Vector bundles and monads on abelian threefolds
aa r X i v : . [ m a t h . AG ] J u l VECTOR BUNDLES AND MONADS ON ABELIANTHREEFOLDS
MARTIN G. GULBRANDSEN
Abstract.
Using the Serre construction, we give examples of sta-ble rank 2 vector bundles on principally polarized abelian three-folds ( X, Θ) with Picard number 1. The Chern classes ( c , c ) ofthese examples realize roughly one half of the classes that are a pri-ori allowed by the Bogomolov inequality and Riemann-Roch (thelatter gives a certain divisibility condition).In the case of even c , we study deformations of these vectorbundles E , using a second description in terms of monads, similarto the ones studied by Barth–Hulek on projective space. By an ex-plicit analysis of the hyperext spectral sequence associated to themonad, we show that the space of first order infinitesimal deforma-tions of E equals the space of first order infinitesimal deformationsof the monad. This leads to the formuladim Ext ( E , E ) = ∆( E ) · Θ + 5(we emphasize that its validity is only proved for special bundles E coming from the Serre construction), where ∆ denotes the dis-criminant 4 c − c .Finally we show that, in the first nontrivial example of the aboveconstruction (where c = 0 and c = Θ ), the infinitesimal iden-tification between deformations of E and of the monad can beextended to a Zariski local identification: this leads to an explicitdescription of a Zariski open neighbourhood of E in its modulispace M (0 , Θ ). This neighbourhood is a ruled, nonsingular vari-ety of dimension 13, birational to a P -bundle over a finite quotientof X × X X × X X , where X is considered as a variety over X via the group law. Contents
1. Introduction 22. The Serre construction 33. Monads 84. Digression on the hyperext spectral sequence 115. Deformations of decomposable monads 136. Birational description of M (0 , Θ ) 23References 26 The author thanks the Max-Planck-Institut f¨ur Mathematik in Bonn for itshospitality and financial support. The bulk of this paper was written during a stayas postdoc at the MPIM in 2008–2009. Introduction
The geometry of moduli spaces for stable vector bundles on Calabi-Yau (by which we just mean having trivial canonical bundle) threefoldsis largely unknown, but of high interest, for instance due to their rel-evance for string theory, and the computation of Donaldson-Thomasinvariants. In lower dimension, vector bundles on Calabi-Yau curves(i.e. elliptic) were classified by Atiyah, and are parametrized by thesame curve. Moduli spaces for stable bundles, or coherent sheaves, onCalabi-Yau surfaces (i.e. K3 or abelian) are holomorphic symplectic va-rieties (Mukai [13], generalizing Beauville [2], generalizing Fujiki [4]).This is a very rare geometric structure, at least on complete varieties.Our leitfaden is the question whether equally interesting geometriesexist in higher dimension.We study here examples of rank 2 vector bundles on abelian three-folds, partly for the intrinsic interest, and partly in the hope that theabelian case may shed light on the case of general Calabi-Yau three-folds, but be more accessible. Our central tool, besides the Serre con-struction, is monads: these are usually put to work on rational varieties,and it may be slightly surprising that they can be useful also in ourcontext. On the other hand, we do not know whether the bundles weconstruct, and their moduli, show typical or exceptional behaviour.1.1.
Notation.
We work over an algebraically closed field k of char-acteristic zero. Our terminology regarding (semi-) stable sheaves andtheir moduli follows Simpson [15]; in particular, stability for a coher-ent O X -module on a polarized projective variety ( X, H ) is defined usingthe normalized Hilbert polynomial with respect to the polarization H .Stable sheaves admit a coarse moduli space M , with a compactifica-tion M parametrizing S-equivalence classes of semistable sheaves. Thesheaves we construct in this text will in fact have the stronger prop-erty of µ -stability, in the sense of Mumford and Takemoto, which ismeasured by the slope, i.e. the ratio of degree with respect to H tothe rank, and which implies stability. Conversely, semistability implies µ -semistability.The words line bundles and vector bundles are used as synonyms ofinvertible and locally free sheaves. In particular, an inclusion of vectorbundles means an inclusion as sheaves, i.e. the quotient need not belocally free. We take Chern classes to live in the Chow ring modulonumerical equivalence.Let ( X, Θ) be a principally polarized abelian variety. If x ∈ X is apoint, we write T x : X → X for the translation map, and define Θ x as T x (Θ) = Θ + x . We identify X with its dual Pic ( X ) by associatingwith x the line bundle P x = O X (Θ − Θ x ). The normalized Poincar´eline bundle on X × X is denoted P ; its restriction to X × { x } is P x . ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 3 The Serre construction
In this section we apply the standard Serre construction to producerank 2 vector bundles on principally polarized abelian threefolds, in-cluding examples with small c . This is the content of Theorem 2.3.These examples (in the case of even c ) will be our objects of study forthe rest of this paper.2.1. The bundles/curves correspondence.
Let E be rank 2 vectorbundle on a projective variety X , and let s ∈ Γ( X, E ) be a section. Ifthe vanishing locus V ( s ) has codimension 2, then: (1) it is a locallycomplete intersection, and (2) its canonical bundle is ( ω X ⊗ V E ) (cid:12)(cid:12) V ( s ) .The Serre construction says (under a cohomological condition on V E )that any codimension two subscheme Y ⊂ X with these two propertiesis of the form V ( s ). More precisely: Theorem 2.1.
Let X be a projective variety with a line bundle L satisfying H p ( X, L − ) = 0 for p = 1 , . Let Y ⊂ X be a codimen-sion two locally complete intersection subscheme with canonical bundleisomorphic to ( ω X ⊗ L ) | Y . Then there is a canonical isomorphism Hom( ( ω X ⊗ L ) | Y , ω Y ) ∼ = Ext ( I Y ⊗ L , O X ) which is functorial in Y with respect to inclusions, and such that iso-morphisms on the left correspond to locally free extensions on the right. For the proof we refer to Hartshorne [7, Thm. 1.1 and Rem. 1.1.1],who attributes “all essential ideas” to Serre [14].It follows that, whenever we choose an isomorphism ( ω X ⊗ L ) | Y ∼ = ω Y , the theorem gives an extension(1) 0 ✲ O X s ✲ E ✲ I Y ⊗ L ✲ E locally free, and hence Y = V ( s ) as required. Definition 2.2.
We say that E and Y corresponds if there is a shortexact sequence (1).Note that, if Y has several connected components, there may beseveral non-isomorphic bundles E corresponding to Y . See Proposition2.8.2.2. Construction of bundles.
For the rest of this paper, we fixa principally polarized abelian threefold ( X, Θ). We assume that itsPicard number is one, although this assumption is not essential in latersections. Thus every divisor is numerically equivalent to an integralmultiple of Θ. Moreover (see e.g. Debarre [3]), an application of theendomorphism construction of Morikawa [11] and Matsusaka [10] showsthat every 1-cycle is numerically equivalent to an integral multiple ofΘ / MARTIN G. GULBRANDSEN
So fix classes c = m Θ and c = n Θ /
2, where m and n are integers.If these are the Chern classes of a rank two vector bundle E , then, byRiemann-Roch χ ( E ) = ( c − c c ) = m − nm, so either m or n is even. Moreover, if E is µ -semistable, then Bogo-molov’s inequality reads m ≤ n . Theorem 2.3.
Let ( X, Θ) be a principally polarized abelian threefoldof Picard number , and let c = m Θ and c = n Θ / , with m and n integers. Assume (1) the strict Bogomolov inequality holds, i.e. m < n , and (2) n is even and mn is divisible by .Then there exist µ -stable rank vector bundles with Chern classes c and c . Remark 2.4.
For each c ∈ NS( X ), the theorem realizes every second c that is allowed by (strict) Bogomolov and Riemann-Roch. The otherhalf seems much more subtle. In fact, we do not know any example ofa rank 2 vector bundle, stable or not, that violates condition (2). Thesituation in which equality occurs in the Bogomolov inequality will beanalysed in Proposition 2.6.Before proving the theorem, we rephrase µ -stability for E as a con-dition on the corresponding curve Y . The argument is similar to thatof Hartshorne [7, Prop. 3.1] in the case of P . Lemma 2.5.
Let ( X, Θ) be as in the theorem, and E be a rank vectorbundle corresponding to a curve Y ⊂ X . Let c ( E ) = m Θ . Then thefollowing are equivalent. (1) E is µ -stable. (2) m > and Y is not contained in any translate of any divisorin the linear system | k Θ | , where k is the round down of m/ .Proof. Since E has a section, it is clear that m > µ -stability. Write ⌊ m/ ⌋ and ⌈ m/ ⌉ for the round down and round upof m/
2. The bundle E fails µ -stability if and only if it contains a linebundle P x ( l Θ) ⊂ E with l ≥ m/
2. Since P x ( l Θ) has global sectionsfor l positive, it suffices to test with l = ⌈ m/ ⌉ . Thus E is µ -stable ifand only if(2) H ( X, E ( −⌈ m/ ⌉ Θ) ⊗ P x )) = 0 for all x ∈ X .Now twist the short exact sequence (1) with −⌈ m/ ⌉ Θ and take coho-mology. Since H i ( X, O X ( −⌈ m/ ⌉ Θ))) = 0 for i = 0 ,
1, and the deter-minant of E has the form P a ( m Θ) for some a ∈ X , we find that thevanishing (2) is equivalent to the vanishing of H ( X, I Y ( ⌊ m/ ⌋ Θ) ⊗ P x ) for all x ∈ X . Since Θ is ample, this is equivalent to H ( X, I Y ⊗ T ∗ x O X ( ⌊ m/ ⌋ Θ)) = 0 for all x ∈ X ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 5 which is condition (2). (cid:3)
Proof of Theorem 2.3.
Since µ -stability, and the conditions (1) and (2)in the statement of the theorem, are preserved under tensor productwith line bundles, it suffices to prove the theorem for m = 2 and m = 3.When m = 2, the theorem claims that there are µ -stable rank 2bundles with c = 2Θ and c = N Θ for all integers N ≥
2. For this,choose N generic points a i ∈ X and let Y = S Ni =1 Y i , Y i = Θ a i ∩ Θ − a i . We want to apply the Serre construction to this curve.First we claim that the Y i ’s are pairwise disjoint, for a i chosen gener-ically. In fact, for i = j write Y i ∩ Y j = (Θ a i ∩ Θ a j ) | {z } V ∩ (Θ − a i ∩ Θ − a j ) | {z } W , where V and W have codimension 2. By an easy moving lemma forabelian varieties [9, Lemma 5.4.1], a general translate V + x intersects W properly, hence empty. Thus (replacing x by a “square root” x/ V + x and W − x are disjoint. So Y i and Y j will be disjoint aftera small perturbation a i a i + x , a j a j + x .The normal bundle of each Y i ⊂ X is O Y i (Θ a i ) ⊕ O Y i (Θ − a i ), hencethe canonical bundle ω Y i is O Y i (Θ a i + Θ − a i ). The theorem of the squareshows that Θ a i + Θ − a i is linearly equivalent to 2Θ. Since the Y i ’s aredisjoint, we conclude that Y is a locally complete intersection withcanonical bundle O Y (2Θ). The Serre construction produces a bundle E with determinant O X (2Θ) and second Chern class [ Y ] = P i [ Y i ] = N Θ .Next we show µ -stability. We claim that the only theta-translatescontaining Y i are Θ a i and Θ − a i . This is a standard result: the inter-section of two theta-translates are never contained in a third one. Infact, consider the Koszul complex:0 → O X ( − Θ a i − Θ − a i ) → O X ( − Θ a i ) ⊕ O X ( − Θ − a i ) → I Y i → . Twist with an arbitrary theta-translate Θ x and apply cohomology toobtain an isomorphism H ( X, O X (Θ x − Θ a i )) ⊕ H ( X, O X (Θ x − Θ − a i )) ∼ = H ( X, I Y i (Θ x )) . Thus Θ x contains Y i if and only if x = ± a i as claimed. It followsthat, for N ≥
2, no theta-translate contains Y , and so E is µ -stable byLemma 2.5.In the case m = 3, we take Y = S Ni =1 Y i , Y i = D i ∩ Θ − a i for N generic points a i ∈ X and generic divisors D i ∈ | a i | . A similarargument to the one above shows that the Serre construction produces MARTIN G. GULBRANDSEN a µ -stable rank 2 vector bundle with determinant O X (3Θ) and secondChern class 2 N Θ , for each N ≥ (cid:3) Recall that a vector bundle E is semihomogeneous if, for every x ∈ X ,there exists a line bundle L ∈ Pic ( X ) such that T ∗ x ( E ) is isomorphicto E ⊗ L (in short, homogeneous means translation invariant, andsemihomogeneous means translation invariant up to twist). Semiho-mogeneous bundles are well understood thanks to work of Mukai [12]. Proposition 2.6.
Let ( X, Θ) be as in the theorem, and let c = m Θ and c = n Θ / satisfy m = 2 n , i.e. equality occurs in the Bogomolovinequality. Then E is a non simple, semihomogeneous vector bundle.In particular, it is semistable, but not stable.More precisely, there are a line bundle L and points x, y ∈ X suchthat E = E ⊗ L − is an extension (necessarily split if x = y ) → P x → E → P y → . Proof.
Semihomogenous bundles of rank r are numerically character-ized (Yang [16]) by the property that the Chern roots may be takento be c /r . This means that the Chern character takes the form ch = r exp( c /r ), or, equivalently, the total Chern class is c = (1 + c /r ) r . If r = 2, this is equivalent to c = 4 c . Thus E is semihomogeneous.Now we use several results by Mukai [12] on semihomogeneous bun-dles. Simple semihomogeneous vector bundles are classified, up to twistby homogeneous line bundles, by the element δ = c /r in NS( X ) ⊗ Q .But m is even, since m = 2 n , so there exist line bundles with class c /
2, which rules out the possibility that E is simple. Moreover, anysemihomogeneous bundle is Gieseker-semistable, and it is simple if andonly if it is Gieseker-stable. This proves the first part.For the last part, we use Mukai’s Harder-Narasimhan filtration forsemihomogeneous bundles, which in particular says that any semiho-mogeneous vector bundle with δ = c /r has a filtration whose factorsare simple semihomogeneous bundles with the same invariant δ . Choos-ing L in the (integral) class c /
2, we ensure that E has δ = 0. Since E is semihomogeneous, but not simple, its Harder-Narasimhan factorsare necessarily line bundles with c = 0. (cid:3) A note on the curves Θ a ∩ Θ − a . For later use, we make anobservation regarding the curve obtained by intersecting two generaltheta-translates, which was used as input for the Serre constructionabove (in the even c case).First note that there is a Zariski open subset U ⊂ X such that Θ ∩ Θ x is a nonsingular irreducible curve for all x ∈ U . This is standard: sinceΘ is a nonsingular surface, generic smoothness shows that Θ ∩ Θ x isnonsingular, but possibly disconnected, for generic x (see Hartshorne[6, III 10.8]). On the other hand, Θ ∩ Θ x is an ample divisor on Θ,hence it is connected (see Hartshorne [6, III 7.9]). ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 7
Lemma 2.7.
Let a and b be two points in X and define Y a = Θ a ∩ Θ − a and Y b = Θ b ∩ Θ − b . Then, for a and b generic, no divisor in | | contains both Y a and Y b .Proof. Begin by imposing the conditions on a and b that Y a and Y b aredisjoint irreducible curves, and also that the two curves Θ a ∩ Θ ± b areirreducible. Assume there is a divisor D ∈ | | containing both Y a and Y b . We will prove the lemma by producing a curve C such that C ∩ Θ b = C ∩ Θ − b , and then deduce from this that b is not generic.First we observe that D meets Θ a ∩ Θ b properly. As the latter isirreducible, it suffices to verify that it is not contained in D . In fact, onechecks (determine H ( I Θ a ∩ Θ b (2Θ)) using the Koszul resolution) thatthe linear subsystem of | | , consisting of divisors containing Θ a ∩ Θ b ,is the pencil spanned by Θ a + Θ − a and Θ b + Θ − b . The only element ofthis pencil containing Y a is Θ a + Θ − a , and the only element containing Y b is Θ b + Θ − b , so no element contains both.In particular, D and Θ a intersects properly, so D ∩ Θ a is a curvecontaining Y a . Since D ∩ Θ a has cohomology class 2Θ , and Y a hasclass Θ , there is another effective 1-cycle C of class Θ such that D ∩ Θ a = Y a + C as 1-cycles. We saw above that D ∩ Θ a meets Θ b properly, so weconsider the 0-cycle D ∩ Θ a ∩ Θ b = Y a ∩ Θ b + C ∩ Θ b . The left hand side contains Y b ∩ Θ a . Since Y a and Y b are disjoint, thismeans that C ∩ Θ b contains Y b ∩ Θ a , i.e. their difference is an effectivecycle. But these are 0-cycles of the same degree, so they are equal.None of the arguments given distinguish between b and − b , so we findthat also C ∩ Θ − b equals Y b ∩ Θ a . Thus we have established C ∩ Θ b = C ∩ Θ − b . To conclude, we apply the endomorphism construction of Morikawa[11] and Matsusaka [10], which we briefly recall. The endomorphism α = α ( C, Θ) associated to C and Θ is defined by α ( x ) = X ( C · Θ x ) − X ( C · Θ)where each term means the sum, using the group law, of the pointsin the intersection cycle appearing. This is well defined as a pointin X , although the intersection cycle is only defined up to rationalequivalence. The constant term is included to force α (0) = 0, i.e. tomake α a group homomorphism. We have just established that C intersects Θ b and Θ − b properly, and the two intersections are equalalready as cycles. In particular α ( b ) = α ( − b ), so all we need to knowto prove the lemma is that α is not constant, so that α (2 b ) = 0 defines anonempty Zariski open subset. But in fact, a theorem of Matsusaka [10] MARTIN G. GULBRANDSEN tells us that α is multiplication by 2 (the intersection number C · Θ = 3!divided by dim X = 3), so the condition required is just that 4 b = 0,i.e. b is not a 4-torsion point. (cid:3) As an immediate consequence of the Lemma, we find that if E corre-sponds to a curve with at least two components of the form Θ a i ∩ Θ − a i ,for sufficiently general points a i , then the short exact sequence0 ✲ O X s ✲ E ✲ I Y (2Θ) ✲ , shows that H ( X, E ) is spanned by s . Proposition 2.8.
Fix N ≥ general points a i ∈ X , and let Y bethe union of the curves Y i = Θ a i ∩ Θ − a i . Then the vector bundles E corresponding to the union Y of Y i = Θ a i ∩ Θ − a i form an ( N − -dimensional family, parametrized by G N − m .Proof. The Serre construction gives a one-one correspondence betweenisomorphisms ω Y ∼ = O Y (2Θ) modulo scale, and isomorphism classes ofvector bundles E which admit a section vanishing at Y : the choice of asection can be left out, since we just observed that it is unique moduloscale. But isomorphisms ω Y ∼ = O Y (2Θ) constitute a homogeneous G Nm -space, as Y has N connected components. Dividing by scale, we areleft with G Nm / G m ∼ = G N − m . (cid:3) Monads
It turns out that the vector bundles constructed in Theorem 2.3admit more deformations than are visible in the Serre construction,i.e. more deformations than those obtained by varying the curve Y and the isomorphism ω Y ∼ = O X (2Θ). In this section we rephrase theconstruction in terms of certain monads (Proposition 3.2). This newviewpoint is then used in the remaining sections to analyse first orderdeformations. Definition 3.1 (Barth–Hulek [1]) . A monad is a composable pair ofmaps of vector bundles A φ ✲ B ψ ✲ C such that ψ ◦ φ is zero, ψ is surjective and φ is an embedding of vectorbundles (i.e. injective as a homomorphism of sheaves, and with locallyfree cokernel).Thus E = Ker( ψ ) / Im( φ ) is a vector bundle, and we say that themonad is a monad for E .We will also use chain complex notation ( M • , d ) for monads, so that M − = A , M = B , M = C and M i is zero otherwise, and thedifferential d consists of two nonzero components d − = φ and d = ψ .Thus M • is exact except in degree zero, where its cohomology is E = H ( M • ). ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 9
Decomposable monads.
Consider rank 2 vector bundles E withtrivial determinant V E ∼ = O X on the principally polarized abelianthreefold ( X, Θ). From the construction in Theorem 2.3, we have aseries of such vector bundles, such that E (Θ) corresponds to a curve Y = S i Y i , where Y i = Θ a i ∩ Θ − a i . (The assumption that X hasPicard number 1 is not needed here; this was only needed to establish µ -stability of E , which is not relevant in this section.)We now show that, corresponding to the decomposition of Y intoits connected components Y i , there is a way of building up E from theKoszul complexes (3) ξ i : 0 ✲ O X ( − Θ) ϑ + i ϑ − i ! ✲ P a i ⊕ P − a i ( ϑ − i − ϑ + i ) ✲ I Y i (Θ) ✲ ϑ ± i are nonzero global sections of O X (Θ ± a i ). This can be conve-niently phrased in terms of a monad. Proposition 3.2.
Let a , . . . , a N ∈ X be generically chosen points and Y i = Θ a i ∩ Θ − a i . Then E (Θ) corresponds to Y = S Ni =1 Y i if and only if E is isomorphic to the cohomology of a monad ( N − O X ( − Θ) φ ✲ L Ni =1 ( P a i ⊕ P − a i ) ψ ✲ ( N − O X (Θ) where, if we decompose φ and ψ into pairs φ ± : ( N − O X ( − Θ) → L Ni =1 P ± a i ψ ± : L Ni =1 P ± a i → ( N − O X (Θ) then we have φ ± = ϑ ± ϑ ± . . . ϑ ± N − ϑ ± N ϑ ± N · · · ϑ ± N , ψ ± = ± ( φ ∓ ) ∨ for nonzero sections ϑ ± i ∈ Γ( X, O X (Θ ± a i )) .Proof. One immediately verifies that homomorphisms φ and ψ of thisform do define a monad.The statement that E (Θ) and Y correspond means that E is anextension ξ : 0 ✲ O X ( − Θ) ✲ E ✲ I Y (Θ) ✲ . Giving such an extension is, by Theorem 2.1, equivalent to giving anisomorphism O Y (2Θ) ∼ = ω Y . The obvious decompositionHom( O Y (2Θ) , ω Y ) ∼ = L Ni =1 Hom( O Y i (2Θ) , ω Y i ) Here and elsewhere, whenever f : F → F is a homomorphism of sheaves, weuse the same symbol to denote any twist f : F ( D ) → F ( D ). gives, when applying Theorem 2.1 also to each Y i , a correspondingdecomposition(4) Ext ( I Y (Θ) , O X ( − Θ)) ∼ = L Ni =1 Ext ( I Y i (Θ) , O X ( − Θ)) , which sends ξ to an N -tuple of extensions ξ i . Each Hom( O Y i (2Θ) , ω Y i )is one dimensional, since Y i is connected, so Ext ( I Y i (Θ) , O X ( − Θ)) isone dimensional, too. This shows that each ξ i is of the form (3).From the functoriality in Theorem 2.1, it follows that the inclusionof each direct summand in (4) is the natural map, induced by theinclusion I Y ⊂ I Y i . Thus ξ is obtained from the ξ i ’s by pullingthem back over this inclusion of ideals, and adding the results inExt ( I Y (Θ) , O X ( − Θ)). By definition of (Baer) addition in Ext-groups,this means that there is a commutative diagram0 ✲ N O X ( − Θ) ✲ L Ni =1 ( P a i ⊕ P − a i ) ✲ L Ni =1 I Y i (Θ) ✲ ✲ O X ( − Θ) β ❄❄ ✲ F ❄❄ ✲ L Ni =1 I Y i (Θ) ✲ ✲ O X ( − Θ) ✲ E ∪ ✻ ✲ I Y (Θ) α ∪ ✻ ✲ L i ξ i , the bottom row is ξ , the top left square ispushout over the N -fold addition β , the bottom right square is pullbackalong the inclusion α , and F is just an intermediate sheaf (in fact avector bundle) that we do not care about. This diagram presents E asthe middle cohomology of a complexKer( β ) φ ✲ L Ni =1 ( P a i ⊕ P − a i ) ψ ✲ Coker( α ) . Now identify Ker( β ) with ( N − O X ( − Θ) by means of the monomor-phism( N − O X → N O X , ( f , . . . , f N − ) ( f , . . . , f N − , − P i f i )and similarly identify Coker( α ) with ( N − O X (Θ) by means of theepimorphism N O X → ( N − O X , ( f , . . . , f N ) ( f − f N , . . . , f N − − f N )(the latter is surjective even when restricted to L i I Y i because the Y i ’sare pairwise disjoint). Via these identifications, the homomorphisms φ and ψ are represented by the matrices as claimed, except that ϑ ± N appears with opposite sign. Change its sign, and we are done. (cid:3) Definition 3.3.
A monad is decomposable if it is isomorphic, as acomplex, to a monad of the form appearing in Proposition 3.2.With this terminology, a rank 2 vector bundle E can be resolved bya decomposable monad if and only if E (Θ) corresponds to a disjointunion Y = S i Y i , where Y i = Θ a i ∩ Θ − a i , via the Serre construction. ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 11
Remark 3.4.
The symmetry seen in the decomposable monads is noaccident, but reflects the self duality of E corresponding to the naturalpairing ∧ on E with values in V ( E ) ∼ = O X . See Barth–Hulek [1].4. Digression on the hyperext spectral sequence
Our basic aim is to understand first order deformations of the bun-dles E appearing as the cohomology of a decomposable monad. Thestrategy is to analyse Ext ( E , E ) using the first hyperext spectral se-quence associated to the monad. This is in principle straight forward,but requires some honest calculation. As preparation, we collect in thissection a few standard constructions in homological algebra, for easeof reference. We fix an abelian category A with enough injectives andinfinite direct sums, and denote by K ( A ) the homotopy category ofcomplexes and by D ( A ) the derived category.4.1. The spectral sequence.
Let ( M • , d M ) and ( N • , d N ) denote com-plexes in A , and assume that N • is bounded from below. The firsthyperext spectral sequence is a spectral sequence E pq = L i Ext q ( M i , N i + p ) ⇒ Ext p + q ( M • , N • ) . Briefly, take a double injective resolution N • → I •• with I •• concen-trated in the upper half plane (for instance a Cartan-Eilenberg res-olution), and form the double complex Hom •• ( M • , I •• ). The requiredspectral sequence is the first spectral sequence associated to this doublecomplex.4.2. The edge map.
Along the axis q = 0, the first sheet of the spec-tral sequence in Section 4.1 has the usual hom-complex Hom • ( M • , N • ).Its cohomology is E p, = Hom K ( A ) ( M • , N • [ p ])where the right hand side denotes homotopy classes of morphisms ofcomplexes. Since all differentials emanating from E p, r for r ≥ edge maps E p, ։ E p, ∞ ⊂ Ext p ( M • , N • )Viewing the right hand side as the group Hom D ( A ) ( M • , N • [ p ]) of mor-phisms in the derived category, it is reasonable to expect, and not hardto verify, that the edge map is in fact the canonical mapHom K ( A ) ( M • , N • [ p ]) → Hom D ( A ) ( M • , N • [ p ]) . Thus the image of E p, in the limit object Ext p ( M • , N • ), consistsof those p -extensions that can be realized by actual morphisms M • → N • [ p ] between complexes, without inverting quasi-isomorphisms. Differentials at E . For q = 1, it is convenient to view elementsof Ext ( M i , N i + p ) as extensions, in the sense of short exact sequences,and this viewpoint leads to the following interpretation of the differen-tials d p at the E -level: Lemma 4.1.
Let ξ ∈ E p be given as a collection of extensions ξ i : 0 → N i + p → X i → M i → . (1) We have d p ( ξ ) = 0 if and only if there are maps f i such thatthe diagram · · · ✲ N i + p − d N ✲ N i + p d N ✲ N i + p +1 ✲ · · ·· · · ✲ X i − ❄ f i − ✲ X i ❄ f i ✲ X i +1 ❄ ✲ · · ·· · · ✲ M i − ❄ d M ✲ M i ❄ d M ✲ M i +1 ❄ ✲ · · · commutes. (2) If we have such a collection of maps ( f i ) , then ξ represents anelement of E p , and the differential d p : E p → E p +2 , = Hom K ( A ) ( M • , N • [ p + 2]) sends ξ to the morphism having components M i − → N i + p +1 induced by f i ◦ f i − . In particular d p ( ξ ) = 0 if and only if thereexists a collection ( f i ) making the middle row in the diagramin (1) a complex.Proof. This is straight forward, although tedious, to verify directly fromthe construction of the spectral sequence. (cid:3)
Serre duality.
Let X be a scheme of pure dimension d over afield, with a dualizing sheaf ω X such that Grothendieck-Serre dual-ity holds. Let M • be a bounded below complex of coherent O X -modules. We obtain two spectral sequences from (5): one abuttingto Ext n ( O X , M • ) = H n ( X, M • ), which we denote by E , and one abut-ting to Ext n ( M • , ω X ), which we denote by ˆ E . Then E is nothing butthe first hypercohomology spectral sequence, and the E -levels of E and ˆ E are Grothendieck-Serre dual. We need to know that the dualityextends to all sheets. Lemma 4.2.
The two spectral sequences E and ˆ E are dual in thefollowing sense: (1) There are canonical dualities between the vector spaces E pqr and ˆ E − p,d − qr for all p, q, r , extending the Grothendieck-Serre dualitybetween H q ( X, M p ) and Ext n − q ( M p , ω ) for r = 1 . ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 13 (2)
The differentials d pqr : E pqr → E p + r,q − r +1 r ˆ d − p − r,d − q + r − r : ˆ E − p − r,d − q + r − r → ˆ E − p,d − qr are dual maps. (The statement can be extended to give a full duality between thetwo spectral sequences, including the filtrations on the abutments andall maps involved. The above is sufficient for our needs.) Proof.
This seems to be well known. We include a sketch, followingHerrera–Liebermann [8] (they work in a context where the complexeshave differentials that are differential operators of degree one; this de-mands more care than in our situation). Firstly, for any three com-plexes L • , M • , N • , the Yoneda pairingExt i ( L • , M • ) × Ext j ( M • , N • ) → Ext i + j ( L • , N • )can be defined on hyperext groups by resolving M • and N • by injec-tive double complexes, and taking the double hom complex. On this“resolved” level, the Yoneda pairing is given by composition, and thereis an induced pairing of hyperext spectral sequences in the appropri-ate sense, which specializes to the usual Yoneda pairing between extgroups of the individual objects L l , M m , N n at the E -level. Special-ize to the situation L • = O X and N • = ω X to obtain a morphism ofspectral sequences from E to the dual of ˆ E , in the above sense. Atthe E -level this is the Grothendieck-Serre duality map, hence an iso-morphism, which is enough to conclude that it is an isomorphism ofspectral sequences [5, Section 11.1.2]. (cid:3) Remark 4.3. If M • and N • denote two complexes of vector bundles,then we may apply the Lemma to the complex ( M • ) ∨ ⊗ N • to ob-tain a duality between the two hyperext spectral sequences abutting toExt n ( M • , N • ) and Ext n ( N • , M • ⊗ ω X ), respectively.5. Deformations of decomposable monads
We now apply the homological algebra from the previous sectionto analyse first order deformations of vector bundles E which can beresolved by a decomposable monad. Firstly, we find that deforma-tions obtained by varying the isomorphism ω Y ∼ = O Y (2Θ) in the Serreconstruction coincides with the deformations obtained by varying thedifferential in the monad, while keeping the objects fixed. Secondly,and this is the nontrivial part, we find that all first order deformationsof E can be obtained by also deforming the objects in the monad, andthere are more of these deformations than those obtained by varying Y in the Serre construction. Since the objects in the monad are sums ofline bundles, their first order deformations are easy to understand, sowe are able to compute the dimension of Ext ( E , E ), in Theorem 5.7. E − , ✲ E − , ✲ E , E , E , E , ✲ E , ✲ E , Figure 1.
The first sheet in the spectral sequence for Ext i ( E , E )5.1. Calculations in the spectral sequence.
Let E be the rank 2vector bundle given as the cohomology of a decomposable monad M • : A φ ✲ B ψ ✲ C given explicitly in Proposition 3.2. In particular, C = A ∨ , and B isself dual. If we fix the self duality ι : B → B ∨ , given by the direct sumof the skew symmetric (cid:18) −
11 0 (cid:19) : P a i ⊕ P − a i → P − a i ⊕ P a i , then ψ = φ ∨ ◦ ι . More generally, for any map f : A → B , we defineits transpose f t : B → C by f t = f ∨ ◦ ι. Thus ψ is the transpose of φ . The spectral sequence from Section 4.1 gives(5) E pq = L i Ext q ( M i , M i + p ) ⇒ Ext p + q ( E , E ) . Using that, for any x ∈ X , line bundles of the form P x ( m Θ) have sheafcohomology concentrated in degree 0 when m > m <
0, we see that the nonzero terms in the first sheet have theshape depicted in Figure 1. It follows that all differentials at level E r vanish for r = 3 and r >
4. Also, the duality of Section 4.4, applied to M • ⊗ ( M • ) ∨ , shows that each term E pqr is dual to E − p, − qr , and similarlyfor the differentials. In this section we analyse the E -sheet, and getas a consequence that the spectral sequence in fact degenerates at the E -level. One can show that, in the affine space of all homomorphisms f : A → B ,the locally closed subset U defined by (1) f is an embedding of vector bundles,and (2) the composition f t ◦ f is zero, has an irreducible connected componentcorresponding to decomposable monads. It seems plausible that this component isall of U . ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 15
The objects E pq . By duality, it suffices to consider the lowerhalf of Figure 1. The only nonzero differentials in this area, at the E -level, are in the lower row q = 0. We observed in Section 4.2that the cohomology groups of this row are the groups of morphisms M • → M • [ p ] modulo homotopy. Lemma 5.1.
The dimensions of E p, for p = 0 , , are , N − and N − − N + 2 , respectively.Proof. The vector spaces in question are the cohomologies of the com-plex 0 ✲ E , d , ✲ E , d , ✲ E , ✲ , where dim E , = dim (cid:0) Hom( A , A ) ⊕ Hom( B , B ) ⊕ Hom( C , C ) (cid:1) = 2( N − + 2 N dim E , = dim (cid:0) Hom( A , B ) ⊕ Hom( B , C ) (cid:1) = 4 N ( N − E , = dim Hom( A , C )= 8( N − (6)(using that the space of global sections of O X (Θ ± a i ) has dimension 1,and the space of global sections of O X (2Θ) has dimension 8). Thus itsuffices to compute the dimensions of the kernels of the two differentials d , and d , , i.e. the vector spaces of morphisms of degree 0 and 1 fromthe monad to itself.One checks immediately that any morphism M • → M • (of degree 0)is multiplication with a scalar, so(7) dim E , = 1 . Next we compute the dimension of the space of morphisms M • → M • [1]. Since C = A ∨ , such a morphism is given by an element ofHom( A , B ) ⊕ Hom( B , A ∨ ) , which we may write as ( µ, − ν t ), where both µ and ν are homomor-phisms A → B . The sign on − ν t is inserted to compensate for thesign on the differential in the shifted complex M • [1]; thus ( µ, − ν t ) de-fines a morphism M • → M • [1] if and only if ν t ◦ φ = φ t ◦ µ .As in Proposition 3.2, we decompose these homomorphisms into pairs µ ± and ν ± , and then ν t ◦ φ = ( ν − ) ∨ ◦ φ + − ( ν ) + ∨ ◦ φ − φ t ◦ µ = ( φ − ) ∨ ◦ µ + − ( φ + ) ∨ ◦ µ − . (8)Choosing generators ϑ ± i ∈ Γ( X, O X (Θ ± a i )), we may represent µ by amatrix with entries µ ± ij ϑ ± i , where µ ± ij are scalars. Similarly for ν . Then the two compositions (8) are given by ( N − × ( N −
1) scalar matriceswith entries( ν t ◦ φ ) ij = ( µ + ij − µ − ij ) ϑ + i ϑ − i + ( µ + Nj − µ − Nj ) ϑ + N ϑ − N ( φ t ◦ µ ) ij = ( ν + ji − ν − ji ) ϑ + j ϑ − j + ( ν + Ni − ν − Ni ) ϑ + N ϑ − N . (9)Recall that the Kummer map X → | | sends a i ∈ X to the divisorΘ a i + Θ − a i . This implies that, for sufficiently general points a i , and i = j , the three elements ϑ + i ϑ − i , ϑ + j ϑ − j and ϑ + N ϑ − N are linearly independent inΓ( X, O X (2Θ)). It follows easily that the two expressions in (9) coincidefor all i and j if and only if there are equalities of scalar ( N − × ( N − µ + ij ) − ( µ − ij ) = ( ν + ij ) − ( ν − ij ) = c c . . . c N − c N c N · · · c N , where c , . . . , c N are arbitrary scalars. Thus the vector space of mor-phisms M • → M • [1] has a basis corresponding to the ( µ + ij ), ( ν + ij ) and( c i ), hence has dimension 2 N ( N −
1) + N . The expressions for dim E p, follow from this, together with (6) and (7). (cid:3) The differentials d pq . The only nonzero differentials at the E -level are d , and its dual d − , . So it suffices to analyse d , . This is, byLemma 4.1, an obstruction map for equipping first order infinitesimaldeformations of the objects M i with differentials, and will henceforthbe denoted ob.The domain(10) E , = L i Ext ( M i , M i )of ob = d , is canonically isomorphic to a direct sum of a large numberof copies of H ( X, O X ). More precisely, for each i and j from 1 to N − ( − , − ) to the i ’th projection A → O X ( − Θ) inthe first argument and the j ’th inclusion O X ( − Θ) → A in the secondargument. This defines an inclusion f ij : H ( X, O X ) ∼ = Ext ( O X ( − Θ) , O X ( − Θ)) ֒ → Ext ( A , A )and clearly the direct sum of all the f ij ’s is an isomorphism. Similarly,for all i and j from 1 to N −
1, we define inclusions h ij : H ( X, O X ) ∼ = Ext ( O X (Θ) , O X (Θ)) ֒ → Ext ( C , C )whose direct sum is an isomorphism. Finally, for all i from 1 to N , andeach sign ± , define inclusions g ± i : H ( X, O X ) ∼ = Ext ( P ± a i , P ± a i ) ֒ → Ext ( B , B ) ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 17 induced by projection to and inclusion of the summand P ± a i of B .Note that also the direct sum of the g ± i ’s is an isomorphism, sinceExt ( P x , P y ) = H ( X, P y − x ) vanishes unless x = y .The obstruction map ob takes values in homotopy classes of mor-phisms M • → M • [2] of complexes, modulo homotopy. Such a mor-phism is given by a single homomorphism from A to C , which can bepresented as an ( N − × ( N −
1) matrix with entries in Γ( X, O X (2Θ)).We now give such a matrix representative for the homotopy class ob( ξ ),for any element ξ in each summand H ( X, O X ) of E , . Lemma 5.2.
For every i , the boundary map of the long exact coho-mology sequence associated to the Koszul complex ✲ O X ϑ + i ϑ − i ! ✲ O X (Θ a i ) ⊕ O X (Θ − a i ) ( ϑ − i − ϑ + i ) ✲ I Y i (2Θ) ✲ induces an isomorphism H ( X, I Y i (2Θ)) / h ϑ + i ϑ − i i ∼ = H ( X, O X ) where h ϑ + i ϑ − i i denotes the one dimensional vector space spanned by thesection ϑ + i ϑ − i .Proof. Since H ( X, O X (Θ ± a i )) = 0, there is an induced right exactsequence H ( X, O X (Θ a i )) ⊕ H ( X, O X (Θ − a i )) ✲ H ( X, I Y i (2Θ)) ✲ H ( X, O X ) ✲ . Each summand H ( X, O X (Θ ± a i )) is spanned by ϑ ± i , which is sent to ∓ ϑ + i ϑ − i in H ( X, I Y i (2Θ)). (cid:3) Proposition 5.3.
Let ξ ∈ H ( X, O X ) . The obstruction map ob doesthe following on each summand in its domain: (1) Lift ξ to sections u and v of I Y i (2Θ) and I Y N (2Θ) , respectively,using the lemma. Then ob( f ij ( ξ )) is represented by the ( N − × ( N − matrix having j ’th column (the transpose of ) ( v · · · v u + v v · · · v ) ↑ (entry i ) and zeros everywhere else. (2) Lift ξ to a section u of I Y i (2Θ) . If i = N , then ob( g ± i ( ξ )) isrepresented by the ( N − × ( N − matrix having u at entry ( i, i ) , and zeros everywhere else. The remaining case ob( g ± N ( ξ )) is represented by the ( N − × ( N − matrix having all entriesequal to u . (3) Lift ξ to sections u and v of I Y j (2Θ) and I Y N (2Θ) , respectively.Then ob( h ij ( ξ )) is represented by the ( N − × ( N − matrixhaving i ’th row ( v · · · v u + v v · · · v ) ↑ (entry j ) and zeros everywhere else.Proof. Notation: In the commutative diagrams that follow, we will usedotted arrows roughly to indicate maps that are not given to us, butneed to be filled in by some construction.
Part 1:
We view ξ as an extension in Ext ( O X ( − Θ) , O X ( − Θ)).Writing out the description of ob from Lemma 4.1 in this situation,one arrives at the diagram(11) 0 ✲ O X ( − Θ) ✲ F ✲ O X ( − Θ) ✲ A ❄ ∩ φ ✲ B ˆ φ i ❄ ψ ✲ φ i ✲ C ❄ constructed as follows: The top row is the extension ξ and the bottomrow is the monad. The leftmost vertical map is inclusion of the i ’thsummand, so in terms of matrices, φ i is the i ’th column of φ . We arerequired to extend φ i to a map ˆ φ i making the left part of the diagramcommute: one way of doing this is detailed below. The induced verticalmap on the right, precomposed with projection A → O X ( − Θ) on the j ’th summand, is a representative for ob( f ij ( ξ )).The assumption that u is a lifting of ξ , means that there is a com-mutative diagram0 ✲ O X ( − Θ) ϑ + i ϑ − i ! ✲ P a i ⊕ P − a i ( ϑ − i − ϑ + i ) ✲ I Y i (Θ) ✲ ✲ O X ( − Θ) ✲ F ˜ u ✻ ✲ O X ( − Θ) u ✻ ✲ v fits in the pullback diagram:0 ✲ O X ( − Θ) ϑ + N ϑ − N ! ✲ P a N ⊕ P − a N ( ϑ − N − ϑ + N ) ✲ I Y N (Θ) ✲ ✲ O X ( − Θ) ✲ F ˜ v ✻ ✲ O X ( − Θ) v ✻ ✲ φ i to be(˜ u, ˜ v ) : F ✲ ( P a i ⊕ P − a i ) ⊕ ( P a N ⊕ P − a N ) ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 19 followed by the appropriate inclusion to B . One verifies immediatelythat ˆ φ i extends φ i in (11), and that the induced map in the rightmostpart of that diagram is given by the vector as claimed in part 1. Part 2:
We view ξ as an extension in Ext ( P ± a i , P ± a i ). In this sit-uation, the description of ob from Lemma 4.1 boils down to a diagram0 ✲ P ± a i ✲ G ✲ P ± a i ✲ C ψ ± i ❄ ✛ A φ ± i ✻ ✛ constructed as follows: The top row is ξ and the two vertical maps φ ± i and ψ ± i denote the i ’th row of φ ± and the i ’th column of ψ ± . The taskis to lift φ ± i to the rightmost dotted arrow, and to extend ψ ± i to theleftmost dotted arrow. The composition of the two dotted arrows isthen a representative for ob( g ± i ( ξ ))).First consider the problem of lifting ∓ ϑ ± i and extending ϑ ∓ i to maps s and t as in the following diagram:(12) 0 ✲ P ± a i ✲ G ✲ P ± a i ✲ O X ( − Θ) ϑ ∓ i ❄ t ✛ O X (Θ) ∓ ϑ ± i ✻ s ✛ Suppose such a diagram is given. If i < N , then ∓ (0 , . . . , , s, , . . . , φ ± i and ± (0 , . . . , , t, , . . . ,
0) would extend ψ ± i . Their com-position is the matrix having t ◦ s in entry ( i, i ), and zeros elsewhere. If i = N , then similarly ∓ ( s, . . . , s ) and ± ( t, . . . , t ) would be the requiredlift and extension. Their composition is the matrix having all entriesequal to t ◦ s . Thus part 2 of the proposition will be established oncewe have constructed such maps s and t having composition t ◦ s = u .Now use that ξ is the pullback of the Koszul complex for Y i along u .This enables us to construct the commutative diagram0 ✲ O X ( − Θ) ϑ + i ϑ − i ! ✲ P a i ⊕ P − a i ( ϑ − i − ϑ + i ) ✲ I Y i (Θ) ✲ P ∓ a i ✛ ϑ ∓ i ✲ P ∓ a i ∓ ϑ ± i ✲ ✛ ✲ O X ( − Θ) ✲ G ′ ✻ ✲ O X ( − Θ) u ✻ ✲ P ∓ a i ✛ ϑ ∓ i ✲ P ∓ a i ( − u ✻ ∓ ϑ ± i ✲ ✛ as follows: The top row is the Koszul complex. In the top part, theunlabelled diagonal arrows are the canonical inclusion of and projectionto the summand P ∓ a i . In particular their composition is the identity map. Pull back along u to get the short exact sequence in the lowerpart of the diagram. Thus this sequence coincides with ξ , twisted by P ∓ a i ( − Θ). There are now uniquely determined dotted arrows makingthe diagram commute, and their composition is u . Twisting back by P ± a i (Θ), the lower part of the diagram is thus the required diagram(12). This ends the proof of part 2. Part 3 is essentially dual to part 1, and is left out. (cid:3)
By Lemma 2.7, we have(13) H ( I Y i (2Θ)) ⊕ H ( I Y j (2Θ)) = H ( O X (2Θ))for all i = j (the Lemma gives an inclusion of the left hand side into theright hand side, and by Riemann-Roch, the two sides have the samedimension). This decomposition of sections of O X (2Θ), together withthe explicit description of ob in the proposition, enables us to conclude: Corollary 5.4.
The obstruction map ob is surjective.Proof. We show that any ( N − × ( N −
1) matrix of sections of O X (2Θ) represents an element in the image of ob. Let i and j bearbitrary indices between 1 and N − v be a section of I Y N (2Θ). Let ξ ∈ H ( X, O X ) bethe image of v under the boundary map in Lemma 5.2, and lift ξ toanother section u of I Y i (2Θ). By parts 1 and 2 of the Proposition,ob( f ii ( ξ ) − g ± i ( ξ )) is represented by the matrix having zeros except forin column i , where all elements equal v . Similarly ob( h ii ( ξ ) − g ± i ( ξ ))is represented by a matrix having all entries of row i equal to v , andzeros elsewhere.Step 2: Let u be a section of I Y i (2Θ). By part 1 of the Propositionand the previous step, we can find a matrix representing an element inthe image of ob, with u as entry ( i, j ) and zeros elsewhere. Similarly,for any section u of I Y j (2Θ), combining part 3 of the proposition withthe previous step, we obtain a matrix having u as entry ( i, j ) and zeroselsewhere.Step 3: If i = j , the previous step and (13) enables us to constructa matrix with arbitrary entries outside the diagonal. Combining thiswith step 1, we can construct a matrix having any given section of I Y N (2Θ) at entry ( i, i ), and zeros elsewhere. By step 2 we can alsoconstruct a matrix having any given section of I Y i (2Θ) at ( i, i ), andzeros elsewhere. By (13) with j = N , this enables us to hit arbitraryelements along the diagonal, too. (cid:3) Corollary 5.5.
The spectral sequence (5) degenerates at E .Proof. The previous corollary implies E , = 0. By duality also E − , =0. It follows from the shape of the first sheet, Figure 1, that all differ-entials vanish at the E -level and beyond. (cid:3) ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 21
First order deformations.
From the calculations in the previ-ous section, we can understand infinitesimal deformations of the vectorbundle E in terms of its monad. Let k [ ǫ ] be the ring of dual numbers.By a first order deformation of M • , we mean a monad over X ⊗ k k [ ǫ ],with M • as fibre over ǫ = 0, modulo isomorphism. Theorem 5.6.
Let M • be a decomposable monad with cohomology E .The vector spaces of first order infinitesimal deformations of M • and of E are isomorphic via the natural map, sending a first order deformationof M • to its cohomology.Proof. Since the hyperext spectral sequence associated to the monaddegenerates at E , and the only E pq terms with p + q = 1 are E , and E , , there is a short exact sequence(14) 0 ✲ E , ✲ Ext ( E , E ) ✲ E , ✲ . Let D ( M • ) be the vector space of first order deformations of M • . Thusthe claim is that the natural map D ( M • ) → Ext ( E , E ) is an isomor-phism. It suffices to show that D ( M • ) → E , is surjective, and thatits kernel maps isomorphically to E , .Now E , is the kernel of the obstruction map ob = d , . By Lemma4.1, this is the space of those first order deformations of the objectsin M • , that allow the differential d M to extend (non uniquely) to thedeformed objects. Via this identification, D ( M • ) → E , is the naturalforgetful map, so it is surjective. Moreover, its kernel is the space offirst order deformations of the differential in M • , keeping the objectsfixed. It remains to see that this space gets identified with E , .By the shape of the spectral sequence (Figure 1) we have E , = E , ,and, by Lemma 5.1, this is E , = Hom K ( X ) ( M • , M • [1]) . The inclusion of E , into Ext ( E , E ) is the edge map discussed inSection 4.2, i.e. the canonical map(15) Hom K ( X ) ( M • , M • [1]) → Hom D ( X ) ( M • , M • [1]) . This can be factored as follows: a morphism of complexes in an ar-bitrary abelian category f : X • → Y • [1] gives rise to a short exactsequence of complexes(16) 0 ✲ Y • β ✲ Z • α ✲ X • ✲ Z • = C ( f [ − f [ − X i ⊕ Y i in degree i and differential ( x, y ) ( dx, f ( x )+ dy ). The maps α and β are the canonical ones. Moreover, the usual Yoneda constructionof elements in Ext from short exact sequences (of objects) can beextended to complexes, by associating to any short sequence (16) the roof C ( β ) X • qism ✛ Y • [1] ✲ where C ( β ) is the mapping cone, with objects Y i +1 ⊕ Z i in degree i and differential ( y, z ) ( − dy, β ( y ) + dz ). The leftmost map is givenby projection, and is a quasi-isomorphism, whereas the rightmost mapis projection followed by α . This roof defines a morphism X • → Y • [1]in the derived category. Moreover, the diagram obtained from the roofby adding the negative of the map f : X • → Y • [1] we started with,is commutative up to homotopy, so the roof and − f defines the samemap in the derived category.Thus we have factored the edge map (15) via short exact sequences,by sending f : M • → M • [1] to the short exact sequence0 ✲ M • ✲ C ( f [ − ✲ M • ✲ . The associated element in Hom D ( X ) ( M • , M • [1]) corresponds, up tosign, to the Yoneda class in Ext ( E , E ) obtained by taking the H cohomology of each complex in this short exact sequence. To phrasethis in terms of first order deformations, we rewrite the cone C ( f [ − M • ⊗ k k [ ǫ ] equipped with the differential d M ⊗ f ⊗ ǫ .The corresponding deformation of E is the H cohomology of this com-plex. But this is the required result, since any differential on M • ⊗ k k [ ǫ ]that specializes to d M for ǫ = 0 has the form d M ⊗ f ⊗ ǫ , for some f satisfying ( d M ⊗ f ⊗ ǫ ) = 0 . Since d M = 0 and ǫ = 0, this says that f ◦ d M + d M ◦ f = 0, i.e. f defines a morphism M • → M • [1]. This gives the required identificationbetween E , and deformations of the differential. (cid:3) Next, we give the dimension formula for Ext ( E , E ), which we phrasein a twist invariant way. Theorem 5.7.
Let E be a rank vector bundle obtained as the coho-mology of a decomposable monad, or the twist of such a bundle by aline bundle. Then dim Ext ( E , E ) = ∆( E ) · Θ + 5 where ∆ denotes the discriminant c − c .Proof. Both sides of the equation are invariant under twist, so it sufficesto verify the formula when E is the cohomology of a decomposablemonad. Consider again the short exact sequence (14). ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 23
The space E , is the kernel of the map ob = d , studied in Section5.1.2. Its domain (10) has dimension(2( N − + 2 N ) dim H ( X, O X ) = 6( N − + 6 N and its codomain E , has dimension 6( N − − N + 2, by Lemma 5.1.Since ob is surjective, the dimension of its kernel E , is thus 7 N − E , = E , is N − ( E , E ) has dimension 8 N − E (Θ)has Chern classes c = 2Θ and c = N Θ , and thus discriminant(4 N − . The formula follows. (cid:3) Remark 5.8.
The space of first order deformations obtained by vary-ing the isomorphism ω Y ∼ = O Y (2Θ), coincides with the space of firstorder deformations of the differential in M • . In fact, it is trivial thatthe former is contained in the latter, and these spaces have the samedimension N −
1, using Proposition 2.8.
Remark 5.9.
The short exact sequence (14), and its interpretationgiven in the proof of Theorem 5.6, is not intrinsic to E , but resultsfrom our choice of representing E by a decomposable monad. How-ever, deformation of the differential in M • , or equivalently, variationof the isomorphism O Y (2Θ) ∼ = ω Y in the Serre construction, defines arational ( N − E in its moduli space,whose tangent space is E , . It seems plausible that this ( N − E .6. Birational description of M (0 , Θ )As before, let ( X, Θ) be a principally polarized abelian threefold withPicard number 1. We write M ( c , c ) for the coarse moduli space ofstable rank 2 vector bundles on X with the indicated Chern classes.The main point in the preceding section is that all first order in-finitesimal deformations of the vector bundles constructed in Section4.4, in the case of even c , can be realized as first order infinitesimaldeformations of a monad. In this section we show that in the first non-trivial example, corresponding to N = 2, this statement holds not onlyinfinitesimally, but Zariski locally: by deforming the monad, we realizea Zariski open neighbourhood of the vector bundle in its moduli space.In terms of the Serre construction, this is the case corresponding tocurves Y ∪ Y with two components Y i = Θ a i ∩ Θ − a i . Theorem 6.1.
Let E be the rank cohomology vector bundle of adecomposable monad, as in Proposition 3.2 for N = 2 . Then, Zariskilocally around E , the moduli space M (0 , Θ ) is a uniruled, nonsingularvariety of dimension . More precisely, there is a Zariski open neighbourhood around E whichis isomorphic to a nonsingular Zariski open subset of a P -bundle overa finite quotient of X × X X × X X , where X is considered as ascheme over X via the group law.Proof. We write down a parameter space for the family of monads(17) 0 ✲ P b ′ ( − Θ) φ ✲ L i =1 ( P a i ⊕ P a ′ i ) ψ ✲ P b (Θ) ✲ a i , a ′ i , b, b ′ are sufficiently general points in X satisfying(18) a + a ′ = a + a ′ = b + b ′ , and φ = ( ϑ , ϑ ′ , ϑ , ϑ ′ ) , ψ = ( ϑ ′ , − ϑ , ϑ ′ , − ϑ )and where the ϑ ’s are required to be nonzero, but otherwise arbitrary.Viewing X as a variety over X via the group law, the fibred product X × X X × X X is the subvariety of X defined by (18). Let T ⊂ X × X X × X X be the open subset consisting of sixtuples ( a , a ′ ; a , a ′ ; b, b ′ ) where theleading four entries are all distinct. Later we may have to shrink T further. With the help of the Poincar´e line bundle on X × X it isclear that, on T × X , there exist vector bundles A , B , C whose fi-bres over a sixtuple in T are the three objects in (17). The sixtuples( a , a ′ ; a , a ; b, b ′ ) in T corresponding to the same three objects con-stitute an orbit for the action of(19) G = ( Z / (2) ⊕ Z / (2)) ⋊ Z / (2)on T , where the action of the first semidirect factor is given by thetranspositions a ↔ a ′ and a ↔ a ′ , and the last factor acts by( a , a ′ ) ↔ ( a , a ′ ). Thus T /G is a parameter space for the objectsin (17).Next we parametrize the maps φ and ψ , which are given by fournonzero sections(20) ϑ ∈ Γ( X, P a − b ′ (Θ)) ϑ ′ ∈ Γ( X, P a ′ − b ′ (Θ)) ϑ ∈ Γ( X, P a − b ′ (Θ)) ϑ ′ ∈ Γ( X, P a ′ − b ′ (Θ))There exist line bundles L , L ′ , L , L ′ on T , whose fibres over a sixtuple( a , a ′ ; a , a ′ ; b, b ′ ) are these (one dimensional) spaces of global sections.Thus, writing F = L i =1 ( L i ⊕ L ′ i ) p −→ T, a point of F , whose four entries are all nonzero, corresponds to a monad(17). More precisely, writing p X for the product p × id X : F × X → T × X , there exists a monad p ∗ X A Φ ✲ p ∗ X B Ψ ✲ p ∗ X C ECTOR BUNDLES AND MONADS ON ABELIAN THREEFOLDS 25 on F × X , whose restriction to the point in F given by (20) is (17). Let F ′ ⊂ F be the open subset consisting of quadruples with only nonzeroentries. The cohomology of the “universal” monad above is a family ofvector bundles over F ′ , giving rise to a morphism of schemes(21) φ : F ′ → M (0 , Θ ) . To make this morphism an embedding, we will divide by the group G to get rid of the ambiguity in the parametrization of the objects by T ,and further divide by another group Γ to take care of distinct maps φ , ψ which give isomorphic monads.For a fixed base point in T , and hence fixed objects in (17), the tuples( ϑ , ϑ ′ , ϑ , ϑ ′ ) which define isomorphic monads constitute orbits underthe following group action on F : view G m as a variety over G m via themultiplication map, and letΓ = G m × G m G m . Its closed points are tuples ( λ , λ ′ ; λ , λ ′ ) satisfying λ λ ′ = λ λ ′ . Theaction on the fibres of F is given on closed points by( ϑ , ϑ ′ , ϑ , ϑ ′ ) ( λ ϑ , λ ′ ϑ ′ , λ ϑ , λ ′ ϑ ′ ) . There is a short exact sequence of group varieties1 ✲ G m ✲ Γ ✲ G m ✲ λ , λ ′ ) to ( λ , λ − , λ , λ − ) and the projec-tion sends ( λ , λ ′ , λ , λ ′ ) to λ i λ ′ i . Correspondingly, we determine F/ Γin two steps. Firstly, the categorical quotient by the G m -action is F/ G m ∼ = L i =1 ( L i ⊗ L ′ i )and the quotient map F → F/ G m corresponds to multiplication in thefibres (locally on T , this is the product of two copies of the quotientSpec R [ x, y ] → Spec R [ xy ] for the G m -action ( x, y ) → ( λx, λ − x ) on A R over an arbitrary ring R ). The induced action of Γ / G m ∼ = G m onthe rank two vector bundle F/ G m is multiplication in the fibres, so thequotient P = { F \ } / Γ is P = P ( L i =1 ( L i ⊗ L ′ i ) ∨ ) , which is a P -bundle over T . The image P ′ ⊂ P of F ′ ⊂ F is anopen subset; in fact it is the complement of the two natural sectionscorresponding to the subbundles L i ⊗ L ′ i of F/ G m . The restrictedquotient map F ′ → F ′ / Γ = P ′ is a geometric quotient; in particular its fibres are orbits in F ′ . It isclear that the morphism (21) is invariant with respect to the Γ-actionon F ′ , so there is an induced morphism φ : P ′ → M (0 , Θ ) . Moreover, the (free) action (19) of G on T has a canonical lift to P , and P ′ is G -invariant. Again φ is invariant under this action, so we obtainthe P -bundle P/G over
T /G , together with an open subset P ′ /G andan induced morphism φ : P ′ /G → M (0 , Θ ) . By construction, the domain P ′ /G parametrizes isomorphism classesof monads of the form (17). Given two such monads M • and M • , withcohomology E and E , the first hyperext spectral sequence gives anisomorphism Hom( M • , M • ) ∼ → Hom( E , E ). Here, the domain is the E , -term in the spectral sequence, which is the group of morphisms ofcomplexes (there are no homotopies, since E − , vanishes). It followsthat M • and M • are isomorphic as complexes if and only if E and E are isomorphic vector bundles. In other words φ is injective on closedpoints. Shrinking T if necessary, we may apply Theorem 5.6 to see that φ is ´etale at points where b = b ′ = 0. By shrinking its domain furtherif necessary, we can assume that it is ´etale everywhere. An ´etale andinjective morphism is an open embedding, so we are done. (cid:3) References
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