: \[\label{eq:ses} 0 \longrightarrow E_{r-1} \xrightarrow{\iota_r} E_r \xrightarrow{\pi_r} P \longrightarrow 0.\tag{2}\]
Lemma 10 (Chern Data of \(E_r\)). For every \(r \ge 1\), \(\operatorname{rk}(E_r) = r\) and \(c_1(E_r) = 0\) in \(N^1(A)_\mathbb{Q}\).
Proof. We proceed by induction on \(r\).
Base case (\(r=1\)): By definition, \(E_1 = P\). Since \(P\) is a line bundle, \(\operatorname{rk}(E_1) = 1\). Because \(P \in \operatorname{Pic}^0(A)\), its first Chern class is numerically trivial, so \(c_1(E_1) = 0\).
Inductive step: Assume \(\operatorname{rk}(E_{r-1}) = r-1\) and \(c_1(E_{r-1}) = 0\). In any short exact sequence of locally free sheaves, rank and first Chern class are additive (see, e.g., Hartshorne, Appendix A). Applying this to 2 yields: \[\begin{align} \operatorname{rk}(E_r) &= \operatorname{rk}(E_{r-1}) + \operatorname{rk}(P) = (r-1) + 1 = r, \\ c_1(E_r) &= c_1(E_{r-1}) + c_1(P) = 0 + 0 = 0. \end{align}\] This completes the induction. ◻
Lemma 11 (Total Cohomology Vanishing at \(m=0\)). For every \(r \ge 1\), \(H^i(A, E_r) = 0\) for all \(0 \le i \le g\).
Proof. We proceed by induction on \(r\).
Base case (\(r=1\)): \(E_1 = P \in \operatorname{Pic}^0(A) \setminus \{\mathcal{O}_A\}\). By Mumford’s Vanishing Theorem (Theorem 4), \(H^i(A, P) = 0\) for all \(0 \le i \le g\).
Inductive step: Assume \(H^i(A, E_{r-1}) = 0\) for all \(0 \le i \le g\). The long exact cohomology sequence associated to 2 at degree \(i\) is: \[\cdots \longrightarrow H^i(A, E_{r-1}) \longrightarrow H^i(A, E_r) \longrightarrow H^i(A, P) \longrightarrow \cdots\] The left term is zero by the inductive hypothesis, and the right term is zero by Mumford’s Vanishing Theorem. Hence, \(H^i(A, E_r) = 0\) for all \(0 \le i \le g\). ◻
Lemma 12 (Existence of Non-split Inductive Extensions). Let \((A, L)\) be a polarized abelian variety of dimension \(g \ge 2\) over an algebraically closed field \(k\) of characteristic zero, and let \(P \in \mathrm{Pic}^0(A) \setminus \{\mathcal{O}_A\}\). For every \(r \ge 2\), \[\mathrm{Ext}^1_A(P, E_{r-1}) \cong H^1(A, P^\vee \otimes E_{r-1}) \neq 0.\]
Proof. The proof proceeds in six steps.
Step 1: The \(\operatorname{Ext}\)–\(H^1\) isomorphism.
Since \(P\) is locally free of rank one, all higher local \(\mathcal{E}xt\) sheaves vanish: \[\mathcal{E}xt^q_{\mathcal{O}_A}(P, E_{r-1}) = 0 \quad \text{for all } q > 0,\] and \(\mathcal{H}om(P, E_{r-1}) \cong P^\vee \otimes E_{r-1}\). The local-to-global Ext spectral sequence therefore gives canonical isomorphisms \[\operatorname{Ext}^i_A(P, E_{r-1}) \;\cong\; H^i(A,\, P^\vee \otimes E_{r-1}) \quad \text{for all } i \geq 0.\] In particular, \[\operatorname{Ext}^1_A(P, E_{r-1}) \;\cong\; H^1(A,\, P^\vee \otimes E_{r-1}).\] It suffices to prove \(H^1(A, P^\vee \otimes E_{r-1}) \neq 0\).
Step 2: The unipotent bundle \(U_s\).
Define \(U_s := P^\vee \otimes E_s\) for each \(s \geq 1\). We claim that \(U_s\) is a unipotent vector bundle of rank \(s\), i.e.it admits a filtration \[0 = U_s^{(0)} \subset U_s^{(1)} \subset \cdots \subset U_s^{(s)} = U_s\] with each successive quotient isomorphic to \(\mathcal{O}_A\).
We verify this by induction on \(s\).
Base case (\(s=1\)): \(U_1 = P^\vee \otimes P \cong \mathcal{O}_A\).
Inductive step (\(s \geq 2\)): Tensoring 2 (for index \(s\)) with the locally free sheaf \(P^\vee\) preserves exactness, giving \[\label{eq:Us-seq} 0 \;\to\; U_{s-1} \;\to\; U_s \;\to\; \mathcal{O}_A\;\to\; 0.\tag{3}\] By induction \(U_{s-1}\) is unipotent of rank \(s-1\), so \(U_s\) is unipotent of rank \(s\).
Step 3: Fourier–Mukai transform of \(U_s\).
Claim 17. Let \(U_s := P^{\vee} \otimes E_s\) for each \(s \ge 1\). Then there exists an isomorphism \[\Phi_{\mathcal{P}}(U_s) \simeq F_s[-g]\] in \(D^b(\operatorname{Coh} \hat{A})\), where \(F_s\) is a non-zero coherent sheaf on \(\hat{A}\) of finite length \(s\) whose set-theoretic support is \(\{0_{\hat{A}}\}\).
Proof. We argue by induction on \(s\).
\(U_1 = P^{\vee} \otimes P \cong \mathcal{O}_A\). The Fourier–Mukai transform of the structure sheaf of an abelian variety is the skyscraper sheaf at the origin of the dual (with a cohomological shift): \[\Phi_{\mathcal{P}}(\mathcal{O}_A) \simeq k(0_{\hat{A}})[-g].\] Thus \(F_1 := k(0_{\hat{A}})\) has length \(1\) and support \(\{0_{\hat{A}}\}\).
Tensoring the exact sequence \(0 \to E_{s-1} \to E_s \to P \to 0\) with the flat sheaf \(P^{\vee}\) yields \[0 \longrightarrow U_{s-1} \longrightarrow U_s \longrightarrow \mathcal{O}_A \longrightarrow 0.\] Applying the exact functor \(\Phi_{\mathcal{P}}\) gives a distinguished triangle in \(D^b(\hat{A})\): \[\Phi_{\mathcal{P}}(U_{s-1}) \longrightarrow \Phi_{\mathcal{P}}(U_s) \longrightarrow \Phi_{\mathcal{P}}(\mathcal{O}_A) \longrightarrow \Phi_{\mathcal{P}}(U_{s-1})[1].\] By the induction hypothesis, this is \[F_{s-1}[-g] \longrightarrow B \longrightarrow k(0_{\hat{A}})[-g] \longrightarrow F_{s-1}[-g+1], \] where \(B := \Phi_{\mathcal{P}}(U_s)\).
Since \(F_{s-1}[-g]\) and \(k(0_{\hat{A}})[-g]\) are concentrated in cohomological degree \(g\), the long exact sequence of cohomology sheaves associated to (4) has the following shape: \[\cdots \to 0 \to H^q(B) \to 0 \to \cdots \quad\text{for } q \le g-1 \text{ and } q \ge g+2,\] which forces \(H^q(B)=0\) in those ranges. At \(q=g+1\) the relevant segment is \[k(0_{\hat{A}}) \to 0 \to H^{g+1}(B) \to 0,\] and exactness at \(H^{g+1}(B)\) gives \(H^{g+1}(B)=0\). At \(q=g\) we obtain the short exact sequence \[0 \to F_{s-1} \to H^g(B) \to k(0_{\hat{A}}) \to 0. \] Thus \(H^q(B)=0\) for all \(q \neq g\).
Since \(B\) has cohomology only in degree \(g\), it is quasi-isomorphic to its sole cohomology sheaf placed in that degree: \[B \simeq H^g(B)[-g].\] Define \(F_s := H^g(B)\). Then \(B \simeq F_s[-g]\), i.e. \[\Phi_{\mathcal{P}}(U_s) \simeq F_s[-g].\]
From the short exact sequence (5) we obtain \[0 \longrightarrow F_{s-1} \longrightarrow F_s \longrightarrow k(0_{\hat{A}}) \longrightarrow 0.\] Additivity of length gives \[\ell(F_s) = \ell(F_{s-1}) + \ell\bigl(k(0_{\hat{A}})\bigr) = (s-1) + 1 = s.\] Moreover, \[\operatorname{Supp}(F_s) \subseteq \operatorname{Supp}(F_{s-1}) \cup \operatorname{Supp}\bigl(k(0_{\hat{A}})\bigr) = \{0_{\hat{A}}\}.\] Since \(s \ge 1\), we have \(F_s \neq 0\), and therefore \(\operatorname{Supp}(F_s) = \{0_{\hat{A}}\}\).
This completes the induction. ◻
Step 4: Cohomology via the derived fiber formula.
The derived fiber of \(\Phi_{\mathcal{P}}(\mathcal{G})\) at a closed point \(\beta \in \widehat{A}\) is \[\Phi_{\mathcal{P}}(\mathcal{G}) \otimes^{\mathbf{L}}_{\mathcal{O}_{\widehat{A}}} k(\beta) \;\simeq\; R\Gamma(A,\, \mathcal{G} \otimes P_\beta).\] Taking \(\beta = 0_{\widehat{A}}\) and using \(P_0 \cong \mathcal{O}_A\): \[R\Gamma(A,\, \mathcal{G}) \;\simeq\; \Phi_{\mathcal{P}}(\mathcal{G}) \otimes^{\mathbf{L}}_{\mathcal{O}_{\widehat{A}}} k(0_{\widehat{A}}).\] Applying this to \(\mathcal{G} = U_{r-1}\) and substituting Claim 17: \[R\Gamma(A,\, U_{r-1}) \;\simeq\; \mathcal{F}_{r-1}[-g] \otimes^{\mathbf{L}}_{\mathcal{O}_{\widehat{A}}} k(0_{\widehat{A}}).\] Let \(R := \mathcal{O}_{\widehat{A}, 0}\) be the local ring at \(0_{\widehat{A}}\) and \(\kappa := R/\mathfrak{m}_0 \cong k\) its residue field. Since \(\mathcal{F}_{r-1}\) has finite-length support at \(0_{\widehat{A}}\), it corresponds to a finite-length \(R\)-module, and the derived tensor product is computed entirely over \(R\): \[R\Gamma(A,\, U_{r-1}) \;\simeq\; \mathcal{F}_{r-1} \otimes^{\mathbf{L}}_R \kappa\,[-g].\] Passing to cohomology: \[\label{eq:Hi-Tor} H^i(A,\, U_{r-1}) \;\cong\; \mathop{\mathrm{Tor}}^R_{g-i}(\mathcal{F}_{r-1},\, \kappa) \quad \text{for all } i.\tag{4}\] Setting \(i = 1\): \[H^1(A,\, P^\vee \otimes E_{r-1}) \;=\; H^1(A,\, U_{r-1}) \;\cong\; \mathop{\mathrm{Tor}}^R_{g-1}(\mathcal{F}_{r-1},\, \kappa).\]
Step 5: Nonvanishing via the Auslander–Buchsbaum formula and the minimal free resolution.
Since \(\widehat{A}\) is a smooth abelian variety of dimension \(g\), the local ring \(R = \mathcal{O}_{\widehat{A}, 0}\) is a regular local ring of dimension \(g\). The module \(\mathcal{F}_{r-1}\) is a nonzero finitely generated \(R\)-module of finite length \(\ell(\mathcal{F}_{r-1}) = r - 1 \geq 1\). A nonzero finite-length module over a positive-dimensional local ring has depth zero: \[\mathop{\mathrm{depth}}_R(\mathcal{F}_{r-1}) = 0.\] Since \(R\) is regular local (hence Cohen–Macaulay), every finitely generated \(R\)-module has finite projective dimension, so the Auslander–Buchsbaum formula applies: \[\mathop{\mathrm{pd}}_R(\mathcal{F}_{r-1}) + \mathop{\mathrm{depth}}_R(\mathcal{F}_{r-1}) = \mathop{\mathrm{depth}}(R) = g,\] giving \(\mathop{\mathrm{pd}}_R(\mathcal{F}_{r-1}) = g\).
Let \[0 \;\to\; R^{\beta_g} \;\xrightarrow{d_g}\; R^{\beta_{g-1}} \;\to\; \cdots \;\to\; R^{\beta_1} \;\xrightarrow{d_1}\; R^{\beta_0} \;\to\; \mathcal{F}_{r-1} \;\to\; 0\] be the minimal free resolution of \(\mathcal{F}_{r-1}\) over \(R\). Minimality means every differential \(d_j\) has its matrix entries in \(\mathfrak{m}_0\). In particular, since \(\mathop{\mathrm{pd}}_R(\mathcal{F}_{r-1}) = g\), we have \[\beta_g \;\neq\; 0.\] Exactness of the resolution at the leftmost term \(R^{\beta_g}\) requires the map \(d_g : R^{\beta_g} \to R^{\beta_{g-1}}\) to be injective. If \(\beta_{g-1} = 0\), then \(d_g\) maps into \(0\), so it cannot be injective unless \(R^{\beta_g} = 0\), contradicting \(\beta_g \neq 0\). Therefore: \[\beta_{g-1} \;\neq\; 0.\] Since the resolution is minimal, tensoring with \(\kappa = R/\mathfrak{m}_0\) kills all differentials, yielding \[\mathop{\mathrm{Tor}}^R_j(\mathcal{F}_{r-1},\, \kappa) \;\cong\; \kappa^{\beta_j} \quad \text{for all } 0 \leq j \leq g.\] In particular, \[\mathop{\mathrm{Tor}}^R_{g-1}(\mathcal{F}_{r-1},\, \kappa) \;\cong\; \kappa^{\beta_{g-1}} \;\neq\; 0.\]
Step 6: Conclusion.
Combining the results of Steps 1–5: \[\operatorname{Ext}^1_A(P,\, E_{r-1}) \;\cong\; H^1(A,\, P^\vee \otimes E_{r-1}) \;=\; H^1(A,\, U_{r-1}) \;\cong\; \mathop{\mathrm{Tor}}^R_{g-1}(\mathcal{F}_{r-1},\, \kappa) \;\neq\; 0.\] Hence there exists a nonzero class \(\xi_r \in \operatorname{Ext}^1_A(P, E_{r-1}) \setminus \{0\}\), and the corresponding non-split extension \[0 \;\to\; E_{r-1} \;\to\; E_r \;\to\; P \;\to\; 0\] exists at every inductive step \(r \geq 2\). ◻
Proposition 18. Let \(A\) be an abelian variety of dimension \(g \ge 2\) over an algebraically closed field \(k\) of characteristic zero, and let \(P \in \operatorname{Pic}^0(A) \setminus \{\mathcal{O}_A\}\). For every integer \(r \ge 1\), the vector space of sheaf homomorphisms from \(E_r\) to \(P\) is one-dimensional. Equivalently, \[\dim_k \operatorname{Hom}_A(E_r, P) = 1 \quad \text{or} \quad h^0(A, E_r^\vee \otimes P) = 1.\]
Proof. We proceed by induction on the rank \(r\).
Base Case (\(r = 1\)): By definition, \(E_1 = P\). Since \(P\) is a line bundle over a proper variety, its only endomorphisms are scalar multiplications, hence \(\operatorname{Hom}_A(P, P) \cong k\). Equivalently, \(E_1^\vee \otimes P \cong P^\vee \otimes P \cong \mathcal{O}_A\). Because \(A\) is a proper scheme over an algebraically closed field, \(H^0(A, \mathcal{O}_A)\) consists strictly of constant functions, yielding \(h^0(A, \mathcal{O}_A) = 1\). The base case holds.
Inductive Step (\(r \ge 2\)): Assume by the inductive hypothesis that \(h^0(A, E_{r-1}^\vee \otimes P) = 1\).
By the construction of \(E_r\), we have a non-split short exact sequence: \[0 \longrightarrow E_{r-1} \xrightarrow{\iota_r} E_r \xrightarrow{\pi_r} P \longrightarrow 0\] Let \(\xi_r \in \operatorname{Ext}^1_A(P, E_{r-1})\) denote the associated non-zero extension class.
Applying the exact contravariant dualizing functor \(\mathcal{H}om_A(-, \mathcal{O}_A)\) and tensoring the resulting sequence with the locally free sheaf \(P\) yields the twisted dual sequence: \[0 \longrightarrow \mathcal{O}_A \longrightarrow E_r^\vee \otimes P \longrightarrow E_{r-1}^\vee \otimes P \longrightarrow 0\] where we have applied the canonical trivialization \(P^\vee \otimes P \cong \mathcal{O}_A\). Let \(c \in \operatorname{Ext}^1_A(E_{r-1}^\vee \otimes P, \mathcal{O}_A)\) be the extension class of this sequence.
Taking the long exact sequence in sheaf cohomology produces: \[0 \longrightarrow H^0(A, \mathcal{O}_A) \xrightarrow{\beta} H^0(A, E_r^\vee \otimes P) \xrightarrow{\alpha} H^0(A, E_{r-1}^\vee \otimes P) \xrightarrow{\delta} H^1(A, \mathcal{O}_A)\] Since \(A\) is proper, \(H^0(A, \mathcal{O}_A) \cong k\). By the inductive hypothesis, \(H^0(A, E_{r-1}^\vee \otimes P)\) is also a 1-dimensional \(k\)-vector space, generated by a global section \(\phi\). Under the tensor-hom adjunction, this generator \(\phi\) corresponds precisely to the canonical surjection \(\pi_{r-1} \colon E_{r-1} \to P\) bounding the preceding extension.
To determine the dimension of the middle term, we must evaluate the kernel of the connecting homomorphism \(\delta\).
Claim 19. The connecting homomorphism \(\delta \colon H^0(A, E_{r-1}^\vee \otimes P) \to H^1(A, \mathcal{O}_A)\) is an injective map.
Proof. By the inductive hypothesis, \(h^0(A, E_{r-1}^\vee \otimes P) = 1\). Under the canonical isomorphism \(H^0(A, E_{r-1}^\vee \otimes P) \cong \operatorname{Hom}_A(E_{r-1}, P)\), this \(1\)-dimensional \(k\)-vector space is generated by the non-zero boundary map \(\pi_{r-1} \colon E_{r-1} \to P\). We therefore set our generator to be \(\pi_{r-1}\).
Let \(c \in \operatorname{Ext}^1_A(E_{r-1}^\vee \otimes P, \mathcal{O}_A)\) be the extension class of the twisted sequence \(0 \to \mathcal{O}_A \to E_r^\vee \otimes P \to E_{r-1}^\vee \otimes P \to 0\). Under the Yoneda correspondence, the action of the connecting homomorphism \(\delta\) on \(\pi_{r-1}\) evaluates exactly to the pullback of this twisted extension class: \[\delta(\pi_{r-1}) = \pi_{r-1}^*c \in \operatorname{Ext}^1_A(\mathcal{O}_A, \mathcal{O}_A).\]
Because tensoring with \(P^\vee\) is an exact equivalence of categories on \(A\), it commutes with all homological operations. Consequently, the pullback of \(c\) along \(\pi_{r-1}\) is canonically isomorphic to the pullback of the dual extension class \(\xi_r^\vee \in \operatorname{Ext}^1_A(E_{r-1}^\vee, P^\vee)\) along the dualized boundary map \(\pi_{r-1}^\vee \colon P^\vee \to E_{r-1}^\vee\). Thus, we have the exact identification: \[\delta(\pi_{r-1}) \cong (\pi_{r-1}^\vee)^*\xi_r^\vee.\]
To show this class is non-trivial, observe that the contravariant functor \(\operatorname{Hom}_A(-, \mathcal{O}_A)\) transforms fibered coproducts (pushouts) into fibered products (pullbacks). Therefore, the pullback of the dual class vanishes if and only if the dual of the pushout vanishes: \[(\pi_{r-1}^\vee)^*\xi_r^\vee = 0 \iff (\pi_{r-1})_*\xi_r = 0 \in \operatorname{Ext}^1_A(P, P).\]
Lemma 13 (Choice of the inductive extension class). For each integer \(r \geq 2\), one may choose the extension class \(\xi_r \in \operatorname{Ext}^1_A(P,\, E_{r-1})\) so that the induced pushout map is nonzero: \[(\pi_{r-1})_*(\xi_r) \;\neq\; 0 \quad\text{in}\quad \operatorname{Ext}^1_A(P,\, P).\]
Proof. We reduce the geometric assertion to a strict monotonicity statement for Betti numbers of finite-length modules over a regular local ring, then derive a contradiction from the assumption of vanishing.
Setup: translation to local algebra. For each integer \(s \geq 1\) define the unipotent bundle \(U_s := P^\vee \otimes E_s\). Tensoring the defining extension \(0 \to E_{s-1} \to E_s \to P \to 0\) by the locally free sheaf \(P^\vee\) preserves exactness and gives \[\label{eq:Us-ses} 0 \;\longrightarrow\; U_{s-1} \;\longrightarrow\; U_s \;\longrightarrow\; \mathcal{O}_A \;\longrightarrow\; 0.\tag{5}\] By Claim 17, the Fourier–Mukai transform satisfies \(\Phi_{\mathcal{P}}(U_s) \simeq F_s[-g]\) in \(D^b(\widehat{A})\), where \(F_s\) is a coherent sheaf of finite length \(s\) supported scheme-theoretically at the origin \(0_{\widehat{A}}\), and the sheaves \(F_s\) fit into the short exact sequences \[\label{eq:Fs-ses} 0 \;\longrightarrow\; F_{s-1} \;\longrightarrow\; F_s \;\longrightarrow\; \kappa(0_{\widehat{A}}) \;\longrightarrow\; 0.\tag{6}\] Set \(R := \mathcal{O}_{\widehat{A},\,0}\), which is a regular local ring of dimension \(g\) with maximal ideal \(\mathfrak{m}\) and residue field \(\kappa = R/\mathfrak{m} \cong k\). We record the canonical chain of identifications \[\label{eq:chain} d_s \;:=\; \dim_k \operatorname{Ext}^1_A(P,\, E_s) \;=\; \dim_k H^1(A,\, U_s) \;=\; \dim_k \mathop{\mathrm{Tor}}_{g-1}^R(F_s,\, \kappa) \;=\; \beta_{g-1}(F_s),\tag{7}\] where the first equality is the local-to-global identification \(\operatorname{Ext}^1_A(P,E_s) \cong H^1(A, P^\vee \otimes E_s) = H^1(A, U_s)\) (valid because \(P\) is locally free of rank one), the second is the derived fiber formula of Lemma 12 at \(i = 1\) giving \(H^1(A, U_s) \cong \mathop{\mathrm{Tor}}_{g-1}^R(F_{s}, \kappa)\), and the third is the definition of the \((g-1)\)-st Betti number \(\beta_{g-1}(F_s) := \dim_\kappa \mathop{\mathrm{Tor}}_{g-1}^R(F_s, \kappa)\).
Step 1: Kernel dimension of \(\iota_*\).
Apply the covariant functor \(\operatorname{Hom}_A(P, -)\) to the short exact sequence \(0 \to E_{r-2} \to E_{r-1} \xrightarrow{\pi_{r-1}} P \to 0\). The resulting long exact sequence in \(\operatorname{Ext}^\bullet_A(P,-)\) reads \[\label{eq:les} \begin{align} \cdots \;\longrightarrow\; &\operatorname{Hom}_A(P,\, E_{r-1}) \xrightarrow{\;(\pi_{r-1})^0_*\;} \operatorname{Hom}_A(P,\, P) \xrightarrow{\;\delta_1\;} \operatorname{Ext}^1_A(P,\, E_{r-2}) \\ &\xrightarrow{\;\iota_*\;} \operatorname{Ext}^1_A(P,\, E_{r-1}) \xrightarrow{\;(\pi_{r-1})_*\;} \operatorname{Ext}^1_A(P,\, P) \;\longrightarrow\; \cdots \end{align}\tag{8}\]
Claim 20. The degree-zero pushout map \((\pi_{r-1})^0_* \colon \operatorname{Hom}_A(P,\, E_{r-1}) \to \operatorname{Hom}_A(P,\, P)\) is the zero map.
Proof of claim. By induction hypothesis of Proposition 18, we have \(h^0(A, U_s) = \dim_k \operatorname{Hom}_A(P, E_s) = 1\) for all \(s\leq (r-1)\). In particular, the restriction map \[\operatorname{Hom}_A(P,\, E_{r-2}) \;\longrightarrow\; \operatorname{Hom}_A(P,\, E_{r-1})\] is an injective linear map between two \(1\)-dimensional \(k\)-vector spaces, hence an isomorphism. By exactness of the long exact sequence 8 at \(\operatorname{Hom}_A(P, E_{r-1})\), it follows that \((\pi_{r-1})^0_* = 0\). ◻
Since \((\pi_{r-1})^0_* = 0\), the connecting map \(\delta_1\) is injective. Because \(P\) is simple, \(\operatorname{Hom}_A(P,P) \cong k\) is one-dimensional. By exactness of 8 at \(\operatorname{Ext}^1_A(P,E_{r-2})\), \[\label{eq:ker1} \ker(\iota_*) \;=\; \operatorname{im}(\delta_1), \qquad \dim_k \ker(\iota_*) \;=\; 1.\tag{9}\]
Step 2: Strict monotonicity of the Betti numbers \(\beta_{g-1}(F_s)\).
We prove the quantitative formula \[\label{eq:betti-growth} \beta_{g-1}(F_s) \;=\; \beta_{g-1}(F_{s-1}) + g \qquad \text{for all } s \geq 1,\tag{10}\] which in particular gives \(d_s = d_{s-1} + g > d_{s-1}\).
Projective dimensions. Since \(F_s\) is a nonzero finitely generated \(R\)-module of finite length, \(\mathop{\mathrm{depth}}_R(F_s) = 0\). The Auslander–Buchsbaum formula over the regular local ring \(R\) of depth \(g\) gives \[\mathop{\mathrm{pd}}_R(F_s) \;=\; g - \mathop{\mathrm{depth}}_R(F_s) \;=\; g.\] The same formula gives \(\mathop{\mathrm{pd}}_R(\kappa) = g\).
Minimality of free resolutions. Let \[0 \;\to\; R^{\beta_g} \xrightarrow{d_g} R^{\beta_{g-1}} \to \cdots \to R^{\beta_1} \xrightarrow{d_1} R^{\beta_0} \;\to\; F_s \;\to\; 0\] be the minimal free resolution of \(F_s\) over \(R\). Minimality means every matrix of \(d_j\) has entries in \(\mathfrak{m}\). Tensoring with \(\kappa = R/\mathfrak{m}\) kills all differentials, so \[\label{eq:tor-betti} \mathop{\mathrm{Tor}}_j^R(F_s,\, \kappa) \;\cong\; \kappa^{\,\beta_j(F_s)} \quad\text{for all } 0 \leq j \leq g.\tag{11}\]
Vanishing of the connecting map. Apply \(\mathop{\mathrm{Tor}}^R_\bullet(-,\kappa)\) to the short exact sequence 6 . The resulting long exact sequence at the relevant degrees is \[\label{eq:tor-les} \begin{align} \cdots \;\to\; &\mathop{\mathrm{Tor}}_g^R(F_{s-1},\kappa) \;\to\; \mathop{\mathrm{Tor}}_g^R(F_s,\kappa) \;\to\; \mathop{\mathrm{Tor}}_g^R(\kappa,\kappa) \xrightarrow{\;\partial\;} \mathop{\mathrm{Tor}}_{g-1}^R(F_{s-1},\kappa) \\ &\;\to\; \mathop{\mathrm{Tor}}_{g-1}^R(F_s,\kappa) \;\to\; \mathop{\mathrm{Tor}}_{g-1}^R(\kappa,\kappa) \;\to\; \cdots \end{align}\tag{12}\] The connecting morphism \(\partial \colon \mathop{\mathrm{Tor}}_g^R(\kappa,\kappa) \to \mathop{\mathrm{Tor}}_{g-1}^R(F_{s-1},\kappa)\) is induced, via the Yoneda pairing, by the extension class of 6 in \(\operatorname{Ext}^1_R(\kappa, F_{s-1})\). By the inductive construction in Claim 17, the sequence 6 arises as the \(\mathcal{H}^g\)-level output of the distinguished triangle \[F_{s-1}[-g] \longrightarrow \Phi_{\mathcal{P}}(U_s) \longrightarrow \kappa(0_{\widehat{A}})[-g] \longrightarrow F_{s-1}[-g+1].\] The connecting morphism in that triangle maps into \(\mathcal{H}^{g+1}(F_{s-1}[-g]) = \mathcal{H}^1(F_{s-1})\), which vanishes because \(F_{s-1}\) is a sheaf concentrated in cohomological degree zero. Under the derived fiber identification this forces \[\label{eq:partial-zero} \partial \;=\; 0.\tag{13}\]
Additivity. With \(\partial = 0\), the long exact sequence 12 splits into short exact sequences at every homological degree. At degree \(g-1\) we obtain \[0 \;\to\; \mathop{\mathrm{Tor}}_{g-1}^R(F_{s-1},\kappa) \;\to\; \mathop{\mathrm{Tor}}_{g-1}^R(F_s,\kappa) \;\to\; \mathop{\mathrm{Tor}}_{g-1}^R(\kappa,\kappa) \;\to\; 0.\] By the Koszul complex on a regular system of parameters for \(R\), \(\mathop{\mathrm{Tor}}_{g-1}^R(\kappa,\kappa) \cong \Lambda^{g-1}(k^g) \cong k^g\). Taking \(k\)-dimensions and applying 11 : \[\beta_{g-1}(F_s) \;=\; \beta_{g-1}(F_{s-1}) \;+\; g,\] which establishes 10 . In terms of the integers \(d_s = \beta_{g-1}(F_s)\) defined in 7 , \[\label{eq:ds-growth} d_s \;=\; d_{s-1} + g \;\;\Longrightarrow\;\; d_s \;>\; d_{s-1} \quad \text{for all } s \geq 1.\tag{14}\]
Step 3: Conclusion by contradiction.
Suppose for contradiction that \((\pi_{r-1})_* = 0\) for every choice of \(\xi_r \in \operatorname{Ext}^1_A(P, E_{r-1})\). Then exactness of 8 at \(\operatorname{Ext}^1_A(P, E_{r-1})\) forces \(\iota_*\) to be surjective. Applying the rank-nullity theorem and 9 : \[d_{r-1} \;=\; \dim_k \operatorname{im}(\iota_*) \;=\; d_{r-2} - \dim_k \ker(\iota_*) \;=\; d_{r-2} - 1 \;<\; d_{r-2}.\] This contradicts 14 , which asserts \(d_{r-1} = d_{r-2} + g > d_{r-2}\).
Therefore the assumption is false. The map \((\pi_{r-1})_*\) is nonzero, so one may choose \(\xi_r \in \operatorname{Ext}^1_A(P,E_{r-1})\) with \((\pi_{r-1})_*(\xi_r) \neq 0\) in \(\operatorname{Ext}^1_A(P,P)\). ◻
Since \(\delta\) is a linear map acting on a \(1\)-dimensional vector space and its evaluation on the generating morphism does not vanish, \(\ker(\delta) = 0\). The map is strictly injective.
We now apply the injectivity of \(\delta\) to the long exact sequence in cohomology associated with the twisted defining sequence \(0 \to \mathcal{O}_A \xrightarrow{\alpha} E_r^\vee \otimes P \xrightarrow{\beta} E_{r-1}^\vee \otimes P \to 0\): \[0 \to H^0(A, \mathcal{O}_A) \xrightarrow{H^0(\alpha)} H^0(A, E_r^\vee \otimes P) \xrightarrow{H^0(\beta)} H^0(A, E_{r-1}^\vee \otimes P) \xrightarrow{\delta} H^1(A, \mathcal{O}_A) \to \dots\]
Because \(\delta\) is strictly injective, \(\ker(\delta) = 0\). By exactness at \(H^0(A, E_{r-1}^\vee \otimes P)\), the image of \(H^0(\beta)\) is equal to \(\ker(\delta)\), yielding \(\operatorname{Im}(H^0(\beta)) = 0\). This implies \(H^0(\beta)\) is the zero map.
Consequently, by exactness at \(H^0(A, E_r^\vee \otimes P)\), we deduce that \(\operatorname{Im}(H^0(\alpha)) = \ker(H^0(\beta)) = H^0(A, E_r^\vee \otimes P)\), meaning \(H^0(\alpha)\) is a surjective map.
Since \(H^0(\alpha)\) is induced by the injective sheaf morphism \(\alpha \colon \mathcal{O}_A \to E_r^\vee \otimes P\), the map \(H^0(\alpha)\) is inherently injective. Therefore, \(H^0(\alpha)\) is a bijective linear map, hence an isomorphism of \(k\)-vector spaces.
Because \(A\) is an abelian variety, \(h^0(A, \mathcal{O}_A) = 1\). The isomorphism immediately yields: \[h^0(A, E_r^\vee \otimes P) = h^0(A, \mathcal{O}_A) = 1.\]
This completes the induction. ◻
Lemma 14 (Indecomposability of \(E_r\)). For every integer \(r \geq 1\), the vector bundle \(E_r\) is indecomposable.
Proof. We proceed by induction on \(r\).
By definition, \(E_1 = P\), a line bundle on \(A\). Line bundles are indecomposable. Indeed, if \(P \cong F_1 \oplus F_2\) with both \(F_i\) nonzero, then taking ranks gives \(1 = \operatorname{rk}(F_1) + \operatorname{rk}(F_2)\), which forces one of them to have rank 0, a contradiction.
Assume that for some \(r-1 \geq 1\), the bundle \(E_{r-1}\) is indecomposable.
Suppose for contradiction that \[E_r \cong F_1 \oplus F_2,\] where \(F_1\) and \(F_2\) are nonzero vector bundles on \(A\). Let \(r_i = \operatorname{rk}(F_i)\), so \(r_1 + r_2 = r\) with \(r_1, r_2 \geq 1\).
By the previous Proposition 18 on morphisms to \(P\), we have \[\operatorname{Hom}_A(E_r, P) \cong k.\] Since \(E_r \cong F_1 \oplus F_2\), it follows that \[\operatorname{Hom}_A(F_1, P) \oplus \operatorname{Hom}_A(F_2, P) \cong k.\] Because \(k\) is a field, one of the two summands must be zero. Without loss of generality, assume \[\operatorname{Hom}_A(F_2, P) = 0 \quad \text{and} \quad \operatorname{Hom}_A(F_1, P) \cong k.\]
Then the restriction \(\pi_r|_{F_2} : F_2 \to P\) is the zero map. Hence \[F_2 \subseteq \ker(\pi_r) = \operatorname{im}(\iota_r) \cong E_{r-1}.\] So we have an inclusion \(F_2 \hookrightarrow E_{r-1}\).
Let \(\operatorname{pr}_2 : E_r \to F_2\) be the projection onto the second summand. Restricting to \(E_{r-1}\) gives a morphism \[\operatorname{pr}_2|_{E_{r-1}} : E_{r-1} \to F_2,\] which is a left inverse to the inclusion \(F_2 \hookrightarrow E_{r-1}\). Thus the inclusion splits, so \(F_2\) is a direct summand of \(E_{r-1}\): \[E_{r-1} \cong F_2 \oplus Q\] for some vector bundle \(Q\).
By the inductive hypothesis, \(E_{r-1}\) is indecomposable and \(F_2 \neq 0\), so necessarily \(Q = 0\) and \(F_2 \cong E_{r-1}\). In particular, \(\operatorname{rk}(F_1) = 1\), so \(F_1\) is a line bundle.
Now consider \(\pi_r|_{F_1} : F_1 \to P\). This is a surjective morphism between line bundles, hence an isomorphism. Let \(s = (\pi_r|_{F_1})^{-1} : P \to F_1 \subset E_r\). Then \[\pi_r \circ s = \operatorname{id}_P,\] so the short exact sequence \[0 \to E_{r-1} \xrightarrow{\iota_r} E_r \xrightarrow{\pi_r} P \to 0\] splits. This implies that the extension class \(\xi_r = 0\) in \(\operatorname{Ext}^1_A(P, E_{r-1})\).
This contradicts the assumption that \(\xi_r \neq 0\) (the extension is non-split).
Therefore, the assumption that \(E_r\) decomposes nontrivially is false. Hence \(E_r\) is indecomposable.
This completes the induction. ◻
Lemma 15 (Dual of \(E_r\)). For every \(r \ge 1\), the dual bundle \(E_r^\vee\) fits into a non-split short exact sequence: \[\label{eq:dual95ses} 0 \longrightarrow P^{-1} \longrightarrow E_r^\vee \longrightarrow E_{r-1}^\vee \longrightarrow 0.\tag{15}\] Moreover, \(E_r^\vee\) is indecomposable, has \(c_1(E_r^\vee) = 0\), and satisfies \(H^i(A, E_r^\vee) = 0\) for all \(0 \le i \le g\).
Proof. Dual sequence: Applying the exact contravariant functor \(\operatorname{Hom}_A(-, \mathcal{O}_A)\) to the sequence of locally free sheaves 2 yields: \[0 \longrightarrow P^\vee \longrightarrow E_r^\vee \longrightarrow E_{r-1}^\vee \longrightarrow 0.\] Since \(P\) is a line bundle, \(P^\vee = P^{-1}\), giving 15 .
Non-splitting: The extension class of 15 is \(\xi_r^\vee \in \operatorname{Ext}^1_A(E_{r-1}^\vee, P^{-1})\). The duality isomorphism \(\operatorname{Ext}^1_A(P, E_{r-1}) \cong \operatorname{Ext}^1_A(E_{r-1}^\vee, P^{-1})\) sends \(\xi_r\) to \(\xi_r^\vee\). Since \(\xi_r \neq 0\), we have \(\xi_r^\vee \neq 0\), so 15 is non-split.
Indecomposability: Follows by the identical argument as Lemma 14, utilizing the sequence 15 and the fact that \(\operatorname{Hom}_A(E_r^\vee, P^{-1}) \cong k\) (proven analogously to Proposition 18).
Chern class and Vanishing: \(c_1(E_r^\vee) = -c_1(E_r) = 0\). The vanishing \(H^i(A, E_r^\vee) = 0\) follows identically to Lemma 11, using \(P^{-1} \in \operatorname{Pic}^0(A) \setminus \{\mathcal{O}_A\}\). ◻
Lemma 16 (ACM for \(E_r\): Positive Twists). For every \(r \ge 1\), \(m \ge 1\), and all \(i > 0\), \(H^i(A, E_r \otimes L^m) = 0\).
Proof. By induction on \(r\).
Base case (\(r=1\)): \(E_1 \otimes L^m = P \otimes L^m\). Since \(L^m\) is ample and \(P\) is numerically trivial, their tensor \(P \otimes L^m\) is ample (see Lazarsfeld, Positivity in Algebraic Geometry, Example 1.2.8). By Kodaira-Nakano vanishing (which holds in characteristic zero), \(H^i(A, P \otimes L^m) = 0\) for all \(i > 0\).
Inductive step: Assume \(H^i(A, E_{r-1} \otimes L^m) = 0\) for all \(i > 0\) and \(m \ge 1\). Tensor 2 with \(L^m\): \[0 \longrightarrow E_{r-1} \otimes L^m \longrightarrow E_r \otimes L^m \longrightarrow P \otimes L^m \longrightarrow 0.\] The long exact sequence at degree \(i > 0\) is: \[H^i(A, E_{r-1} \otimes L^m) \longrightarrow H^i(A, E_r \otimes L^m) \longrightarrow H^i(A, P \otimes L^m).\] The left term vanishes by the inductive hypothesis; the right term vanishes by Kodaira-Nakano. Hence, \(H^i(A, E_r \otimes L^m) = 0\). ◻
Lemma 17 (ACM for \(E_r\): Negative Twists). For every \(r \ge 1\), \(m \le -1\), and all \(0 < i < g\), \(H^i(A, E_r \otimes L^m) = 0\).
Proof. Fix \(m \le -1\) and \(0 < i < g\). By Serre Duality on an abelian variety (where \(\omega_A \cong \mathcal{O}_A\)): \[H^i(A, E_r \otimes L^m)^\vee \cong H^{g-i}(A, (E_r \otimes L^m)^\vee) = H^{g-i}(A, E_r^\vee \otimes L^{-m}).\] Set \(j := g - i\). Since \(0 < i < g\), we have \(0 < j < g\). Also, \(-m \ge 1\). By Lemma 15, \(E_r^\vee\) satisfies the same inductive structural properties as \(E_r\). Thus, Lemma 16 applied to \(E_r^\vee\) gives: \[H^j(A, E_r^\vee \otimes L^{-m}) = 0 \quad \text{for all } j > 0, -m \ge 1.\] Since \(j = g - i > 0\), we obtain \(H^{g-i}(A, E_r^\vee \otimes L^{-m}) = 0\). Therefore, \(H^i(A, E_r \otimes L^m)^\vee = 0\), implying \(H^i(A, E_r \otimes L^m) = 0\). ◻
Theorem 21 (Indecomposable Rank-\(r\) ACM Bundle). Let \((A, L)\) be a polarized abelian variety of dimension \(g \ge 2\) over an algebraically closed field \(k\) of characteristic zero. For every \(P \in \operatorname{Pic}^0(A) \setminus \{\mathcal{O}_A\}\) and every \(r \ge 1\), the bundle \(E_r\) of Definition 6 satisfies:
\(\operatorname{rk}(E_r) = r\) and \(E_r\) is indecomposable;
\(c_1(E_r) = 0\) in \(N^1(A)\);
\(E_r\) is ACM with respect to \(L\): \(H^i(A, E_r \otimes L^m) = 0\) for all \(m \in \mathbb{Z}\) and \(0 < i < g\).
Proof. All three parts are established by our lemmas:
Rank and indecomposability follow from Lemma 10 and Lemma 14.
The first Chern class is zero by Lemma 10.
The ACM property is verified by checking all integers \(m\):
◻
Let \(A\) be an abelian variety of dimension \(g \geq 2\) defined over an algebraically closed field, and suppose that the Picard number \(\rho(A) = 1\). Let \(H\) be a primitive ample generator of the Néron–Severi group \(\mathrm{NS}(A)\), and let \(L\) be an ample line bundle on \(A\) such that \[L \cong H^{\otimes d}\] for some integer \(d > 0\).
Theorem 22. Every line bundle \(M\) on \(A\) can be written uniquely in the form \[M \cong H^{\otimes m} \otimes \alpha,\] where \(m \in \mathbb{Z}\) and \(\alpha \in \mathrm{Pic}^0(A)\). Moreover, \(M\) is arithmetically Cohen–Macaulay with respect to \(L\) if and only if one of the following holds:
\(\alpha \not\cong \mathcal{O}_A\), or
\(\alpha \cong \mathcal{O}_A\) and \(d\) does not divide \(m\).
Equivalently, \(M\) fails to be arithmetically Cohen–Macaulay with respect to \(L\) if and only if \(M\) is isomorphic to some integer power of \(L\).
Proof. A line bundle \(M\) is said to be arithmetically Cohen–Macaulay (ACM) with respect to \(L\) if \[H^i(A, M \otimes L^{\otimes n}) = 0\] for all integers \(n\) and all \(0 < i < g\).
Since \(\rho(A) = 1\), the Néron–Severi group is free of rank one. Thus the numerical class of any line bundle \(M\) is \(m\) times the class of \(H\) for a unique integer \(m\). This means that \(M \otimes H^{-m}\) has trivial numerical class and therefore lies in \(\mathrm{Pic}^0(A)\). Setting \(\alpha = M \otimes H^{-m}\), we obtain the unique decomposition \(M \cong H^{\otimes m} \otimes \alpha\).
Now fix any integer \(n\). Then \[M \otimes L^{\otimes n} \cong H^{\otimes (m + dn)} \otimes \alpha.\] We study the cohomology groups of this bundle by considering the sign of the exponent \(m + dn\).
If \(m + dn > 0\), then \(H^{\otimes (m+dn)}\) is ample. On an abelian variety, the cohomology of an ample line bundle vanishes in all positive degrees. Hence the middle cohomology groups vanish.
If \(m + dn < 0\), then \(H^{\otimes (m+dn)}\) is anti-ample. By Serre duality (noting that the canonical bundle of \(A\) is trivial), the cohomology is concentrated in degree \(g\). Again, the groups \(H^i\) for \(0 < i < g\) vanish.
The remaining case is \(m + dn = 0\). Here \(M \otimes L^{\otimes n} \cong \alpha\).
If \(\alpha \not\cong \mathcal{O}_A\), then \(\alpha\) is a nontrivial line bundle of degree zero, and all its cohomology groups vanish. So the required vanishing still holds.
If instead \(\alpha \cong \mathcal{O}_A\), then \(M \otimes L^{\otimes n} \cong \mathcal{O}_A\). But on an abelian variety of dimension \(g \geq 2\), the structure sheaf has nontrivial cohomology in every degree between 0 and \(g\). In particular, there exists some \(i\) with \(0 < i < g\) such that \(H^i(A, \mathcal{O}_A) \neq 0\). Therefore \(M\) is not ACM with respect to \(L\).
Putting these cases together, \(M\) is ACM precisely when there is no integer \(n\) satisfying both \(m + dn = 0\) and \(\alpha \cong \mathcal{O}_A\). This is equivalent to the two conditions stated in the theorem.
The equivalent formulation follows immediately: the powers of \(L\) are exactly the line bundles for which \(\alpha \cong \mathcal{O}_A\) and \(d\) divides \(m\). ◻
Remark 23. The decomposition \(M \cong H^{\otimes m} \otimes \alpha\) relies only on the assumption that \(\rho(A) = 1\). It holds for every line bundle, regardless of the ACM condition.
The next result shows that this simple characterization is special to Picard rank one. It already fails for abelian surfaces of Picard rank two.
Proposition 24. There exists an abelian surface \(A\) with \(\rho(A) = 2\), an ample line bundle \(L\) on \(A\), and a line bundle \(M\) on \(A\) such that \(M\) is not isomorphic to any power of \(L\), yet \(M\) is not arithmetically Cohen–Macaulay with respect to \(L\).
Proof. Let \(A = E_1 \times E_2\), where \(E_1\) and \(E_2\) are non-isogenous elliptic curves over an algebraically closed field \(k\). The Néron–Severi group \(\operatorname{NS}(A)\) has rank \(\rho(A) = 2\) and is generated by the numerical equivalence classes of the fibers of the natural projections. Specifically, fix points \(p \in E_1\) and \(q \in E_2\), and define the generators \[F_1 = E_1 \times \{q\}, \quad F_2 = \{p\} \times E_2.\] The intersection form on \(\operatorname{NS}(A)\) is given by \(F_1^2 = F_2^2 = 0\) and \(F_1 \cdot F_2 = 1\).
Define the line bundle \[L = \operatorname{pr}_1^* \mathcal{O}_{E_1}(p) \otimes \operatorname{pr}_2^* \mathcal{O}_{E_2}(q).\] The first Chern class of \(L\) is \(c_1(L) = F_1 + F_2\). Since \(L^2 = (F_1+F_2)^2 = 2 > 0\) and \(L \cdot F_i = 1 > 0\) for \(i=1,2\), the Nakai–Moishezon criterion implies that \(L\) is strictly ample.
Next, define the line bundle \[M = \operatorname{pr}_1^* \mathcal{O}_{E_1}(2p).\] The pullback of a point divisor on \(E_1\) corresponds to the vertical fiber \(F_2\), so \(c_1(M) = 2F_2\). In the basis \(\{F_1, F_2\}\), the classes of \(L\) and \(M\) are \((1,1)\) and \((0,2)\), respectively. Clearly, \(c_1(M)\) is not a scalar multiple of \(c_1(L)\), confirming that \(M\) is not isomorphic to any tensor power of \(L\).
To demonstrate that \(M\) is not arithmetically Cohen–Macaulay (ACM) with respect to \(L\), it suffices to show that the intermediate cohomology group \(H^1(A, M)\) is non-vanishing. By the Künneth formula, we have \[H^1(A, M) \cong \Big( H^0(E_1, \mathcal{O}_{E_1}(2p)) \otimes H^1(E_2, \mathcal{O}_{E_2}) \Big) \oplus \Big( H^1(E_1, \mathcal{O}_{E_1}(2p)) \otimes H^0(E_2, \mathcal{O}_{E_2}) \Big).\] By Riemann–Roch on the elliptic curve \(E_1\), the line bundle \(\mathcal{O}_{E_1}(2p)\) has degree \(2 > 0\), giving \(h^0(E_1, \mathcal{O}_{E_1}(2p)) = 2\) and \(h^1(E_1, \mathcal{O}_{E_1}(2p)) = 0\). Furthermore, for the elliptic curve \(E_2\), \(h^1(E_2, \mathcal{O}_{E_2}) = 1\). Consequently, the first summand is isomorphic to \(k^2 \otimes k \cong k^2\), yielding \[h^1(A, M) = 2 \neq 0.\] As \(H^1(A, M)\) is non-zero, \(M\) fails the vanishing condition required to be ACM with respect to \(L\). ◻
Theorem 25. Let \(A\) be an abelian surface over an algebraically closed field, and let \(L\) be an ample line bundle on \(A\). For a line bundle \(M\) on \(A\), set \[N_n := M\otimes L^{\otimes n}, \qquad n\in \mathbb{Z}.\] Then \(M\) is arithmetically Cohen–Macaulay with respect to \(L\) if and only if for every \(n\in \mathbb{Z}\), the line bundle \(N_n\) satisfies one of the following:
\(N_n^2>0\);
\(N_n\in \operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\);
\(N_n^2=0\), \(N_n\notin \operatorname{Pic}^0(A)\), and in the standard decomposition \[N_n \cong \pi^*Q \otimes P\] with \(\pi:A\to E\) an elliptic quotient, \(Q\in \operatorname{Pic}(E)\) of nonzero degree, and \(P\in \operatorname{Pic}^0(A)\), one has \[P|_F \not\cong \mathcal{O}_F\] for a fiber \(F\) of \(\pi\).
Equivalently, \(M\) fails to be arithmetically Cohen–Macaulay with respect to \(L\) if and only if for some \(n\in \mathbb{Z}\), one of the following holds:
\(N_n \cong \mathcal{O}_A\);
\(N_n^2<0\);
\(N_n^2=0\), \(N_n\notin \operatorname{Pic}^0(A)\), and \[N_n \cong \pi^*T\] for some elliptic quotient \(\pi:A\to E\) and some line bundle \(T\in \operatorname{Pic}(E)\) with \(\deg T\neq 0\).
Proof. Since \(\dim A=2\), the definition of the arithmetically Cohen–Macaulay property reduces to \[M \text{ is ACM with respect to }L \iff H^1(A,M\otimes L^{\otimes n})=0 \text{ for every } n\in \mathbb{Z}.\] Thus it suffices to classify those line bundles \(N\) on \(A\) for which \[H^1(A,N)=0.\] We divide the argument according to the value of \(N^2\).
Step 1: the case \(N^2>0\).
By the Riemann–Roch theorem on an abelian surface, \[\chi(N)=\frac{N^2}{2}.\] Hence \(N^2>0\) implies \(\chi(N)>0\). Since \(\omega_A\cong \mathcal{O}_A\), Serre duality gives \[h^2(A,N)=h^0(A,N^{-1}).\] Therefore \[\chi(N)=h^0(A,N)-h^1(A,N)+h^0(A,N^{-1})>0.\] In particular, at least one of \(h^0(A,N)\) and \(h^0(A,N^{-1})\) is nonzero. Thus either \(N\) or \(N^{-1}\) is effective.
If \(N\simeq\mathcal{O}_A(D)\) with \(D\ge 0\) and \(D^2=N^2>0\), then \(D\) is ample. Indeed, if there were an irreducible curve \(C\subset A\) with \(D\cdot C=0\), the Hodge index theorem would force \(C^2<0\), but an effective irreducible curve on an abelian surface has arithmetic genus \(\ge 1\) (no rational curves), so by adjunction \(C^2=2p_a(C)-2\ge 0\), contradiction. Hence \(D\cdot C>0\) for every irreducible \(C\), and by Nakai–Moishezon \(D\) is ample. Thus \(N\) is ample.
If instead \(N^{-1}\) is effective, the same argument shows that \(N^{-1}\) is ample, i.e. \(N\) is anti-ample.
Now on an abelian variety the cohomology of an ample line bundle is concentrated in degree \(0\), and the cohomology of an anti-ample line bundle is concentrated in top degree. Since \(\dim A=2\), in both cases one has \[H^1(A,N)=0.\]
Step 2: the case \(N\in \operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\).
For a nontrivial algebraically trivial line bundle on an abelian variety, all cohomology groups vanish. Hence \[H^i(A,N)=0 \qquad \text{for all } i,\] and in particular \[H^1(A,N)=0.\]
Step 3: the case \(N^2<0\).
Again Riemann–Roch gives \[\chi(N)=\frac{N^2}{2}<0.\] Using Serre duality, \[h^1(A,N)=h^0(A,N)+h^0(A,N^{-1})-\chi(N).\] Since the first two terms are nonnegative and \(-\chi(N)>0\), it follows that \[h^1(A,N)>0.\] Thus \[H^1(A,N)\neq 0.\]
Step 4: the case \(N^2=0\).
We first treat the algebraically trivial case. If \(N\in \operatorname{Pic}^0(A)\), then either \(N\cong \mathcal{O}_A\), in which case \[H^1(A,\mathcal{O}_A)\neq 0,\] or else \(N\) is nontrivial algebraically trivial, in which case Step 2 applies and \[H^1(A,N)=0.\]
We therefore assume from now on that \[N^2=0, \qquad N\notin \operatorname{Pic}^0(A).\] In this situation \(N\) is a degenerate line bundle in the standard sense. The structure theorem for degenerate line bundles on abelian varieties yields an elliptic quotient \[\pi:A\to E\] and a decomposition \[N \cong \pi^*Q \otimes P,\] where \(Q\in \operatorname{Pic}(E)\) has nonzero degree and \(P\in \operatorname{Pic}^0(A)\).
We now compute \(H^1(A,N)\).
Case 4a: \(P|_F \not\cong \mathcal{O}_F\) for a fiber \(F\) of \(\pi\).
Since \(\pi^*Q\) restricts trivially to every fiber of \(\pi\), one has \[N|_F \cong P|_F.\] By assumption this is a nontrivial degree-zero line bundle on the elliptic curve \(F\). Hence \[H^0(F,N|_F)=H^1(F,N|_F)=0.\] By cohomology and base change, \[\pi_*N=0, \qquad R^1\pi_*N=0.\] The Leray spectral sequence therefore gives \[H^1(A,N)=0.\]
Case 4b: \(P|_{F} \cong \mathcal{O}_{F}\). We claim that \(P\) descends from the base elliptic curve \(E\). Indeed, since \[0 \longrightarrow F \longrightarrow A \xrightarrow{\pi} E \longrightarrow 0\] is an exact sequence of abelian varieties, dualizing yields an exact sequence \[0 \longrightarrow \widehat{E} \xrightarrow{\pi^{*}} \widehat{A} \longrightarrow \widehat{F} \longrightarrow 0,\] where \(\widehat{A}=\operatorname{Pic}^{0}(A)\), \(\widehat{E}=\operatorname{Pic}^{0}(E)\), and \(\widehat{F}=\operatorname{Pic}^{0}(F)\). Thus the kernel of the restriction map \[\operatorname{Pic}^{0}(A)\longrightarrow \operatorname{Pic}^{0}(F), \qquad M \longmapsto M|_{F},\] is precisely \(\pi^{*}\operatorname{Pic}^{0}(E)\). Since \(P|_{F}\cong \mathcal{O}_{F}\), it follows that there exists \(\beta \in \operatorname{Pic}^{0}(E)\) such that \[P \cong \pi^{*}\beta.\]
Therefore \[N \cong \pi^{*}Q \otimes P \cong \pi^{*}Q \otimes \pi^{*}\beta \cong \pi^{*}(Q\otimes \beta)=:\pi^{*}T,\] where \(T\in \operatorname{Pic}(E)\). Since \(\beta\in \operatorname{Pic}^{0}(E)\), we have \[\deg T=\deg(Q\otimes \beta)=\deg Q \neq 0.\]
Now apply the Leray spectral sequence to \(\pi^{*}T\). Because \(\pi:A\to E\) is a quotient of abelian varieties with connected fibers isomorphic to the elliptic curve \(F\), one has \[\pi_{*}\mathcal{O}_{A}\cong \mathcal{O}_{E}, \qquad R^{1}\pi_{*}\mathcal{O}_{A}\cong \mathcal{O}_{E}.\] Hence, by the projection formula, \[\pi_{*}(\pi^{*}T)\cong T\otimes \pi_{*}\mathcal{O}_{A}\cong T, \qquad R^{1}\pi_{*}(\pi^{*}T)\cong T\otimes R^{1}\pi_{*}\mathcal{O}_{A}\cong T.\]
Since \(\dim E=1\), the Leray spectral sequence degenerates to a short exact sequence \[0 \longrightarrow H^{1}(E,\pi_{*}\pi^{*}T) \longrightarrow H^{1}(A,\pi^{*}T) \longrightarrow H^{0}(E,R^{1}\pi_{*}\pi^{*}T) \longrightarrow 0,\] and therefore \[H^{1}(A,\pi^{*}T)\cong H^{1}(E,T)\oplus H^{0}(E,T).\]
Finally, because \(\deg T\neq 0\), at least one of \(H^{0}(E,T)\) and \(H^{1}(E,T)\) is nonzero. More precisely:
if \(\deg T>0\), then \(H^{0}(E,T)\neq 0\) and \(H^{1}(E,T)=0\);
if \(\deg T<0\), then \(H^{0}(E,T)=0\) and \(H^{1}(E,T)\neq 0\).
Thus in either case \[H^{1}(A,N)=H^{1}(A,\pi^{*}T)\neq 0.\] This completes Case 4b.
Combining Cases 4a and 4b, we conclude that when \(N^2=0\) and \(N\notin \operatorname{Pic}^0(A)\), one has \[H^1(A,N)=0 \iff N\cong \pi^*Q\otimes P \text{ with } P|_F\not\cong \mathcal{O}_F.\] Equivalently, \[H^1(A,N)\neq 0 \iff N\cong \pi^*T \text{ for some } T\in \operatorname{Pic}(E) \text{ with } \deg T\neq 0.\]
Step 5: conclusion.
We have completely classified the line bundles \(N\) on \(A\) with \(H^1(A,N)=0\): \[H^1(A,N)=0\] if and only if exactly one of the following holds:
\(N^2>0\);
\(N\in \operatorname{Pic}^0(A)\setminus\{\mathcal{O}_A\}\);
\(N^2=0\), \(N\notin \operatorname{Pic}^0(A)\), and in the decomposition \[N\cong \pi^*Q\otimes P\] one has \(P|_F\not\cong \mathcal{O}_F\).
Applying this criterion to every twist \[N_n=M\otimes L^{\otimes n},\] we obtain the asserted characterization of those \(M\) which are arithmetically Cohen–Macaulay with respect to \(L\). The equivalent description of the non-ACM locus is just the negation of the above three good cases. ◻
In this section, we establish the categorical link between unipotent bundles on an abelian variety \(A\) and the module category of a specific regular local ring. This equivalence serves as the technical bridge between the geometry of the variety and the representation-theoretic definition of wildness.
Let \(A\) be a complex abelian variety of dimension \(g\), and let \(\widehat{A} = \operatorname{Pic}^0(A)\) be its dual abelian variety. We denote by \(\mathcal{P}\) the normalized Poincaré line bundle on \(A \times \widehat{A}\). The Fourier–Mukai functor \(\Phi_\mathcal{P} \colon D^b(A) \to D^b(\widehat{A})\) is defined as: \[\Phi_\mathcal{P}(E) = \mathbf{R}p_{\widehat{A},*} \left( p_A^* E \otimes \mathcal{P}\right).\] According to the seminal work of Mukai [7], \(\Phi_A\) is an equivalence of triangulated categories. For our purposes, we focus on the subcategory of unipotent bundles.
Definition 7. A vector bundle \(U\) on \(A\) is said to be unipotent if it admits a filtration \(0 = U_0 \subset U_1 \subset \dots \subset U_r = U\) such that each successive quotient \(U_i / U_{i-1}\) is isomorphic to the structure sheaf \(\mathcal{O}_A\). We denote the category of such bundles by \(\mathcal{U}\).
Theorem 26. Let \(\mathcal{U}\) be the category of unipotent vector bundles on \(A\). There exists an equivalence of categories: \[\mathcal{U}\simeq \operatorname{flmod}\left( k[[x_1, \dots, x_g]] \right),\] where \(\operatorname{flmod}(R)\) denotes the category of finite-length modules over the ring \(R\).
Proof. The proof proceeds by identifying the transform of the structure sheaf and utilizing the smoothness of the dual variety.
It is a fundamental result [7] that \(\Phi_A(\mathcal{O}_A) \cong k(0)[-g]\), where \(k(0)\) is the skyscraper sheaf supported at the identity \(0 \in \widehat{A}\). To simplify the degree bookkeeping, we define the shifted functor \(\Psi = \Phi_A[g]\). This construction ensures that \(\Psi(\mathcal{O}_A) \cong k(0)\).
Let \(\operatorname{Coh}_0(\widehat{A})\) be the category of coherent sheaves on \(\widehat{A}\) supported set-theoretically at \(\{0\}\). We show that \(\Psi\) restricts to an equivalence \(\mathcal{U}\to \operatorname{Coh}_0(\widehat{A})\). If \(U \in \mathcal{U}\), we proceed by induction on the rank \(r\). For \(r=1\), \(U \cong \mathcal{O}_A\) and \(\Psi(U) \cong k(0)\). For \(r > 1\), consider the extension \(0 \to U_{r-1} \to U \to \mathcal{O}_A \to 0\). By the exactness of \(\Phi_A\), we obtain the following distinguished triangle in \(D^b(\widehat{A})\): \[\Psi(U_{r-1}) \to \Psi(U) \to k(0) \to \Psi(U_{r-1})[1].\] By the induction hypothesis, \(\Psi(U_{r-1})\) is a coherent sheaf concentrated in degree \(0\) and supported at \(\{0\}\). Since \(k(0)\) is likewise a sheaf in degree \(0\), the long exact sequence of cohomology sheaves shows that \(H^i(\Psi(U)) = 0\) for \(i \neq 0\), and \(H^0(\Psi(U))\) is supported at \(\{0\}\).
Conversely, any \(F \in \operatorname{Coh}_0(\widehat{A})\) admits a filtration with successive quotients isomorphic to \(k(0)\). Since \(\Psi\) is an equivalence of categories, \(\Psi^{-1}(F)\) has a filtration with quotients \(\Psi^{-1}(k(0)) \cong \mathcal{O}_A\), proving \(\Psi^{-1}(F) \in \mathcal{U}\).
Generally, the category of coherent sheaves on a scheme \(X\) supported at a closed point \(x\) is equivalent to the category of finite-length modules over the local ring \(\mathcal{O}_{X,x}\) [11]. Thus, \(\operatorname{Coh}_0(\widehat{A}) \simeq \operatorname{flmod}(\mathcal{O}_{\widehat{A},0})\). Since \(\widehat{A}\) is a smooth variety of dimension \(g\), the completion of its local ring \(\widehat{\mathcal{O}}_{\widehat{A},0}\) is isomorphic to \(k[[x_1, \dots, x_g]]\) by the Cohen Structure Theorem. Because finite-length modules are automatically complete, we have: \[\operatorname{flmod}(\mathcal{O}_{\widehat{A},0}) \simeq \operatorname{flmod}(\widehat{\mathcal{O}}_{\widehat{A},0}) \cong \operatorname{flmod}(k[[x_1, \dots, x_g]]).\] Combining these identifications yields the desired equivalence. ◻
Corollary 10. For \(g \ge 2\), the category of unipotent bundles on \(A\) is of wild representation type.
Proof. It is a classical result in representation theory that finite length modules over regular local ring \(k[[x,y]]\) is wild. Consequently, for \(g \ge 2\), \(\operatorname{flmod}(k[[x_1, \dots, x_g]])\) is wild. By the equivalence in Theorem 3.2, \(\mathcal{U}\) is wild. ◻
Lemma 18. Let \(A\) be a complex abelian variety of dimension \(g\), let \(L\) be an ample line bundle on \(A\), let \[0=U_0\subset U_1\subset\cdots\subset U_r=U\] be a unipotent vector bundle on \(A\), so that \[U_j/U_{j-1}\cong \mathcal{O}_A \quad \text{for all } j = 1,\dots, r,\] and let \(P\in \operatorname{Pic}^0(A)\) be a nontrivial line bundle. Then the vector bundle \[E := P \otimes U\] is arithmetically Cohen–Macaulay (ACM) with respect to the polarization defined by \(L\), i.e., \[H^i(A, E \otimes L^n) = 0 \quad \text{for all } n\in \mathbb{Z} \text{ and } 0<i<g.\]
Proof. We prove the claim by analyzing the cohomology groups \(H^i(A, P \otimes U \otimes L^n)\) for all integers \(n\) and degrees \(0 < i < g\). The strategy is to reduce to the cohomology of the filtration quotients \(P \otimes L^n\), and then to use standard vanishing theorems on abelian varieties.
Let us fix a polarization \(L\) on \(A\), which we assume to be very ample (after replacing if necessary by a suitable power). Let \(P \in \operatorname{Pic}^0(A)\) be a nontrivial line bundle, and let \(U\) be a unipotent vector bundle. This means that there exists a filtration \[0 = U_0 \subset U_1 \subset \dots \subset U_r = U\] where each quotient \(U_j/U_{j-1}\) is isomorphic to \(\mathcal{O}_A\). Our goal is to show that \[H^i(A, P \otimes U \otimes L^n) = 0 \quad \text{for all } n \in \mathbb{Z} \text{ and } 0 < i < g.\]
For each \(j = 1,\dots, r\), we have a short exact sequence \[0 \to U_{j-1} \to U_j \to \mathcal{O}_A \to 0.\] Tensoring with \(P \otimes L^n\), we obtain a short exact sequence \[0 \to P \otimes U_{j-1} \otimes L^n \to P \otimes U_j \otimes L^n \to P \otimes L^n \to 0,\] since tensoring with a line bundle preserves exactness.
By the associated long exact sequence in cohomology, we have \[\cdots \to H^i(A, P \otimes U_{j-1} \otimes L^n) \to H^i(A, P \otimes U_j \otimes L^n) \to H^i(A, P \otimes L^n) \to \cdots\] Thus, if we know that \(H^i(A, P \otimes U_{j-1} \otimes L^n) = 0\) and \(H^i(A, P \otimes L^n) = 0\) for \(0 < i < g\), then the middle term must also vanish: \[H^i(A, P \otimes U_j \otimes L^n) = 0 \quad \text{for } 0 < i < g.\]
So it suffices to prove the following two claims:
\(H^i(A, P \otimes L^n) = 0\) for all \(n \in \mathbb{Z}\) and \(0 < i < g\);
Apply this inductively along the filtration of \(U\).
We now proceed to prove these two claims.
We distinguish three cases depending on the sign of \(n\).
Since \(L\) is ample, so is \(L^n\). Because \(P \in \operatorname{Pic}^0(A)\) is algebraically equivalent to \(0\), the tensor product \(P \otimes L^n\) remains ample. By a well-known vanishing theorem due to Mumford, we have: \[H^i(A, M \otimes Q) = 0 \quad \text{for all } i > 0,\] where \(M\) is ample and \(Q \in \operatorname{Pic}^0(A)\). Applying this with \(M = L^n\) and \(Q = P\), we deduce: \[H^i(A, P \otimes L^n) = 0 \quad \text{for all } i > 0.\] In particular, the intermediate cohomology groups vanish: \[H^i(A, P \otimes L^n) = 0 \quad \text{for all } 0 < i < g.\]
In this case, we are computing \(H^i(A, P)\) for a nontrivial \(P \in \operatorname{Pic}^0(A)\). We appeal to the following classical result:
Fact 1. Let \(P \in \operatorname{Pic}^0(A)\) be a nontrivial line bundle on a complex abelian variety \(A\). Then: \[H^i(A, P) = 0 \quad \text{for all } i.\]
Therefore, in particular: \[H^i(A, P) = 0 \quad \text{for all } 0 < i < g.\]
Let \(n = -m\) where \(m > 0\). We compute the cohomology \(H^i(A, P \otimes L^{-m})\) using Serre duality. Since \(K_A \cong \mathcal{O}_A\), we have: \[H^i(A, P \otimes L^{-m}) \cong H^{g-i}(A, (P \otimes L^{-m})^\vee)^\vee.\] Note that \[(P \otimes L^{-m})^\vee \cong P^{-1} \otimes L^m,\] because \((L^{-m})^\vee \cong L^m\). Hence: \[H^i(A, P \otimes L^{-m}) \cong H^{g-i}(A, P^{-1} \otimes L^m)^\vee.\] Now observe that \(P^{-1} \in \operatorname{Pic}^0(A)\) and \(L^m\) is ample. By Case 1 applied to \(P^{-1} \otimes L^m\), we know: \[H^{g-i}(A, P^{-1} \otimes L^m) = 0 \quad \text{for } g - i > 0.\] For \(0 < i < g\), we have \(0 < g - i < g\), hence: \[H^{g-i}(A, P^{-1} \otimes L^m) = 0.\] Therefore: \[H^i(A, P \otimes L^{-m}) = 0 \quad \text{for } 0 < i < g.\]
This completes the verification that: \[H^i(A, P \otimes L^n) = 0 \quad \text{for all } n \in \mathbb{Z} \text{ and } 0 < i < g.\]
We now prove by induction on \(j\) that: \[H^i(A, P \otimes U_j \otimes L^n) = 0 \quad \text{for all } n \in \mathbb{Z} \text{ and } 0 < i < g.\]
Since \(U_1/U_0 \cong \mathcal{O}_A\), necessarily \(U_1 \cong \mathcal{O}_A\). Thus: \[P \otimes U_1 \otimes L^n \cong P \otimes L^n,\] and the assertion follows from already discussed above.
Suppose this holds for \(j - 1\). Consider the short exact sequence: \[0 \to P \otimes U_{j-1} \otimes L^n \to P \otimes U_j \otimes L^n \to P \otimes L^n \to 0.\] The associated long exact sequence in cohomology yields: \[H^i(A, P \otimes U_{j-1} \otimes L^n) \to H^i(A, P \otimes U_j \otimes L^n) \to H^i(A, P \otimes L^n).\] By the induction hypothesis, the left-hand side is zero. By (\(*\)), the right-hand side is also zero. Hence: \[H^i(A, P \otimes U_j \otimes L^n) = 0 \quad \text{for all } n \in \mathbb{Z} \text{ and } 0 < i < g.\]
This completes the inductive proof.
Taking \(j = r\), we conclude: \[H^i(A, P \otimes U \otimes L^n) = 0 \quad \text{for all } n \in \mathbb{Z} \text{ and } 0 < i < g.\]
This verifies the ACM property for \(E = P \otimes U\). ◻
Theorem 27. Let \(A\) be a complex abelian variety of dimension \(g \ge 2\), and let \(L\) be an ample line bundle on \(A\). The category of Arithmetically Cohen-Macaulay (ACM) vector bundles on \(A\) with respect to the polarization defined by \(L\) is of wild representation type.
Proof. Let \(\mathcal{U}\) be the category of unipotent bundles on \(A\). The Fourier-Mukai transform yields an equivalence \(\mathcal{U} \simeq \mathrm{flmod}(k[[x_1, \dots, x_g]])\), which is classically known to be of wild representation type for \(g \ge 2\). Fix a nontrivial line bundle \(P \in \mathrm{Pic}^0(A)\). The exact functor \(\Phi: \mathcal{U} \to \mathrm{Coh}(A)\) defined by \(U \mapsto P \otimes U\) is fully faithful, embedding \(\mathcal{U}\) as a subcategory. By Lemma 18 , the essential image of \(\Phi\) consists entirely of ACM bundles. Since a wild category embeds fully faithfully into the category of ACM bundles, the latter is strictly wild. ◻
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