Let X be a smooth projective complex variety, and let G be an algebraic
reductive complex group. We define the notion of principal G-sheaf, that
generalises the notion of principal G-bundle. Then we define a notion of
semistability, and construct the projective moduli space of semistable
principal G-sheaves on X. This is a natural compactification of the moduli
space of principal G-bundles.
  This is the announcement note presented by the second author in the
conference held at Catania (11-13 April 2001), dedicated to the 60th birthday
of Silvio Greco. Detailed proofs will appear elsewhere.Comment: 10 pages, LaTeX2e. Submitted to the conference proceedings of
  "Commutative Algebra and Algebraic Geometry", Catania, April 200

Gomez, Tomas L.

Sols, Ignacio

English

arXiv

Tomás L. Gomez

Ignacio Sols

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Projective moduli space of semistable principal sheaves for a reductive group

Gomez, Tomás L.

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LV (2000)  Fasc. II, pp. 437446PROJECTIVE MODULI SPACE OF SEMISTABLE PRINCIPALSHEAVES FOR A REDUCTIVE GROUPTOMA´S L. GO´MEZ - IGNACIO SOLSDedicated to Silvio Greco in occasion of his 60-th birthday.1. Introduction.This contribution to the homage to Silvio Greco is mainly an announce-ment of results to appear somewhere in full extent, explaining their developmentfrom our previous article [5] on conic bundles.In [11] and [15] Narasimhan and Seshadri de�ned stable bundles on acurve and provided by the techniques of Geometric Invariant Theory (GIT) de-veloped by Mumford [10] a projective moduli space of the stable equivalenceclasses of semistable bundles. Then Gieseker [4] and Maruyama [8] [9] gener-alized this construction to the case of a higher-dimensional projective variety,obtaining again a projective moduli space by also allowing torsion-free sheaves.Ramanathan [12] [13] has provided the moduli space of semistable principalbundles on a connected reductive group G , thus generalizing the Narasimhanand Seshadri notion and construction, which then becomes the particular caseG = Gl(n,C).Faltings [3] has considered the moduli stack of principal bundles onsemistable curves. For G orthogonal or symplectic he considers a torsion-freeBoth authors are members of EAGER (EC FP5 Contract no. HPRN-CT-2000-00099). T. G. was supported by a postdoctoral fellowship of Ministerio de Educacio´n yCultura (Spain)438 TOMA´S L. GO´MEZ - IGNACIO SOLSsheaf with a quadratic form, and he also de�nes a notion of stability. For gen-eral reductive group G he uses the approach of loop groups. Sorger [19] hadconsidered a similar problem. He works on a curve C (not necessarily smooth)on a smooth surface S , and constructs the moduli space of torsion free sheaveson C together with a symmetric form taking values on the dualizing sheaf ωC .In the talk open problems on principal bundles closing the conference onvector bundles on algebraic curves and Brill-Noether theory at Bad Honeff2000, prof. Narasimhan proposed the problem of generalizing the work ofthe late Ramanathan to the case of higher-dimensional varieties and to thecase of positive characteristics. We solve the �rst problem by providing asuitable de�nition of principal sheaf on a higher-dimensional projective varietyX over the complex �eld, and a de�nition of its (semi)stability, which in casedim X = 1 is that of Ramanathan, and for which a projective moduli space canbe obtained.We start by recalling our notion of (semi)stable conic bundles, i.e. sym-metric (2, 0)-tensors ϕ : E ⊗ E → OC of rank E = 3 on an algebraic curveC , and their projective moduli space, notion and moduli space which have beengeneralized to the case of (s, 0) tensors on a curve by Schmitt [14] with the pur-pose of dealing with (semi)stable objects ϕ : Eρ → M , where Eρ is the vectorbundle associated to a vector bundle E and an arbitrary representation ρ ofG = Gl(n,C), and M is a line bundle. In case the symmetric (2, 0) tensor is ofmaximal rank at all points and det(E) ∼= OC , i.e. the case when (E, ϕ) is just aprincipal SO(3,C)-bundle, our notion of (semi)stability is drastically simpli�edand becomes equivalent to Ramanathans notion of (semi)stability. We then gen-eralize to higher dimension, with techniques of Simpson [18] and Huybrechts-Lehn [7], the notion and coarse projective moduli space of (semi)stable (s, 0)-tensors, by allowing E to be a torsion free sheaf and those symmetric or anti-symmetric and nowhere degenerate provide thus the moduli space of principalsheaves on G = O(n,C), Sp(n,C), SO(2n + 1,C), the remaining classicalgroup SO(2n,C) requiring a special treatment which fortunately does not alterthe notion of (semi)stability.Then we cope with the problem of an arbitrary connected reductive groupG , by de�ning principal sheaves as (2, 1) tensors, i.e. torsion free sheavesE and ϕ : E ⊗ E → E∗∗, which on the points of the open set UE whereE is locally free are isomorphic to the structure tensor ϕg� : g� ⊗ g� → g�of the Lie algebra g� tangent to the commutator G � = [G,G], together witha G → Aut (g�) reduction of the associated principal bundle on UE . The(semi)stability is de�ned as the δ-(semi)stability of the (2, 1) tensor (E, ϕ)for a polynomial parameter δ with degree exactly dim X − 1, and it leads toa coarse projective moduli space. Furthermore, it reduces to RamanathansPROJECTIVE MODULI SPACE OF SEMISTABLE PRINCIPAL. . . 439(semi)stability and moduli space in case dim X = 1.This announcement note consists mainly of the precise de�nitions andstatements of such objects and results.2. Conic bundles.Let X be a complete, smooth, connected curve, and �x a positive rationalτ > 0. A conic bundle on X of degree d is a rank 3 symmetric (2, 0)-tensoron X , i.e. a vector bundle E of rank 3 and degree d , together with a nonzerohomomorphismϕ : E2 = Sym2E → LWhere L is a line bundle. We say it is (semi)stable if1) For any subbundle F ⊆ E , it isdegF − cϕ(F)τrankF (≤)degE − 2τrankEwherecϕ(F) =2, if ϕ(F2) �= 0 (i.e. F not isotropic)1, if ϕ(F2) = 0 and ϕ(FE) �= 00, if ϕ(FE) = 0 (i.e. F singular)2) For all critical �ags F1 ⊆ F2 ⊆ E ,degF1 + degF2 (≤) degE,where a critical �ag is a �ag with F1 isotropic of rank 1, F2 of rank 2,ϕ(F1F) �= 0, ϕ(F2F2) �= 0 and ϕ(F1F2) = 0 (at a general point of X theseare a point of the conic and its tangent line). In other words, choosing a basisadapted to the �ltration on the �ber of E over a general point, the matrix formof ϕ is of the form � 0 0 ×0 × ·× · ·�where × is a nonzero entry, and  ·  is arbitrary.By the expression (semi)stable we alwaysmean both semistable and stable,and then by the symbol (≤) we mean ≤ and <, respectively. As usual, thereis a notion of stable equivalence classes of semistable objects (see [5] for thede�nition), and then it is proved in [5], by the use of GIT, the following.440 TOMA´S L. GO´MEZ - IGNACIO SOLSTheorem 1. There is a projective coarse moduli space of stable equivalenceclasses of semistable conic bundles of degree d and parameter τ , on a smooth,complete, connected curve.If det(E) ∼= OX , L ∼= OX and ϕ is nowhere degenerate, i.e. suchthat rank ϕ(x) = 3 for all x ∈ X , which amounts to a principal SO(3) −bundle on X , then the condition 2), independent of the parameter τ is enoughfor the de�nition of (semi)stability, thus leading to a projective coarse modulispace as in Theorem 1.Recall that in [12], [13], a de�nition of (semi)stable principal bundle Pon a curve, for a connected, reductive group G was already given: if for allreduction P(H ) of P to a maximal parabolic subgroup H ⊆ G , the vectorbundle P(H, h) associated to P by the adjoint representation of H in its tangentLie algebra h, hasdegP(H, h) (≤) 0In fact Ramanathan obtains in [13] a projective coarse moduli space of stableequivalence classes of semistable principal G-bundles of �xed topologicaltype and our result for SO(3)-bundles on X becomes a particular case ofRamanathans result, because it is proved in [5] that condition 2 is equivalentto the notion of Ramanathan.Rank 2 bundles correspond, after projectivization, to geometrically ruledsurfaces, and properties of the (semi)stable objects have been largely studiedsince their de�nition in [11] and [15]. Our de�nition of (semi)stable conicbundles opens analogous problems. For instance we would like to express herethe following conjecture. It has been proved in [2], for a semistable scroll of Prof degree d and irregularity q which is special (i.e. r distinct from the Riemann-Roch number d + 1− 2q ), the existence of a hyperplane containing r − 1 linesof the ruling, which amounts to the upper bound d − (r − 1) for the degree ofa unisecant curve of the ruled surface, a problem posed by Severi in [17] (theanalogous bound being trivial in the nonsemistable case). Most probably, for aspecial semistable conic bundle of Pr there is a hyperplane containing �r−22 � ofits conics, thus leading to an analogous upper bound of the minimal degree of abisecant curve of the surface (and so on).3. Principal sheaves for a classical group.Let X be a smooth, projective complex variety of dimension n.De�nition 2. A tensor �eld, or just a tensor, on X , is a pair (E, ϕ) consistingPROJECTIVE MODULI SPACE OF SEMISTABLE PRINCIPAL. . . 441of a torsion free sheaf E and an homomorphismϕ : ⊗s E → OX ,the rank and Chern classes of the tensor being called those of E . Let δ be apositive rational polynomial of degree at most n − 1 (i.e. rational coef�cients,and positive leading coef�cient). The tensor is said to be δ-(semi)stable if for allweighted �ltration (E., m.) of E , i.e. subsheaves E1 ⊂ . . . ⊂ Et ⊂ Et+1 = Eand positive integers m1, . . . ,mt , it is�mi (rχEi − riχE)+ δ µ(E .,m., ϕ)(≤)0where r, ri , χE , χEi are the ranks and Hilbert polynomials of E, Ei , and µ isde�ned asµ = min{λi1 + . . .+ λis |ϕ(Ei1 ⊗ . . .⊗ Eis ) �= 0}where λ1 < . . . < λs are integers with λi − λi−1 = mir and�λi rank(Ei/Ei−1) = 0.In [6] the de�nition is slightly more general, and we prove the followingTheorem 3. There is a coarse projective moduli space of δ-stable equivalenceclasses of δ-semistable tensors on a projective variety X , of �xed Chern classesand rank.The proof has two parts: �rst, show that the family consisting of suchobjects is bounded (remark that for δ-semistable (E, ϕ), the torsion free sheaf Eneeds not be semistable). Second, proceed with the techniques of Simpson [15]and Huybrechts-Lehn [7], starting by considering an integer m � 0 such thatall torsion free sheaves in the family are generated by global sections and haveH 0(E(m)) = χE(m). For each member of the family choose an isomorphismβ of H 0(E(m)) with a �xed complex vector space V of dimension χ(E(m)),thus obtaining a quotientV ⊗OX (−m) � H 0(E(m)) ⊗OX (−m) −→ Einducing, for l high enough, a quotientq : V ⊗ H 0(OX (l −m)) −→ H 0(E(l)).442 TOMA´S L. GO´MEZ - IGNACIO SOLSConsider also the induced homomorphismψ : V⊗s −→ H 0(E(m)⊗s) −→ H 0(OX (sm))We then obtain an element ofP��χE (l)(V ∗ ⊗ H 0(OX (l − m))∗)�× P�V ∗⊗s ⊗ H 0(OX (sm)�which we consider included in projective space by the linear system |O(n1, n2)|with n2n1 =χE (l)δ(m) − δ(l)χE (m)χE(m) − sδ(m)This assignation embeds in a projective space P the scheme R of triples(E, ϕ, β), with (E, ϕ) being a δ-semistable tensor of the given rank and Chernclasses and β a choice of basis as above. Quotienting by GIT with the naturalaction of Sl(V ) on R, induced from its natural action on P, we obtain thewantedprojective coarse moduli space.De�nition 4. Let G = O(r,C) or Sp(r,C). A principal G-sheaf on X isa tensor ϕ : E ⊗ E −→ OX symmetric or antisymmetric which induces anisomorphism E|U −→ E∗|U on the open set U where E is locally free. We callit (semi)stable if for all isotropic subsheaves F ⊆ E it isχF + χF⊥(≤)χETheorem 5. For any positive polynomial δ of degree exactly n − 1, a principalG-sheaf on X (G = O(r,C)) or Sp(r,C) is δ-(semi)stable if and only if it is(semi)stable, so there is a coarse projective moduli space of stable-equivalenceclasses of semistable principal G-sheaves.The remaining classical group. G = SO(r,C). De�ne a principal SO(r,C)-sheaf to be a triple (E, ϕ, ψ), where (E, ϕ) is a principal O(r,C)-sheaf and ϕis an isomorphism between det (E) and OX such that det (ϕ) = ψ2. Note thatfor each O(r,C)-sheaf (E, ϕ), there is at most two distinct SO(r,C)-sheaves,namely (E, ϕ, ψ) and (E, ϕ,−ψ). If rank (E) is odd, these two objects areisomorphic. This is why for SO(2m + 1,C) we can forget the third datum ψ .But if rank (E) is even, these two objects might not be isomorphic. With thesame de�nition of (semi)stability as in De�nition 4 , Theorem 5 still holds inthis case (i.e. the added datum does not alter the GIT notion of stability) so weobtain a coarse projective moduli space in the case G is any classical group.PROJECTIVE MODULI SPACE OF SEMISTABLE PRINCIPAL. . . 4434. Principal sheaves on a reductive group.Tensors considered in Section 3 were all (s, 0) tensors, but with the samemachinery we could have worked with (semi)stability and coarse projectivemoduli space of (s, 1)-tensors. In particular we need in this section (2, 1)-tensors ϕ : E ⊗ E −→ E∗∗ , for which δ-(semi)stability is de�ned by the factthat for all weighted �ltration (E1 ⊂ . . . ⊂ Et ,m1, . . . ,mt > 0) of E , it is�mi (rχEi − riχE )+ δ µ(E .,m., ϕ) (≤) 0whereµ = min{λi + λj − λk |0 �= ϕ : Ei ⊗ Ej −→ E∗∗/E∗∗k−1}For �xed value of rank and Chern classes, there is a projective coarse modulispace of stable equivalence classes of δ-semistable (2, 1) tensors on X .De�nition 6. Let X be a projective variety, and G an algebraic group. Aprincipal G-sheaf P is a triple (E, ϕ, ξ) where (E, ϕ) is a (2, 1)-tensor onXϕ : E ⊗ E −→ E∗∗such that for the points x of the open set UE where E is locally free, ϕ(x) isisomorphic to the structure tensor ϕg� : g� ⊗ g� −→ g� of the Lie algebra g�tangent to the commutator subgroup G � = [G,G] (in particular, there is anassociated Aut (g�)-bundle PUE on UE ), and ξ is a reduction of PUE to G, viaAd : G −→ Aut (g�).Obviously, if E is locally free, we recover the usual notion of principalG-bundle.De�nition 7. Let G be a connected reductive group. We say that a principalG-sheaf P = (E, ϕ, ξ) is (semi)stable if the tensor (E, ϕ) is δ-(semi)stable,where δ is a polinomial of degree exactly n − 1.In order to characterize this notion, we de�ne the Hilbert polynomial χE•of a �ltration E• of E (understood as Z-indexed, with E−∞ = 0 and E+∞ = E )asχE• =�i∈Z�rank(E)χEi − rank(Ei)χE �and say the �ltration is balanced if �i∈Zirank(Ei/Ei−1) = 0444 TOMA´S L. GO´MEZ - IGNACIO SOLSProposition 8. A principal G-sheaf P = (E, ϕ, ξ) is (semi)stable if and onlyif for all balanced algebra �ltrations E• ⊆ E it isχE• (≤) 0Theorem 9. There is a projective coarse moduli space of stable-equivalenceclasses of semistable principal G-sheaves on X of �xed topological type.Comment on the proof. It is a long proof, parallel to the proof of Ramanathan[13], which will appear published elsewhere. Because of the nondegeneracy ofthe Killing form of the semisimple Lie algebra g�, the factor µ(E .,m., ϕ) isalways nonpositive. Although our notion of (semi)stability is equivalent to theδ-(semi)stability of the (2, 1) tensor (E, ϕ), it does not assure the existence of amoduli space, because it must also be checked that the extra datum of reductionξ does not alter the (semi)stability in the sense of GIT of the correspondingpoint of the Sl(V )-acted projective space, which is the main bulk of the proof.The case dim X=1. Finally, we need some considerations on root spaces inorder to show that our notion of (semi)stability coincides with Ramanathanswhen dim X = 1. Recall from [1] that a t�-root decompositiong� =�α∈Rt� ∪{0}g�αof the Lie algebra g� arises whenever a toral algebra t� ⊆ g� is given, notnecessarily a Cartan algebra, in particular for the center t� = z(l(h�)) of theLevi component l(h�) of any parabolic subalgebra h� ⊂ g�. In this case a systemof simple t�-roots (or decomposition Rt� = R+t� ∪ R−t� ) is naturally given, so theset Rt� ∪ {0} has a natural partial ordering (α ≤ β if β is the sum of α with asum of simple t�-roots). Denote g�(≤α) = ⊕β≤αg�β and analogously g�(<α) . Wealso write Rh� for Rt� . Both are invariant by the adjoint action of h�, thus by theinner automorphism action of the corresponding parabolic subgroup H � of thegroup G �, so the analogous subalgebras g�(≤α) and g�(<α) of the Lie algebra g� arealso H �-invariant.Let P = (E, ϕ, ξ) be a principal G-bundle on X , having a further H �→ Greduction to a parabolic subgroup H , let H � = H ∩ G �, and let α ∈ Rh� ∪ {0}where h� = Lie(H �) as before. We de�ne E(≤α) and E(<α) as the subbundle ofE associated to this reduction by the above representation of H � on g�(≤α) andg�(<α) , and de�ne Eα as E(≤α)/E(<α) .PROJECTIVE MODULI SPACE OF SEMISTABLE PRINCIPAL. . . 445Proposition 10. A principal G-bundle P = (E, ϕ, ξ) on a curve is semistableif and only if E is semistable. It is furthermore stable if there is no reductionP(H ) of P to a parabolic subgroup H of G such that deg Eα = 0 for allroots α ∈ Rh� ∪ {0}, i.e. such that the degree of the line bundle associated to theprincipal H -bundle P(H ) by any of the characters of H is zero.Corollary 11. In case dim X = 1, a principal G-bundle is (semi)stable, in oursense, if and only if it is (semi)stable in the sense of Ramanathan [12, 13] .REFERENCES[1] A. Borel, Linear algebraic groups, New York: W. A. Benjamin Inc. 1969.[2] J. Cilleruelo - I. Sols, The Severi bound on sections of rank two semistable bundleson a Riemann surface, to appear in Ann. of Math., Nov. 2001.[3] G. Faltings, Moduli stacks for bundles on semistable curves, Math. Ann., 304(1996), pp. 489515.[4] D. Gieseker, On the moduli of vector bundles on an algebraic surfaces, Ann.Math., 106 (1977), pp. 4560.[5] T. Go´mez - I. Sols, Stability of conic bundles, Internat. J. Math., 11 (2000),pp. 10271055.[6] T. Go´mez - I. Sols, Stable tensor �elds and moduli space of principal G-sheavesfor classical groups, (math. AG/0103150).[7] D. Huybrechts - M. Lehn, Framed modules and their moduli, Internat. J. Math.,6 (1995), pp. 297324.[8] M. Maruyama, Moduli of stable sheaves. I, J. Math. Kyoto Univ., 17 (1977),pp. 91126.[9] M. Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ., 18 (1978),pp. 557614.[10] D. Mumford - J. Fogarty - F. Kirwan, Geometric invariant theory, (Third edi-tion) Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag,Berlin, 1994.[11] M.S. Narasimhan - C.S. Seshadri, Stable and unitary vector bundles on a compactRiemann surface, Ann. Math., 82 (1965), pp. 540567.[12] A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math.Ann., 213 (1975), pp. 129152.[13] A. Ramanathan, Moduli for principal bundles over algebraic curves: I and II,Proc. Indian Acad. Sci. (Math. Sci.), 106 (1996), pp. 301326 and pp. 421449.446 TOMA´S L. GO´MEZ - IGNACIO SOLS[14] A. Schmitt, A universal construction for moduli spaces of decorated vector bun-dles over curves, preprint 2000, math. AG/0006029.[15] C.S. Seshadri, Space of unitary vector bundles on a compact Riemann sur-face, Ann. Math., 85 (1967), pp. 303336.[16] C.S. Seshadri, Quotient spaces modulo reductive algebraic groups, Ann. Math.,95 (1972), pp. 511556.[17] F. Severi, Sulla classi�cazione delle rigate algebriche, Rend. Mat., 2 (1941),pp. 132.[18] C. Simpson, Moduli of representations of the fundamental group of a smoothprojective variety I, Publ. Math. I.H.E.S., 79 (1994), pp. 47129.[19] C. Sorger, Theta-characte´ristiques des courbes trace´es sur une surface lisse , J.reine angew. Math., 435 (1993), pp. 83118.Toma´s L. Go´mezSchool of Mathematics,Tata Institute of Fundamental Research,Bombay 400 005 (INDIA)e-mail: tomas@math.tifr.res.inIgnacio SolsDepartamento de Algebra,Facultad de Matema´ticas,Universidad Complutense de Madrid,28040 Madrid (SPAIN)e-mail: sols@eucmax.sim.ucm.es

Dedicated to Silvio Greco on the occasion of his 60th birthday (Catania, 2001).Let X be a smooth projective complex variety, and let G be an algebraic reductive complex group. We define the notion of principal G-sheaf, that   generalises the   notion of principal  G-bundle.  Then  we define   a notion  of semistability, and construct the projective moduli space of semistable principal G-sheaves on X. This is a natural compactification of the moduli  space  of principal G-bundles. This is the announcement note presented by the second author in the conference held at Catania  (11-13 April 2001), dedicated to the 60th birthday of  Silvio Greco. Detailed proofs will appear  elsewhere.Ministerio de Educación y Cultura (Spain)EAGERDepto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Gómez, Tomás L.

Docta Complutense

Projective moduli space of semistable principal sheaves for a reductive group.

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Projective moduli space of semistable principal sheaves for a reductive group

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