Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of
complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple
$\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of
algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some
remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$
(Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it
was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small
resolution of singularities. The aim of the present note is to study the
singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps'
space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$
is irreducible. The proof is based on the factorization property of Quasimaps'
space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to
Q^D_\alpha$.Comment: 8 pages, AmsLatex 1.

Finkelberg, Michael

Kuznetsov, Alexander

Mirković, Ivan

English

arXiv

Mirkoviæ, Ivan

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstMathematics and Statistics Department FacultyPublication Series Mathematics and Statistics1997The Singular Supports of IC Sheaves onQuasimaps' Spaces are IrreducibleMichael FinkelbergAlexander KuznetsovIvan MirkoviæFollow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubsPart of the Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted forinclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. Formore information, please contact scholarworks@library.umass.edu.Recommended CitationFinkelberg, Michael; Kuznetsov, Alexander; and Mirkoviæ, Ivan, "The Singular Supports of IC Sheaves on Quasimaps' Spaces areIrreducible" (1997). Mathematics and Statistics Department Faculty Publication Series. 11.Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/11arXiv:alg-geom/9705003v1  1 May 1997THE SINGULAR SUPPORTS OF IC SHEAVES ON QUASIMAPS’ SPACES AREIRREDUCIBLEMICHAEL FINKELBERG, ALEXANDER KUZNETSOV, AND IVAN MIRKOVIC´1. Introduction1.1. Let C be a smooth projective curve of genus 0. Let B be the variety of complete flags in ann-dimensional vector space V . Given an (n − 1)-tuple α ∈ N[I] of positive integers one can considerthe space Qα of algebraic maps of degree α from C to B. This space is noncompact. Some remarkablecompactifications QDα (Quasimaps), QLα (Quasiflags) of Qα were constructed by Drinfeld and Laumonrespectively. In [Ku] it was proved that the natural map π : QLα → QDα is a small resolution ofsingularities. The aim of the present note is to study the singular support of the Goresky-MacPhersonsheaf ICα on the Quasimaps’ space QDα .Namely, we prove that this singular support SS(ICα) is irreducible. The proof is based on the factoriza-tion property of Quasimaps’ space and on the detailed analysis of Laumon’s resolution π : QLα → QDα .We are grateful to P.Schapira for the illuminating correspondence.This note is a sequel to [Ku] and [FK]. In fact, the local geometry of QDα was the subject of [Ku]; theglobal geometry of QDα was the subject of [FK], while the microlocal geometry of QDα is the subject ofthe present work. We will freely refer the reader to [Ku] and [FK].2. Reductions of the main theorem2.1. Notations.2.1.1. We choose a basis {v1, . . . , vn} in V . This choice defines a Cartan subgroup H ⊂ G = SL(V ) =SLn of matrices diagonal with respect to this basis, and a Borel subgroup B ⊂ G of matrices uppertriangular with respect to this basis. We have B = G/B.Let I = {1, . . . , n − 1} be the set of simple coroots of G = SLn. Let R+ denote the set of positivecoroots, and let 2ρ =∑θ∈R+ θ. For α =∑aii ∈ N[I] we set |α| :=∑ai. Let X be the lattice of weightsof G,H . Let X+ ⊂ X be the set of dominant (with respect to B) weights. For λ ∈ X+ let Vλ denote theirreducible representation of G with the highest weight λ.Recall the notations of [Ku] concerning Kostant’s partition function. For γ ∈ N[I] a Kostant partition ofγ is a decomposition of γ into a sum of positive coroots with multiplicities. The set of Kostant partitionsof γ is denoted by K(γ).There is a natural bijection between the set of pairs 1 ≤ q ≤ p ≤ n−1 and R+, namely, (p, q) correspondsto iq+iq+1+. . .+ip. Thus a Kostant partition κ is given by a collection of nonnegative integers (κp,q), 1 ≤q ≤ p ≤ n− 1. Following loc. cit. (9) we define a collection µ(κ) as follows: µp,q =∑r≤q≤p≤s κs,r.M.F. and A.K. were partially supported by CRDF grant RM1-265. M.F. was partially supported by INTAS-94-4720.I.M. was partially supported by NSF.12 MICHAEL FINKELBERG, ALEXANDER KUZNETSOV, AND IVAN MIRKOVIC´Recall that for γ ∈ N[I] we denote by Γ(γ) the set of all partitions of γ, i.e. multisubsets (subsets withmultiplicities) Γ = {{γ1, . . . , γk}} of N[I] with∑kr=1 γr = γ, γr > 0 (see e.g. [Ku], 1.3).The configuration space of colored effective divisors of multidegree γ (the set of colors is I) is denotedby Cγ . The diagonal stratification Cγ = ⊔Γ∈Γ(γ)CγΓ was introduced e.g. in loc. cit. Recall that forΓ = {{γ1, . . . , γk}} we have dimCγΓ = k.2.1.2. For the definition of Laumon’s Quasiflags’ space QLα the reader may consult [La] 4.2, or [Ku] 1.4.It is the space of complete flags of locally free subsheaves0 ⊂ E1 ⊂ · · · ⊂ En−1 ⊂ V ⊗OC =: Vsuch that rank(Ek) = k, and deg(Ek) = −ak.It is known to be a smooth projective variety of dimension 2|α|+ dimB.2.1.3. For the definition of Drinfeld’s Quasimaps’ space QDα the reader may consult [Ku] 1.2. It is thespace of collections of invertible subsheaves Lλ ⊂ Vλ ⊗OC for each dominant weight λ ∈ X+ satisfyingPlu¨cker relations, and such that degLλ = −〈λ, α〉.It is known to be a (singular, in general) projective variety of dimension 2|α|+ dimB.The open subspace Qα ⊂ QDα of genuine maps is formed by the collections of line subbundles (as opposedto invertible subsheaves) Lλ ⊂ Vλ⊗OC . In fact, it is an open stratum of the stratification by the type ofdegeneration of QDα introduced in [Ku] 1.3:QDα =Γ∈Γ(α−β)⊔β≤αDβ,ΓαWe have Dα,∅ = Qα, and Dβ,Γα = Qβ × Cα−βΓ (see loc. cit. 1.3.5).The space QDα is naturally embedded into the product of projective spacesPα =∏1≤p≤n−1P(Hom(OC(−〈ωp, α〉), Vωp ⊗OC))and is closed in it (see loc. cit. 1.2.5). Here ωp stands for the fundamental weight dual to the coroot ip.The fundamental representation Vωp equals ΛpV .2.2. We will study the characteristic cycle of the Goresky-MacPherson perverse sheaf (or the corre-sponding regular holonomic D-module) ICα on QDα . As QDα is embedded into the smooth space Pα, wewill view this characteristic cycle SS(ICα) as a Lagrangian cycle in the cotangent bundle T∗Pα. A prioriwe have the following equality:SS(ICα) = T ∗QαPα +Γ∈Γ(α−β)∑β<αmβ,Γα T∗Dβ,ΓαPα,closures of conormal bundles with multiplicities.Theorem. SS(ICα) = T ∗QαPα is irreducible.In the following subsections we will reduce the Theorem to a statement about geometry of Laumon’sresolution.32.3. We fix a coordinate z on C identifying it with the standard P1. We denote by Q∞α ⊂ QDα the opensubspace formed by quasimaps which are genuine maps in a neighbourhood of the point∞ ∈ C. In otherwords, (Lλ ⊂ Vλ⊗OC)λ∈X+ ∈ Q∞α iff for each λ the invertible subsheaf Lλ ⊂ Vλ⊗OC is a line subbundlein some neighbourhood of ∞ ∈ C.Evidently, Q∞α intersects all the strata Dβ,Γ. Thus it suffices to prove the irreducibility of the singularsupport of Goresky-MacPherson sheaf of Q∞α .There is a well-defined map of evaluation at ∞ ∈ C:Υα : Q∞α −→ BIt is compatible with the stratification of Q∞α and realizes Q∞α as a (stratified) fibre bundle over B. Ineffect, G acts naturally both on Q∞α (preserving stratification) and on B; the map Υα is equivariant, andB is homogeneous. We denote the fiber Υ−1α (B) over the point B ∈ B by Zα.It inherits the stratificationZα =Γ∈Γ(α−β)⊔β≤αZDβ,Γαfrom Q∞α and QDα . It is just the transversal intersection of the fiber Υ−1α (B) with the stratification ofQ∞α . As in [Ku] 1.3.5 we have ZDβ,Γα∼−→ Zβ × (C −∞)α−βΓ .Hence it suffices to prove the irreducibility of the singular support SS(IC(Zα)) of Goresky-MacPhersonsheaf IC(Zα) of Zα.2.4. Factorization. The Theorem 6.3 of [FM] admits the following immediate Corollary. Let(φβ , γ1x1, . . . , γkxk) = φα ∈ Zβ × (C − ∞)α−βΓ = ZDβ,Γα ⊂ Zα. Consider also the points(φr , γrxr) = φγr ∈ Z0 × (C −∞)γr{{γr}}= ZD0,{{γr}}γr ⊂ Zγr , 1 ≤ r ≤ k.Proposition. There is an analytic open neighbourhood Uα (resp. Uβ, resp. Uγr , 1 ≤ r ≤ k) of φα (resp.φβ , resp. φγr , 1 ≤ r ≤ k) in Zα (resp. Zβ , resp. Zγr , 1 ≤ r ≤ k) such thatUα∼−→ Uβ ×∏1≤r≤kUγr2Recall the nonnegative integers mβ,Γα introduced in 2.2. The Proposition implies the following Corollary.Corollary. mβ,Γα =∏1≤r≤km0,{{γr}}γr . 2Thus to prove that all the multiplicities mβ,Γα vanish, it suffices to check the vanishing of m0,{{γ}}γ forarbitrary γ > 0.2.5. It remains to prove that the conormal bundle T ∗D0,{{γ}}γPα to the closed stratum of Qγ enters thesingular support SS(ICα) with multiplicity 0. To this end we choose a point (B, γ0) = φ ∈ B × C =Q0 ×Cγ{{γ}} = D0,{{γ}}γ ⊂ Qγ ⊂ Pγ . We also choose a sufficiently generic meromorphic function f on Pγregular around φ and vanishing on D0,{{γ}}γ . According to the Proposition 8.6.4 of [KS], the multiplicityin question is 0 iff Φf (ICγ)φ = 0, i.e. the stalk of vanishing cycles sheaf at the point φ vanishes.To compute the stalk of vanishing cycles sheaf we use the following argument, borrowed from [BFL] §1.As π : QLγ −→ QDγ is a small resolution of singularities, up to a shift, ICα = π∗Q. By the proper basechange, Φfπ∗Q = π∗Φf◦πQ. So it suffices to check that Φf◦πQ|π−1(φ) = 0.4 MICHAEL FINKELBERG, ALEXANDER KUZNETSOV, AND IVAN MIRKOVIC´Let us denote the differential of the function f at the point φ by ξ so that (φ, ξ) ∈ T ∗D0,{{γ}}γPγ . Then thesupport of Φf◦πQ|π−1(φ) is a priori contained in the microlocal fiber over (φ, ξ) which we define presently.2.5.1. Definition. Let ̟ : A→ B be a map of smooth varieties. For a ∈ A let d∗a̟ : T∗̟(a)B −→ T∗aAdenote the codifferential, and let (b, η) be a point in T ∗B. Then the microlocal fiber of ̟ over (b, η) isdefined to be the set of points a ∈ ̟−1(b) such that d∗a̟(η) = 0.2.5.2. Thus we have reduced the Theorem 2.2 to the following Proposition.Proposition. For a sufficiently generic ξ such that (φ, ξ) ∈ T ∗D0,{{γ}}γPγ , the microlocal fiber of Laumon’sresolution π over (φ, ξ) is empty. Equivalently, the cone ∪E•∈π−1(φ)Ker(d∗E•π) is a proper subvariety ofthe fiber of T ∗D0,{{γ}}γPγ at φ.2.6. Piecification of a simple fiber. The fiber π−1(φ) was called the simple fiber in [Ku] §2. It wasproved in loc. cit. 2.3.3 that π−1(φ) is a disjoint union of (pseudo)affine spaces S(µ(κ)) where κ runsthrough the set K(γ) of Kostant partitions of γ (for the notation µ(κ) see 2.1.1 or [Ku] (9)). Anotherway to parametrize these pseudoaffine pieces was introduced in [FK] 2.11. Let us recall it here.We define nonnegative integers cp, 1 ≤ p ≤ n− 1, so that γ =∑n−1p=1 cpip.2.6.1. Definition. D(γ) is the set of collections of nonnegative integers (dp,q)1≤q≤p≤n−1 such thata) For any 1 ≤ q ≤ p ≤ r ≤ n− 1 we have dr,q ≤ dp,q;b) For any 1 ≤ p ≤ n− 1 we have∑pq=1 dp,q = cp.2.6.2. Lemma. The correspondence κ = (κp,q)1≤q≤p≤n−1 7→ (dp,q :=∑n−1r=p κr,q)1≤q≤p≤n−1 defines abijection between K(γ) and D(γ). 22.6.3. Using the above Lemma we can rewrite the parametrization of the pseudoaffine pieces of thesimple fiber as follows:π−1(φ) =⊔d∈D(γ)S(d)In these terms the dimension formula of [Ku] 2.3.3 reads as follows: for d = (dp,q)1≤q≤p≤n−1 we havedimS(d) =∑1≤q<p≤n−1 dp,q.Note also that∑1≤q≤p≤n−1 dp,q =∑1≤p≤n−1 cp = |γ|.2.7. Proposition. For arbitrary d = (dp,q)1≤q≤p≤n−1 ∈ D(γ) and arbitrary quasiflag E• ∈ S(d) ⊂π−1(φ) we have dimKer(dE•π) <∑1≤p≤n−1 dp,q +∑1≤q≤p≤n−1 dp,q − 1.This Proposition implies the Proposition 2.5.2 straightforwardly. In effect, codimKer(dE•π) = dimQLγ −dimKer(dE•π) > 2|γ|+ dimB −∑1≤p≤n−1 dp,p −∑1≤q≤p≤n−1 dp,q + 1 = dimB+ 1+∑1≤q<p≤n−1 dp,q.Hence the subspace Ker(d∗E•π) ⊂ T∗φPγ has codimension greater than dimB + 1 +∑1≤q<p≤n−1 dp,q.Recall that dimD0,{{γ}}γ = dimB + 1. Hence the codimension of Ker(d∗E•π) ∩ T∗D0,{{γ}}γPγ in the fiber ofT ∗D0,{{γ}}γPγ at φ is greater than∑1≤q<p≤n−1 dp,q = dimS(d). Hence the cone ∪E•∈S(d)Ker(d∗E•π) is aproper subvariety of the fiber of T ∗D0,{{γ}}γPγ at φ.5The union of these proper subvarieties over d ∈ D(γ) is again a proper subvariety of the fiber of T ∗D0,{{γ}}γPγat φ which concludes the proof of the Proposition 2.5.2.2.8. Fixed points. It remains to prove the Proposition 2.7. To this end recall that the Cartan groupH acts on V and hence on QLα. The group C∗ of dilations of C = P1 preserving 0 and∞ also acts on QLαcommuting with the action of H . Hence we obtain the action of a torus T := H × C∗ on QLα.It preserves the simple fiber π−1(φ) and its pseudoaffine pieces S(d), d ∈ D(γ), for evident reasons. Itwas proved in [FK] 2.12 that each piece S(d), d = (dp,q)1≤q≤p≤n−1 contains exactly one T-fixed pointδ(d) = (E1, . . . , En−1). HereE1 = E1,1E2 = E2,1 ⊕E2,2.........En−1 = En−1,1 ⊕En−1,2 ⊕ . . . ⊕En−1,n−1and Ep,q = O(−dp,q) ⊂ Ovq ⊂ V = V ⊗OC with quotient sheafOO(−dp,q)concentrated at 0 ∈ C.2.8.1. Now the T-action contracts S(d) to δ(d). Since the map π is T-equivariant, and the dimensionof Ker(dE•π) is lower semicontinuous, the Proposition 2.7 follows from the next one.Key Proposition. For arbitrary d = (dp,q)1≤q≤p≤n−1 ∈ D(γ) (γ 6= 0) we have dimKer(dδ(d)π) <∑1≤p≤n−1 dp,p +∑1≤q≤p≤n−1 dp,q − 1.The proof will be given in the next section.2.8.2. Remark. In general, the pieces S(d) of the simple fiber are not equisingular, i.e. dimKer(dE•π) isnot constant along a piece. The simplest example occurs for G = SL3, γ = 2i1 + 2i2. Then the simplefiber is a singular 2-dimensional quadric. Its singular point is the fixed point of the 1-dimensional pieceS(d) where d1,1 = 2, d2,1 = d2,2 = 1. At this point we have dimKer(dδ(d)π) = 3 while at the other pointsin this piece we have dimKer(dE•π) = 2.3. The proof of the Key Proposition3.1. Tangent spaces. Let Ω be the following quiver: Ω = 1 −→ 2 −→ . . . −→ n − 1. Thus the setof vertices coincides with I. A quasiflag (E1 →֒ E2 →֒ . . . →֒ En−1 ⊂ V) ∈ QLγ may be viewed as arepresentation of Ω in the category of coherent sheaves on C. If we denote the quotient sheaf V/Ep byQp, 1 ≤ p ≤ n− 1, we have another representation of Ω in coherent sheaves on C, namely,Q• := (Q1 ։ Q2 ։ . . .։ Qn−1)3.1.1. Exercise. TE•QLγ = HomΩ(E•, Q•) where HomΩ(?, ?) stands for the morphisms in the category ofrepresentations of Ω in coherent sheaves on C.3.1.2. Consider a point L• = (L1, . . . ,Ln−1) ∈ Pγ . Here Lp ⊂ Vωp ⊗OC is an invertible subsheaf, theimage of morphism OC(−〈ωp, γ〉) →֒ Vωp ⊗OC .Exercise. TL•Pγ =∏n−1p=1 Hom(Lp, Vωp ⊗OC/Lp).6 MICHAEL FINKELBERG, ALEXANDER KUZNETSOV, AND IVAN MIRKOVIC´3.1.3. Recall that for E• ∈ QLγ we have π(E•) = L• ∈ Pγ where Lp = ΛpEp for 1 ≤ p ≤ n− 1.Exercise. For h• = (h1, . . . , hn−1) ∈ TE•QLγ we have dE•π(h•) = (Λ1h1,Λ2h2, . . . ,Λn−1hn−1) ∈ TL•Pγ .3.2. From now on we fix γ > 0, d ∈ D(γ), δ(d) =: E•. To unburden the notations we will denote thetangent space TE•QLγ by T . Since QLγ is a smooth (2|γ|+dimB)-dimensional variety it suffices to find asubspace N ⊂ T of dimension2|γ|+ dimB −∑1≤p≤n−1dp,p −∑1≤q≤p≤n−1dp,q + 1 =∑1≤q<p≤n−1(dp,q + 1) + 1such that dE•π|N is injective.3.3. Let N0 = ⊕n−1≥p>q≥1Hom(O(−dp,q),O). We have dimN0 =∑n−1≥p>q≥1(dp,q + 1).Recall that we have canonically T = HomΩ(E•, Q•), whereQp = V/Ep =(p⊕q=1(OO(−dp,q))vq)⊕(n⊕q=p+1Ovq).3.4. Let us define a map ν0 : N0 → T assigning to an element (fp,q) ∈ N0 a morphism ν0(fp,q) := F ∈Hom•(E•, Q•) of graded coherent sheaves, where F |Ep,q =n⊕r=p+1F rp,q, andF rp,q : Ep,q → Ovr ⊂ Qp is defined as the composition Ep,q ⊂ Er,q = O(−dr,q)fr,q−−→ Ovr3.5. Lemma. The map F : E• → Q• is a morphism of representations of the quiver Ω.Proof. We need to check the commutativity of the following diagramEp −−−−→ Ep′Fy FyQp −−−−→ Qp′Since Ep and Qp′ are canonically decomposed into the direct sum it suffices to note that for any q ≤ p ≤p′ < r the following diagramEp,q −−−−→ Ep′,q −−−−→ Er,qF rp,qy F rp′,qy fr,qyOvr Ovr Ovrcommutes and for any q ≤ p < r ≤ p′ the following diagramEp,q −−−−→ Er,q −−−−→ Ep′,qF rp,qy fr,qy 0yOvr Ovr −−−−→(OO(−dr,q))vrcommutes as well.73.6. Let N1 = C. Let p0 = min{1 ≤ p ≤ n − 1 | dp,p > 0} and pick a non-zero element f ∈Hom(O(−dp,p),OO(−dp,p)). Define the map ν1 : N1 → T by assigning to 1 ∈ N1 the element F ∈HomΩ(E•, Q•) defined on Ep,p as the compositionEp,p = O(−dp,p)f−→OO(−dp,p)vp ⊂ Qpand with all other components equal to zero.3.7. Let M(r, d;V) denote the space of rank r and degree d subsheaves in V .3.7.1. Let E ⊂ V be a rank k and degree d subsheaf in the vector bundle V . Let V/E = T ⊕ F be adecomposition of the quotient sheaf into the sum of the torsion T and a locally free sheaf F . Considerthe map det : M(r, d;V) → M(1, d; ΛkV) sending E to ΛkE. Then the restriction of its differentialdE det : TEM(r, d;V) = Hom(E,V/E) → Hom(ΛkE,ΛkV/ΛkE) = TΛkEM(1, d; ΛkV) to the subspaceHom(E,F) ⊂ Hom(E,V/E) factors as Hom(E,F) ∼= Hom(ΛkE,Λk−1E ⊗ F) ⊂ Hom(ΛkE,ΛkV/ΛkE).Therefore it is injective.3.7.2. Let E = O⊕(r−1) ⊕O(−d) be a subsheaf in V = O⊕n. Then the restriction of differential dE detto the subspace Hom(E, T ) ⊂ Hom(E,V/E) is injective.This immediately follows from the following fact. Let E˜ = O⊕r ⊂ V be the normalization of E in V ,that is, the maximal vector subbundle E˜ ⊂ V such that E˜/E is torsion. Then T = E˜/E ∼= ΛkE˜/ΛkE ⊂ΛkV/ΛkE.3.7.3. Clearly, the subsheaves Λk−1E ⊗F ⊂ ΛkV/ΛkE and T ⊂ ΛkV/ΛkE do not intersect.3.7.4. It follows from 3.7.1, 3.7.2 and 3.7.3 that the composition dE•π ◦ (ν0 ⊕ ν1) : N0⊕N1 → Tπ(E•)Pγis injective, hence N := (ν0 ⊕ ν1)(N0 ⊕ N1) ⊂ TE•QDγ enjoys the desired property. Namely, dE•π|N isinjective, and dimN =∑1≤q<p≤n−1(dp,q + 1) + 1.This completes the proof of the Key Proposition 2.8.1 along with the Main Theorem 2.2. 2References[BFL] Bressler P., Finkelberg M., Lunts V., Vanishing cycles on Grassmannians, Duke Mathematical Journal, 61, 3(1990), pp. 763-777.[FK] Finkelberg M., Kuznetsov A., Global Intersection Cohomology of Quasimaps’ spaces, Preprint alg-geom/9702010,pp. 1-20, to appear in IMRN.[FM] Finkelberg M., Mirkovic´ I., Semiinfinite flags. I. Transversal slices, to appear.[KS] Kashiwara M., Schapira P., Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften, 292 (1994).[Ku] Kuznetsov A., Laumon’s resolution of Drinfeld’s compactification is small, Preprint alg-geom/9610019, pp. 1-15,to appear in MRL.[La] Laumon G., Faisceaux Automorphes Lie´s aux Se´ries d’Eisenstein, In: Automorphic Forms, Shimura Varietes, andL-functions, Perspectives in Mathematics, 10, Academic Press, Boston, MA (1990), pp. 227–281.8 MICHAEL FINKELBERG, ALEXANDER KUZNETSOV, AND IVAN MIRKOVIC´Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow 121002 RussiaE-mail address: fnklberg@main.mccme.rssi.ruIndependent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow 121002 RussiaE-mail address: sasha@ium.ips.ras.ruDept. of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst MA 01003-4515, USAE-mail address: mirkovic@math.umass.edu

The Singular Supports of IC Sheaves on Quasimaps\u27 Spaces are Irreducible

https://scholarworks.umass.edu/cgi/viewcontent.cgi?article=2171&context=math_faculty_pubs

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Abstract

Similar works

Full text

Available Versions

ScholarWorks@UMass Amherst