Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes
aa r X i v : . [ m a t h . AG ] J a n FOURIER-MUKAI TRANSFORMATION AND LOGARITHMIC HIGGSBUNDLES ON PUNCTUAL HILBERT SCHEMES
INDRANIL BISWAS AND ANDREAS KRUG
Abstract.
Given a vector bundle E on a smooth projective curve or surface X carryingthe structure of a V -twisted Hitchin pair for some vector bundle V , we observe that theassociated tautological bundle E [ n ] on the punctual Hilbert scheme of points X [ n ] has aninduced structure of a (( V ∨ ) [ n ] ) ∨ -twisted Hitchin pair, where ( V ∨ ) [ n ] is a vector bundle on X [ n ] constructed using the dual V ∨ of V . In particular, a Higgs bundle on X induces alogarithmic Higgs bundle on the Hilbert scheme X [ n ] . We then show that the known resultson stability of tautological bundles and reconstruction from tautological bundles generalizeto tautological Hitchin pairs. Introduction
Let X be a smooth projective variety with dim X ≤
2. In this case, the Hilbert scheme X [ n ] of n points on X is a smooth projective variety of dimension dim X [ n ] = n · dim X . If X is acurve, X [ n ] coincides with the symmetric product X ( n ) := X n / S n , where S n is the group ofpermutations of { , . . . , n } . If X is a surface, then X [ n ] is a resolution of the singularities of X ( n ) via the Hilbert–Chow morphism µ : X [ n ] → X ( n ) . The points parametrizing non-reducedsub-schemes of X form a divisor B n ⊂ X [ n ] . Given a vector bundle E on X , we get a vectorbundle E [ n ] on X [ n ] which is the direct image of the pullback of E to the universal subschemeΞ n ⊂ X × X [ n ] . These vector bundles on X [ n ] play a crucial role in the investigation of thetopology and geometry of the Hilbert schemes [Leh99, LS01, LS03], but are also useful toolsfor studying properties of the bundles on X itself; see, for example [Voi02, EL15, Ago17].Higgs bundles constitute an extensively studied topic; they appear in algebraic geometry,differential geometry, symplectic geometry and also in representation theory [Ngo10]. Werecall that the Higgs bundles on Riemann surfaces were introduced by Hitchin [Hi87] and theHiggs bundles on higher dimensional complex manifolds were introduced by Simpson [Si88].It turns out that a Higgs field E → E ⊗ Ω X of a vector bundle E on X does not, ingeneral, induce a Higgs field of E [ n ] ; see Example 3.3. However we observe that it inducesa logarithmic Higgs field on E [ n ] , that is a homomorphism E [ n ] → E [ n ] ⊗ Ω X [ n ] ( log B n ) or,equivalently, a homomorphism E [ n ] ⊗ T X [ n ] ( − log B n ) → E [ n ] .The construction is as follows. As proved by Stapleton [Sta16, Thm. B], there is an iso-morphism T X [ n ] ( − log B n ) ∼ = ( T X ) [ n ] , identifying the sheaf of logarithmic vector fields on X [ n ] with the tautological bundle associated to the tangent sheaf of X . Now, the logarithmic Higgsfield on E [ n ] associated to a Higgs field ϑ : E ⊗ T X → E is given by the composition E [ n ] ⊗ T X [ n ] ( − log B n ) ∼ = E [ n ] ⊗ T [ n ] X ν −→ ( E ⊗ T X ) [ n ] ϑ [ n ] −−→ E [ n ] Mathematics Subject Classification.
Key words and phrases.
Logarithmic Higgs bundle, Hilbert scheme, Fourier–Mukai transformation, stability. where ν : E [ n ] ⊗ T [ n ] X → ( E ⊗ T X ) [ n ] is the cup product relative to the projection morphism pr X [ n ] : Ξ n → X [ n ] ; compare Subsection 2.7 and Remark 3.1. We denote this induced Higgsfield by ϑ { n } : E [ n ] ⊗ T X [ n ] ( − log B n ) → E [ n ] , and get the tautological logarithmic Higgs bundle ( E, ϑ ) [ n ] := ( E, ϑ ).There is the more general notion of a V -cotwisted Hitchin pair , for V a vector bundle of X , such that a Higgs bundle on X is exactly a T X -cotwisted Hitchin pair, and a logarithmicHiggs bundle on X [ n ] is exactly a T X [ n ] ( − log B n )-cotwisted Hitchin pair; see Subsection 2.1.The above construction generalizes in such a way that for a V -cotwisted Hitchin pair ( E, ϑ )on X , we get a tautological V [ n ] -cotwisted Hitchin pair ( E, ϑ ) [ n ] = ( E [ n ] , ϑ { n } ) on X [ n ] ; seeSubsection 3.2.In this article, we proof that basically all the known results on stability of tautologicalbundles and reconstruction from tautological bundles lift to results for tautological Hitchinpairs. As probably the most interesting special case, we get results on tautological logarithmicHiggs bundles. Concretely, we proof the following results. Theorem 1.1.
Let C be a smooth projective curve of genus g ( C ) ≥ , and let ( E, ϑ ) , ( F, η ) be semi-stable V -cotwisted Hitchin bundles for some vector bundle V ∈ VB ( C ) . Then, forevery n ∈ N , we have ( E, ϑ ) [ n ] ∼ = ( F, η ) [ n ] = ⇒ ( E, ϑ ) ∼ = ( F, η ) . For a scheme Y , by VB ( Y ) and Coh ( Y ) we denote the category of vector bundles andcoherent sheaves respectively on Y . Theorem 1.2.
Let C be an elliptic curve, and let ( E, ϑ ) , ( F, η ) be V -cotwisted Hitchin bundlesfor some vector bundle V ∈ VB ( C ) . Then, for every n ∈ N , we have ( E, ϑ ) [ n ] ∼ = ( F, η ) [ n ] = ⇒ ( E, ϑ ) ∼ = ( F, η ) . Theorem 1.3.
Let X be a smooth quasi-projective variety of dimension dim X ≥ , let V ∈ VB ( X ) , and let ( E, ϑ ) , ( F, η ) be V -cotwisted Hitchin pairs for some vector bundle V ∈ VB ( C ) ,such that the coherent sheaves E and F are reflexive. Then, for every n ∈ N , ( E, ϑ ) [ n ] ∼ = ( F, η ) [ n ] = ⇒ ( E, ϑ ) ∼ = ( F, η ) . Let C be a smooth projective curve. In the group N ( C ( n ) ) of divisors modulo numericalequivalence, we consider the ample class H n that is represented by C ( n − + x ⊂ C ( n ) for any x ∈ C ; see [Kru18b, Sect. 1.3] for details. Theorem 1.4.
Let C be a smooth projective curve, and let ( E, ϑ ) be a Hitchin bundle on C .Then for every n ∈ N , the following two statements hold.(i) If ( E, ϑ ) is stable and µ ( E ) / ∈ [ − , n − , then the logarithmic Higgs bundle ( E [ n ] , ϑ { n } ) is stable with respect to H n .(ii) If ( E, ϑ ) is semi-stable and µ ( E ) / ∈ ( − , n − , then the logarithmic Higgs bundle ( E [ n ] , ϑ { n } ) is semi-stable with respect to H n . The paper is organized as follows. In Section 2, we collect basic notions and results onHitchin pairs needed for later use. In particular, we define the push-forward of Hitchin pairsalong flat and finite morphisms, and define Fourier–Mukai transforms of Hitchin pairs for acertain class of kernels; see Subsection 2.7 and Subsection 2.8. In Section 3, we explain, insome more detail than done in this introduction, how to get induced structures of Hitchin
OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 3 pairs and logarithmic Higgs bundles on tautological bundles on Hilbert schemes of points. Allfour of our results listed above have already been proved in the special case of ordinary sheaveswithout the structure of a Hitchin pair. (A coherent sheaf is the same as a 0-cotwisted Hitchinpair.) For Theorem 1.1, see [BN12] and [BN17, Sect. 2]; for Theorem 1.2, see [Kru18a, Thm.1.5]; for Theorem 1.3, see [Kru18a, Thm. 3.2 & Rem. 3.5]; for Theorem 1.4, see [Kru18b].In all cases, the proofs can be amended in such a way that they work for Hitchin pairs.We follow different approaches in explaining how to amend the proofs. For Theorem 1.1 andTheorem 1.2, we give full proofs in Section 4. The reader should be able to follow these proofswithout going back to [BN12], [BN17] or [Kru18a], though some steps of the arguments mightbe carried out in greater details in these articles. In contrast, for the proofs of Theorem 1.3and Theorem 1.4, we only explain the extra ingredients needed to lift the proofs from thecases of ordinary sheaves to Hitchin pairs, and where to insert these ingredients. Hence, forfollowing these proofs, the reader should at the same time have a look at the relevant partsof [Kru18a] and [Kru18b].In the final Subsection 6.2, we remark that also a result of Stapleton [Sta16] on stabilityof tautological bundles on Hilbert schemes of points on surfaces extends to Hitchin bundles.2.
Preliminaries on Hitchin pairs
Throughout this section, let X be a scheme over the complex numbers C . Usually, we willwork with varieties, but at one point in Subsection 4.1 we will have to consider Hitchin pairson infinitesimal neighborhoods of a diagonal, which is why we choose the greater generalityhere.2.1. Basic Definitions.
Let V ∈ VB ( X ) be a vector bundle. A V -cotwisted Hitchin pairon X is a pair ( E, ϑ ) consisting of a coherent sheaf E ∈ Coh ( X ) and an O X -linear map ϑ : E ⊗ V → V such that the composition (cid:0) ϑ ◦ ( ϑ ⊗ id V ) (cid:1) | V V : E ⊗ ^ V ֒ → E ⊗ V ⊗ V ϑ ⊗ id V −−−−→ E ⊗ V ϑ −→ E , where V V ⊂ V ⊗ V is the second exterior product, vanishes. A Hitchin pair ( E, ϑ ) where E is a vector bundle is also called a Hitchin bundle .A morphism between two V -cotwisted Hitchin pairs ( E, ϑ ) and (
F, η ) on X is an O X -linearmap γ : E → F such that the following diagram commutes E ⊗ V γ ⊗ id V (cid:15) (cid:15) ϑ / / E γ (cid:15) (cid:15) F ⊗ V η / / F .
We denote the category of V -cotwisted Hitchin pairs on X by Hi V ( X ). It is an abeliancategory with kernels and cokernels given by the kernels and cokernels of the underlyingmorphisms of coherent sheaves, equipped with the induced O X -linear homomorphism. Inother words, the forgetful functor For : Hi V ( X ) → Coh ( X )is exact. I. BISWAS AND A. KRUG
Remark 2.1.
In the literature, usually V -twisted Hitchin pairs are considered instead of V -cotwisted Hitchin pairs. A V -twisted Hitchin pair is a pair ( E, ζ ) consisting of E ∈ Coh ( X )and an O X -linear morphism ζ : E → E ⊗ V such that the composition E ζ −→ E ⊗ V ζ ⊗ id V −−−−→ E ⊗ V ⊗ V ։ E ⊗ ^ V vanishes. However, the category Hi V ∨ ( X ) of V ∨ -twisted Hitchin pairs is equivalent to Hi V ( X )via the pair of mutually inverse exact functors Hi V ( X ) → Hi V ∨ ( X ) , ( E, ϑ ) ( E, ϑ ′ ) , ϑ ′ := (cid:0) E id E ⊗ η V −−−−−→ E ⊗ V ⊗ V ∨ ϑ ⊗ id V ∨ −−−−−→ E ⊗ V ∨ (cid:1) , Hi V ∨ ( X ) → Hi V ( X ) , ( E, ζ ) ( E, ζ ′ ) , ζ ′ := (cid:0) E ⊗ V ζ ⊗ id V −−−−→ E ⊗ V ∨ ⊗ V id E ⊗ ε V −−−−−→ E (cid:1) , where η V : O X → V ⊗ V ∨ sends any f to f · id V , and ε V : V ∨ ⊗ V → O X is the trace map.The reason that we prefer to work with V -cotwisted sheaves is that they allow an easierformulation of a push-forward functor (under certain circumstances); see Remark 2.6.2.2. Hitchin subsheaves.
Let (
E, ϑ ) be a V -cotwisted Hitchin pair on X . A coherentsubsheaf A ⊂ E is called a Hitchin subsheaf if ϑ ( A ⊗ V ) ⊂ A . A Hitchin subsheaf carriesitself the structure of a V -cotwisted Hitchin pair ( A, ϑ | A ) which makes the inclusion A ֒ → E amorphism of Hitchin pairs. Furthermore, the quotient E/A carries the structure of a Hitchinpair (
E/A, ¯ ϑ ) such that the quotient map E ։ E/A is a morphism of Hitchin pairs.For later use, we note the following quite obvious statements.
Lemma 2.2.
Let ( E, ϑ ) be a Hitchin pair, and let A ⊂ E be a Hitchin subsheaf.(i) If B ⊂ E is another Hitchin subsheaf, then the intersection A ∩ B ⊂ E is a Hitchinsubsheaf too.(ii) Let ( F, η ) be another Hitchin pair, and let ϕ : E → F be a morphism of Hitchin pairs.Then ϕ ( A ) ⊂ F is a Hitchin subsheaf. Higgs bundles.
Let X be a smooth variety with tangent bundle T X . A Higgs bundle on X is a T X -cotwisted Hitchin pair ( E, ϑ ).Given a reduced divisor D ⊂ X , we consider the sheaf of logarithmic vector fields T X ( − log D ).It is defined as the sheaf of vector fields which along the regular locus D reg of D are tangentto D . A logarithmic Higgs bundle (with respect to the divisor D ) is a T X ( − log D )-cotwistedHitchin pair ( E, ϑ ).Logarithmic Higgs bundles on curves were first considered by Bottacin [Bo95]. He provedthat a moduli space of logarithmic Higgs bundle on a curve has a natural Poisson structure.We note that a logarithmic Higgs bundle is a parabolic Higgs bundle with trivial quasi-parabolic structure. Mochizuki has proved many important results on parabolic Higgs bundles[Mo06], [Mo14].2.4.
Change of the cotwisting bundle.
Let ϕ : V → W be a morphism of vector bundleson X . There is an induced exact functor ϕ : Hi W ( X ) → Hi V ( X ) , ( E, ϑ ) (cid:0) E, ϑ ◦ ( id E ⊗ ϕ ) (cid:1) . Lemma 2.3.
Let ϕ : V → Q be a surjective morphism of vector bundles with kernel K := ker ( p ) .(i) The functor ϕ : Hi Q ( X ) → Hi V ( X ) is fully faithful. OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 5 (ii) A Hitchin pair ( E, ϑ ) ∈ Hi V ( X ) is in the essential image of the functor ϕ if andonly if ϑ ( E ⊗ K ) = 0 .(iii) Let ( E, ϑ ) ∈ Hi Q ( X ) . A subsheaf A ⊂ E is a Hitchin subsheaf of ( E, ϑ ) if and onlyif it is a Hitchin subsheaf of ϕ ( E, ϑ ) .Proof. Let (
E, ϑ ) , ( F, η ) ∈ Hi Q ( X ). By the surjectivity of ϕ , a O X -linear morphism γ : E → F is a morphism between the Q -cotwisted Hitchin pairs ( E, ϑ ) and (
F, η ) if and only if it is amorphism between the V -cotwisted Hitchin pairs ϕ ( E, ϑ ) and ϕ ( F, η ), which proves (i).Part (ii) is straight-forward to prove. Part (iii) follows directly from the surjectivity of ϕ andthe definition of Hitchin subsheaves. (cid:3) Pull-back of Hitchin pairs.
Let f : X ′ → X be a morphism, and let E and V becoherent sheaves on X . There is a natural isomorphism α := α f,E,V : f ∗ E ⊗ f ∗ V ∼ −→ f ∗ ( E ⊗ V ) . Concretely, if f : Spec A ′ → Spec A is a morphism of affine schemes, so that E = f M , V = e N for some A -modules M and N , then α is given by the map( M ⊗ A A ′ ) ⊗ A ′ ( N ⊗ A A ′ ) → ( M ⊗ A N ) ⊗ A A ′ , ( m ⊗ a ) ⊗ ( n ⊗ b ) ( m ⊗ n ) ⊗ ( ab ) . (1)Using the isomorphism α , we can define the pull-back along f on the level of Hitchin pairs as f ∗ : Hi V ( X ) → Hi f ∗ V ( X ′ ) , ( E, ϑ ) ( f ∗ E, f ∗ ϑ ◦ α ) . We often omit the isomorphism α in our notation and simply write f ∗ ϑ : f ∗ E ⊗ f ∗ V → f ∗ E .2.6. Hitchin pairs under tensor products.
Let (
E, ϑ ) ∈ Hi V ( X ) be a Hitchin pair, andlet P ∈ Coh ( X ). We write P ⊗ ( E, ϑ ) for the Hitchin pair ( P ⊗ E, id P ⊗ ϑ ). Clearly, thisdefines a functor P ⊗ : Hi V ( X ) → Hi V ( X ).2.7. Push-forwards of Hitchin pairs.
Let now π : X → Y be a morphism such that π ∗ V isagain a vector bundle. Note that this is always the case if π is flat and finite. In the following,for every ( E, ϑ ) ∈ Hi V ( X ), we will naturally equip π ∗ E with the structure of a π ∗ V -cotwistedHitchin pair. There is a natural morphism ν := ν π,E,V : π ∗ E ⊗ π ∗ V → π ∗ ( E ⊗ V ) . Over an open affine subset
Spec A = U ⊂ X , it is given by the A -linear cup productΓ( π − U, E ) ⊗ A Γ( π − U, V ) ∪ −→ Γ( π − U, E ⊗ V ) . (2) Lemma 2.4.
Let π : X → Y be a morphism of schemes, let ( E, ϑ ) ∈ Hi V ( X ) , and set ˇ ϑ := π ∗ ϑ ◦ ν : π ∗ E ⊗ π ∗ V → π ∗ E .
Then the following vanishing statement holds: (cid:0) ˇ ϑ ◦ ( ˇ ϑ ⊗ id π ∗ V ) (cid:1) | π ∗ E ⊗ V ( π ∗ V ) = 0 . Proof.
From the description (2) of ν over open affine subsets, we see that the diagram π ∗ ( E ⊗ V ) ⊗ π ∗ V ν E ⊗ V,V / / π ∗ ϑ ⊗ id π ∗ V (cid:15) (cid:15) π ∗ ( E ⊗ V ⊗ V ) π ∗ ( ϑ ⊗ id V ) (cid:15) (cid:15) π ∗ E ⊗ π ∗ V ν E,V / / π ∗ ( E ⊗ V ) I. BISWAS AND A. KRUG commutes. Hence, we can rewrite the composition ˇ ϑ ◦ ( ˇ ϑ ⊗ id V ) asˇ ϑ ◦ ( ˇ ϑ ⊗ id V ) = ( π ∗ ϑ ) ◦ ν E,V ◦ ( π ∗ ϑ ⊗ id π ∗ V ) ◦ ( ν E,V ⊗ id π ∗ V )= ( π ∗ ϑ ) ◦ ( π ∗ ( ϑ ⊗ id V )) ◦ ν E ⊗ V,V ◦ ( ν E,V ⊗ id π ∗ V )= ( π ∗ ( ϑ ◦ ( ϑ ⊗ id V ))) ◦ ν E,V ⊗ V ◦ ( id π ∗ E ⊗ ν V,V ) . (3)It follows from (2) that ν V,V ( V ( π ∗ V )) ⊂ π ∗ ( V V ), hence ν E,V ⊗ V ( id π ∗ E ⊗ ν V,V )( π ∗ E ⊗ ^ ( π ∗ V )) ⊂ π ∗ ( E ⊗ ^ V ) . (4)Since ( E, ϑ ) is a Hitchin pair, we have that ( ϑ ◦ ( ϑ ⊗ id V )) | E ⊗ V V = 0, hence( π ∗ ( ϑ ◦ ( ϑ ⊗ id V ))) | π ∗ ( E ⊗ V V ) = 0 . (5)The combination of (3), (4), and (5) gives the desired vanishing. (cid:3) Corollary 2.5.
Let π : X → Y be a morphism of schemes, and let V ∈ VB ( X ) be a vectorbundle with the property that π ∗ V is again a vector bundle. Then there is a push-forwardfunctor π ∗ : Hi V ( X ) → Hi π ∗ V ( Y ) , π ∗ ( E, ϑ ) = ( π ∗ E, ˇ ϑ ) . In particular, if π : X → Y is flat and finite, there exists for every vector bundle V ∈ VB ( X ) a push-forward functor π ∗ : Hi V ( X ) → Hi π ∗ V ( Y ) . Remark 2.6.
Corollary 2.5 is the reason that we are working with cotwisted instead oftwisted Hitchin pairs; compare Remark 2.1. Indeed, in terms of twisted Hitchin pairs, thepush-forward is a functor Hi V ( X ) → Hi ( π ∗ ( V ∨ )) ∨ ( Y ), which makes it a bit inconvenient toformulate things in terms of twisted Hitchin pairs whenever a push-forward occurs.Given a Cartesian diagram of schemes X ′ f ′ / / π ′ (cid:15) (cid:15) X π (cid:15) (cid:15) Y ′ f / / Y (6)and a coherent sheaf E ∈ Coh ( X ), we denote the natural base change morphism by β := β E : f ∗ π ∗ E → π ′∗ f ′∗ E .
In the case that all the schemes involved are affine, which means that (6) is the spectrum ofa diagram of commutative rings of the form B ′ B o o A ′ ϕ ′ O O A , o o ϕ O O (7)with E = f M and V = e N for some B -modules M and N , the map β is given by( M A ) ⊗ A A ′ → ( M ⊗ B B ′ ) A ′ , m ⊗ a m ⊗ ϕ ′ ( a ) . (8) Lemma 2.7.
Let π be flat and finite, and let ( E, ϑ ) ∈ Hi V ( X ) . Then β E : f ∗ π ∗ ( E, ϑ ) ∼ −→ β V π ′∗ f ′∗ ( E, ϑ ) is an isomorphisms of f ∗ π ∗ V -cotwisted Hitchin pairs on Y ′ . OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 7
Proof.
By the assumptions on π , the map β E : f ∗ π ∗ E → π ′∗ f ′∗ E is an isomorphism of coherentsheaves. It remains to proof that it is a morphism of Hitchin pairs, which amounts to checkingthe commutativity of the diagram f ∗ π ∗ E ⊗ f ∗ π ∗ V α / / β E ⊗ id (cid:15) (cid:15) f ∗ ( π ∗ E ⊗ π ∗ V ) f ∗ ν / / f ∗ π ∗ ( E ⊗ V ) f ∗ π ∗ ϑ / / f ∗ π ∗ E β E (cid:15) (cid:15) π ′∗ f ′∗ E ⊗ f ∗ π ∗ V id ⊗ β V / / π ′∗ f ′∗ E ⊗ π ′∗ f ′∗ V ν / / π ′∗ ( f ′∗ E ⊗ f ′∗ V ) π ′∗ α / / π ′∗ f ′∗ ( E ⊗ V ) π ′∗ f ′∗ ϑ / / π ′∗ f ′∗ E .
For this, we may assume that (6) is given by the spectrum of (7), E = f M , and V = e N . Then,using the concrete descriptions (1), (2), and (8) of the maps α , ν , and β , it can be checkedthat both paths of the above diagram are given by( M A ⊗ A A ′ ) ⊗ A ′ ( N A ⊗ A A ′ ) → ( M ⊗ B B ′ ) A ′ , ( m ⊗ a ) ⊗ ( n ⊗ a ) ϑ ( m ⊗ n ) ⊗ ϕ ′ ( a a ) . (cid:3) Later, we usually omit the equivalence β : Hi f ∗ π ∗ V ( Y ′ ) ∼ −→ Hi π ′∗ f ′∗ V ( Y ′ ) when applyingLemma 2.7, and simple write f ∗ π ∗ ( E, ϑ ) ∼ = π ′∗ f ′∗ ( E, ϑ ).2.8.
Fourier–Mukai transforms of Hitchin pairs.
Let X and Y be varieties, and let Z ⊂ X × Y be a closed sub-scheme. We denote by pr X : X × Y → X and pr Y : X × Y → Y the projections to the factors, and write p : Z → X and q : Z → Y for the restrictions of theseprojections to Z . In the following, we will always assume that q : Z → Y is flat and finite.For a coherent sheaf P ∈
Coh ( Z ), we define the Hitchin enhanced Fourier–Mukai transform Φ Z P : Hi V ( X ) → Hi q ∗ p ∗ V ( Y ) as the following composition of the three functors discussed inSubsections 2.5, 2.6, and 2.7:Φ Z P : Hi V ( X ) p ∗ −→ Hi p ∗ V ( Z ) P⊗ −−→ Hi p ∗ V ( Z ) q ∗ −→ Hi q ∗ p ∗ V ( Y ) . Remark 2.8.
Note that, in contrast to the usual convention for Fourier–Mukai transforms,none of our functors are derived. Hence, the functor Φ Z P : Hi V ( X ) → Hi q ∗ p ∗ V ( Y ) needs notto be exact. However, it is exact whenever P is flat over X , as will be the case in all ourapplications; compare [Kru18a, Thm. 1.1]. In this case, the Hitchin enhanced Fourier–Mukaitransform is compatible with the usual Fourier–Mukai transform Φ P : Coh ( X ) → Coh ( Y ) inthe sense that we have For Y ◦ Φ Z P ∼ = Φ P ◦ For X , where For Y : Hi q ∗ p ∗ V ( Y ) → Coh ( Y ) and For X : Hi V ( X ) → Coh ( X ) are the forgetful functors. Example 2.9.
Let f : Y → X be a morphism, and let Γ f = ( f × id Y )( Y ) ⊂ X × Y be itsgraph. Then, for every L ∈ Coh ( Y ) and ( E, ϑ ) ∈ Hi V ( X ), we have a natural isomorphismΦ Γ f ( f × id Y ) ∗ L ∼ = f ∗ ( E, ϑ ) ⊗ L . In particular, Φ Γ f O Γ f ( E, ϑ ) ∼ = f ∗ ( E, ϑ ). Lemma 2.10.
Let → P ′ → P → P ′′ → be a short exact sequence of coherent sheaveson Z . Then, for every Hitchin bundle ( E, ϑ ) ∈ Hi V ( X ) , there is the following short exactsequence in Hi q ′∗ p ′∗ V ( Y ) : → Φ Z P ′ ( E, ϑ ) → Φ Z P ( E, ϑ ) → Φ Z P ′′ ( E, ϑ ) . Proof.
This follows from the fact that tensor products by vector bundles are exact, togetherwith the fact that a short exact sequence of Hitchin pairs is given by a short exact sequenceof the underlying coherent sheaves. (cid:3)
I. BISWAS AND A. KRUG
Let now i : Z ′ ֒ → Z be a closed embedding, and let p ′ = p ◦ i : Z ′ → X , q ′ = q ◦ i : Z ′ → Y be the restrictions of the projections to Z ′ . The canonical surjection O Z → O Z ′ induces a sur-jection ϕ : q ∗ p ∗ V → q ′∗ p ′∗ V . Recall that this yields a functor ϕ : Hi q ′∗ p ′∗ V ( X ) → Hi q ∗ p ∗ V ( X );see Subsection 2.4. Lemma 2.11.
For
Q ∈
Coh ( Z ′ ) there is an isomorphism of functors Φ Zi ∗ Q ∼ = ϕ ◦ Φ Z ′ Q .Proof. This follows by the projection formula for the embedding i : Z ′ ֒ → Z , together withLemma 2.3. (cid:3) Let now f : T → Y be a morphism of schemes, and let Z T := T × Y Z ⊂ T × X be the fiberproduct such that we have a Cartesian diagram Z T f ′ / / (cid:15) (cid:15) Z (cid:15) (cid:15) T f / / Y . (9)
Lemma 2.12.
There is an isomorphism of functors f ∗ ◦ Φ Z P ∼ = Φ Z T f ′∗ P .Proof. This follows from Lemma 2.7. (cid:3)
Stability of Hitchin pairs.
For this subsection, let X be a smooth projective varietyof dimension n . We fix an ample numerical class H ∈ N ( X ). For every sheaf A ∈ Coh ( X ),we define its H -degree and its H -slope by deg H ( A ) = Z X c ( A ) · H n − , µ H ( A ) = deg H ( A ) rank ( A ) . A Hitchin bundle (
E, ϑ ) ∈ Hi V ( X ) is called ( H -slope) stable if for every Hitchin subsheaf A ⊂ E with rank ( A ) < rank ( E ), we have µ ( A ) < µ ( E ). It is called ( H -slope) semi-stable iffor every Hitchin subsheaf A ⊂ E , we have µ ( A ) ≤ µ ( E ).Clearly, if E is (semi-)stable as an ordinary sheaf, the Hitchin pair ( E, ϑ ) is (semi-)stabletoo. However, the converse is not true.
Example 2.13.
Let X be a smooth projective curve, and let L ∈ Pic X be a line bundle ofdegree −
1. We set E := O X ⊕ L . Then E is not stable and has exactly one destabilizingsubsheaf, namely the direct summand O X . Now, let ϑ : E ⊗ L ∼ = L ⊕ L ⊗ id L ! −−−−−−−→ O X ⊕ L ∼ = E , and consider the L -cotwisted Hitchin pair ( E, ϑ ). Then, the subbundle O X ⊂ E is not aHitchin subbundle, hence ( E, ϑ ) is stable.
Remark 2.14.
Let (
E, ϑ ) ∈ Hi V ( X ) be stable (or semi-stable), and let L ∈ Pic X . Then( E, ϑ ) ⊗ L is again stable (or semi-stable). Lemma 2.15.
Let ϕ : V → Q be a surjective morphism of vector bundles on X . A Hitchinpair ( E, ϑ ) ∈ Hi Q ( X ) is (semi-)stable if and only if ϕ ( E, ϑ ) ∈ Hi V ( X ) is (semi-)stable. OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 9
Proof.
By Lemma 2.3, A ⊂ E is a Hitchin subsheaf of ( E, ϑ ) if and only if it is a Hitchinsubsheaf of ϕ ( E, ϑ ). The assertion now follows from the fact that the slope of a Hitchin pairis defined as the slope of the underlying sheaf. (cid:3)
Proposition 2.16.
For every ( E, ϑ ) ∈ Hi V ( X ) , there is a unique filtration, called the Harder–Narasimhan filtration of ( E, ϑ ) , by Hitchin subsheaves E = F E ⊃ F E ⊃ · · · ⊃ F m − E ⊃ F m E = 0 such that all the successive quotients ( F i E/ F i +1 E, ¯ ϑ ) are semi-stable Hitchin pairs with µ ( F i E/ F i +1 E ) < µ ( F i +1 E/ F i +2 E ) for every i = 0 , . . . , m − .Proof. See [Sim94, Sect. 3]. (cid:3)
Remark 2.17.
The Harder–Narasimhan filtration of the Hitchin pair (
E, ϑ ) does not needto agree with the Harder–Narasimhan filtration of E ; compare Example 2.13.2.10. Equivariant Hitchin pairs.
Let G be a finite group acting on a scheme X . A G -equivariant vector bundle ( V, α ) on X consists of a vector bundle V ∈ VB ( X ) and a familyof isomorphisms α = { α g : V ∼ −→ g ∗ V } , called a G -linearization , such that for all g, h ∈ G thefollowing diagram commutes: V α g / / α hg g ∗ V g ∗ α h / / g ∗ h ∗ V ∼ = / / ( hg ) ∗ V .
Remark 2.18.
Let (
V, α ) be a G -equivariant vector bundle. For g ∈ G , we consider thecomposition Hi V ( X ) g ∗ −→ Hi g ∗ V ( X ) α g −−→ Hi V ( X ) , which is an autoequivalence of Hi V ( X ). In the following, we simply denote this autoequiva-lence by g ∗ : Hi V ( X ) ∼ −→ Hi V ( X ). For g, h ∈ G , there is a canonical ismorphism of autoequiv-alences g ∗ h ∗ ∼ = ( hg ) ∗ , which means that we have an action of the group G on the category Hi V ( X ).Given a G -equivariant vector bundle ( V, α ), a G -equivariant V -cotwisted Hitchin pair (cid:0) ( E, ϑ ) , λ (cid:1) consists of a Hitchin pair ( E, ϑ ) ∈ Hi V ( X ) and a family of isomorphisms of Hitchinpairs λ = { λ g : ( E, ϑ ) ∼ −→ g ∗ ( E, ϑ ) } such that for all g, h ∈ G the following diagram commutes: E λ g / / λ hg g ∗ E g ∗ λ h / / g ∗ h ∗ E ∼ = / / ( hg ) ∗ E .
That λ g is a morphism of Hitchin pairs means explicitly that the following diagram commutes: E ⊗ V λ g ⊗ α g / / ϑ (cid:15) (cid:15) g ∗ E ⊗ g ∗ V g ∗ ϑ (cid:15) (cid:15) E λ g / / g ∗ E . A G -equivariant Hitchin subsheaf of (cid:0) ( E, ϑ ) , λ (cid:1) is a Hitchin subsheaf A ⊂ E such that forevery g ∈ G , we have the equality λ g ( A ) = g ∗ A of subsheaves of g ∗ E .Let π : X → X/G be a geometric quotient. For every g ∈ G , we have the equality π ◦ g = π ,which gives an isomorphism of functors g ∗ ◦ π ∗ ∼ = π ∗ . Hence, for every V ∈ VB ( X/G ), the pull-back π ∗ V carries a canonical G -linearisation. Furthermore, for every Hitchin pair( F, η ) ∈ Hi V ( X/G ), the pull-back π ∗ ( F, η ) has canonically the structure of a G -equivariant π ∗ V -cotwisted Hitchin pair. Lemma 2.19.
Let a finite group G act on a smooth projective variety X such that Y = X/G is again smooth and π : X → Y is flat. Let H ∈ N ( Y ) be an ample class, V ∈ VB ( Y ) , and ( F, η ) ∈ Hi V ( Y ) a Hitchin bundle.(i) If µ π ∗ H ( A ) ≤ µ π ∗ H ( π ∗ n F ) holds for all S n -equivariant Hitchin subsheaves A of π ∗ ( F, η ) with rank A < rank F , then F is slope semi-stable with respect to H .(ii) If µ π ∗ H ( A ) < µ π ∗ H ( π ∗ n F ) holds for all S n -equivariant Hitchin subsheaves A of π ∗ ( F, η ) with rank A < rank F , then F is slope stable with respect to H .Proof. The proof is completely analogous to the proof for vector bundles without the structureof a Hitchin pair; see [Kru18b, Lem. 1.1]. (cid:3) Construction of Hitchin pairs on Hilbert schemes of points
Hilbert schemes of points.
Let X be a smooth quasi-projective variety. For every n ∈ N there is a fine moduli space X [ n ] of zero-dimensional sub-schemes of X of length n , calledthe Hilbert scheme of n points on X (also called punctual Hilbert scheme). This means thatthere is a closed sub-scheme Ξ n ⊂ X × X [ n ] which is flat and finite of degree n over X [ n ] , calledthe universal family , such that, for every scheme T and every closed sub-scheme Z ⊂ X × T which is flat and finite of degree n over T , there is a classifying morphism f : T → X [ n ] suchthat Z = ( id X × f ) − Ξ n , where ( id X × f ) − Ξ n ∼ = T × X [ n ] Ξ n is the scheme-theoretic preimage.There is the Hilbert–Chow morphism µ : X [ n ] → X ( n ) to the symmetric product X ( n ) := X n / S n that sends any zero-dimensional sub-scheme ξ ⊂ X to its weighted support: µ ([ ξ ]) = P x ∈ ξ ℓ ( ξ, x ) · x .If X = C is a smooth curve, µ is an isomorphism, hence C [ n ] ∼ = C ( n ) . If X is a smoothsurface, X [ n ] is smooth and µ : X [ n ] → X ( n ) is a crepant resolution of the singularities of thesymmetric product.3.2. Tautological bundles and Hitchin pairs.
Let X be a smooth quasi-projective variety,let n ∈ N , and let X p ←− Ξ n q −→ X [ n ] be the projections from the universal family of the Hilbertscheme X [ n ] . Given a sheaf E ∈ Coh ( X ), the associated tautological sheaf is defined by E [ n ] = q ∗ p ∗ E ∈ Coh ( X [ n ] ). Over a point [ ξ ] ∈ X [ n ] corresponding to a zero-dimensional sub-scheme ξ ⊂ X of length n , the fiber of the tautological sheaf is given by E [ n ] ([ ξ ]) = H ( E | ξ ).If E is a vector bundle, then E [ n ] is again a vector bundle with rank E [ n ] = n rank E . Thisfollows from the fact that q : Ξ n → X [ n ] is flat and finite of degree n .Let V ∈ VB ( X ). If E carries the structure of a V -cotwisted Hitchin pair, we get an inducedstructure of a V [ n ] -cotwisted Hitchin pair on E [ n ] . More precisely, for ( E, ϑ ) ∈ Hi V ( X ), wedefine the associated tautological Hitchin pair , using the Hitchin enhanced Fourier–Mukaitransform as introduced in Subsection 2.8, by( E [ n ] , ϑ { n } ) := ( E, ϑ ) [ n ] := Φ Ξ n O Ξ n ( E, ϑ ) ∈ Hi V [ n ] ( X [ n ] ) . Remark 3.1.
To provide some intuition, let us give a concrete description of the inducedmap ϑ { n } = q ∗ p ∗ ϑ ◦ ν q,E,V : E [ n ] ⊗ V [ n ] → E [ n ] . For a zero-dimensional sub-scheme ξ ⊂ X of OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 11 length n , the fiber of ϑ { n } over the point [ ξ ] ∈ X [ n ] is given by the composition ϑ { n } ([ ξ ]) : ( E [ n ] ⊗ V [ n ] )([ ξ ]) ∼ = H ( E | ξ ) ⊗ H ( E | ξ ) ∪ −→ H (( E ⊗ V ) | ξ ) ϑ | ξ −−→ H ( E | ξ ) ∼ = E [ n ] ([ ξ ]) . The fact that (cid:0) ϑ { n } ◦ ( ϑ { n } ⊗ id V [ n ] ) (cid:1) | E [ n ] ⊗ V V [ n ] = 0, which already follows from Lemma 2.4,can also be read off from the above description of ϑ { n } . Lemma 3.2.
Let T be a scheme, let Z ⊂ X × T be a sub-scheme which is flat and finiteof degree n over T , and let f : T → X [ n ] be the classifying morphism for Z . Then, for ( E, ϑ ) ∈ Hi V ( X ) , we have a natural isomorphism of f ∗ V [ n ] -cotwisted Hitchin pairs f ∗ ( E, ϑ ) [ n ] ∼ = Φ Z O Z ( E, ϑ ) Proof.
This follows by applying Lemma 2.12 to the Cartesian diagram Z / / (cid:15) (cid:15) Ξ n (cid:15) (cid:15) T f / / X [ n ] . (cid:3) Tangent bundle of the Hilbert scheme. If X is a smooth curve or surface, theHilbert scheme X [ n ] is again smooth. In this case, one might hope that, if we start with asheaf E ∈ Coh ( X ) and a non-zero Higgs field E → E ⊗ Ω X , we get an induced non-zero Higgsfield E [ n ] → E [ n ] ⊗ Ω X [ n ] on E [ n ] . However, as the following example shows, there cannot besuch a general construction. Example 3.3.
Let X = P . Then, as Ω X = O P ( − E := O P ( − ⊕ O P (1)has a non-zero Higgs field. There is an isomorphism ( P ) [2] ∼ = ( P ) (2) ∼ = P under which thetautological bundles are given by O P ( − [2] ∼ = O P ( − ⊕ , O P (1) [2] ∼ = O ⊕ P ;see [Nag17, Sect. 3]. Hence, E [2] ∼ = O P ( − ⊕ ⊕O ⊕ P . Using the Euler sequence, one computes H (Ω P ( − H (Ω P ) = H (Ω P (1)) = 0. It follows that Hom ( E [2] , E [2] ⊗ Ω P ) = 0, whichmeans that there is no non-zero Higgs field on E [2] , though there is one on E .However, our construction from the previous subsection equips E [ n ] with the structure of alogarithmic Higgs sheaf, with respect to the boundary divisor B n ⊂ X [ n ] , whenever there is aHiggs field on E . Indeed, let ( E, ϑ ) be a Higgs sheaf, i.e., a T X -cotwisted Hitchin pair. Then( E, ϑ ) [ n ] is a ( T X ) [ n ] -cotwisted Hitchin pair. Hence, our assertion follows from the followingresult. Theorem 3.4 ([Sta16]) . Let X be a smooth curve or surface. Then there is an isomorphism ( T X ) [ n ] ∼ = T X [ n ] ( − log B n ) . Proof.
For X a smooth surface, this is [Sta16, Thm. B]. The proof in the case case where X is a smooth curve is essentially the same. The only difference is that we do not needto restrict to an open subset U ⊂ X [ n ] (as done in the surface case on top of page 1181 of[Sta16].) since, in the curve case, the universal family Ξ n is already smooth everywhere (for X a smooth curve, we have Ξ n ∼ = X × X ( n − ). (cid:3) Reconstruction in the case of curves
Curves of genus g ≥ . In this subsection, we will prove Theorem 1.1. So let C bea smooth projective curve of genus g ( C ) ≥
2, let V ∈ VB ( C ), and let ( E, ϑ ) ∈ Hi V ( C ) bea semi-stable Hitchin bundle. We enhance the reconstruction method of [BN12] and [BN17,Sect. 2], which reconstructs vector bundles E on C from their tautological images E [ n ] on thesymmetric product C ( n ) , to Hitchin bundles.Let C ∼ = ∆ ⊂ C × C be the diagonal with ideal sheaf I ∆ . We denote the ( n − Z := V ( I n ∆ ) ⊂ C × C . Via the projection pr : C × C → C , the sub-scheme Z ⊂ C × C is a flat family of length n sub-schemes of C over C . Hence, we get a classifying morphism ι : C → C ( n ) , which is a closed embedding withimage the small diagonal in C ( n ) . By Lemma 3.2, we have ι ∗ ( E, ϑ ) [ n ] ∼ = Φ Z O Z ( E, ϑ ) . Hence, it is sufficient to show that the isomorphism class of (
E, ϑ ) ∈ Hi V ( C ) is determined byΦ Z O Z ( E, ϑ ) ∈ Hi b ∗ a ∗ V ( C ), where a, b : Z → C are the restrictions of the projections pr , pr : C × C → C .The structure sheaf O Z has the filtration O Z = J ⊃ J ⊃ · · · ⊃ J n − ⊃ J n ∆ = 0 , where J ∆ := I ∆ / I n ∆ . This induces a filtration by Hitchin subsheavesΦ Z O Z ( E, ϑ ) = Φ Z J ( E, ϑ ) ⊃ Φ Z J ( E, ϑ ) ⊃ · · · ⊃ Φ Z J n − ( E, ϑ ) ⊃ Φ Z J n ∆ ( E, ϑ ) = 0 (10)which we will show to be the Harder–Narasimhan filtration of Φ Z O Z ( E, ϑ ) ∈ Hi b ∗ a ∗ V ( C ). Notethat we have short exact sequences0 → J k +1∆ → J k ∆ → i ∗ ω ⊗ kC → i : C ֒ → Z is the closed embedding of the reduced diagonal. By Lemma 2.11 combinedwith Example 2.9, we have Φ Zi ∗ ω ⊗ kC ∼ = ϕ (cid:0) ( E, ϑ ) ⊗ ω ⊗ kC (cid:1) where ϕ : b ∗ a ∗ V ։ V is the canonicalsurjection induced by O Z ։ O ∆ . Hence, applying Lemma 2.10 to (11) gives a short exactsequence 0 → Φ Z J k +1∆ ( E, ϑ ) → Φ Z J k ∆ ( E, ϑ ) → ϕ (cid:0) ( E, ϑ ) ⊗ ω ⊗ kC (cid:1) → . By the assumption that (
E, ϑ ) is semi-stable, Remark 2.14, and Lemma 2.15, we see thatthe factor Φ Z J k ∆ ( E, ϑ ) / Φ Z J k +1∆ ( E, ϑ ) ∼ = ϕ (cid:0) ( E, ϑ ) ⊗ ω ⊗ kC (cid:1) is semi-stable. Furthermore, since deg ( ω C ) > g ( C ), we have strict inequalities µ (cid:16) Φ Z J k ∆ ( E, ϑ ) / Φ Z J k +1∆ ( E, ϑ ) (cid:17) < µ (cid:16) Φ Z J k +1∆ ( E, ϑ ) / Φ Z J k +2∆ ( E, ϑ ) (cid:17) . This means that (10) is indeed the Harder–Narasimhan filtration of Φ Z O Z ( E, ϑ ) ∈ Hi b ∗ a ∗ V ( X );see Proposition 2.16. The uniqueness of the first factor ϕ ( E, ϑ ) of the Harder–Narasimhanfiltration, together with Lemma 2.3(i), now show that the isomorphism class of (
E, ϑ ) isdetermined by Φ Z O Z ( E, ϑ ) ∼ = ι ∗ ( E, ϑ ) [ n ] . Remark 4.1.
In analogy with [BN17, Sect. 2], we can relax the assumptions of Theorem 1.1as follows. Let (
E, ϑ ) ∈ Hi V ( X ) be a Hitchin bundle with Harder–Narasimhan filtration E = F E ⊃ F E ⊃ · · · ⊃ F m − E ⊃ F m E = 0 . OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 13
We set µ min ( E, ϑ ) := µ ( E/ F E ) and µ max ( E, ϑ ) := µ ( F m − E ). Then, instead of assuming inTheorem 1.1 that ( E, ϑ ) and (
F, η ) are semi-stable, it is sufficient to assume that µ max ( E, ϑ ) − µ min ( E, ϑ ) < g ( C ) −
1) and µ max ( F, ϑ ) − µ min ( F, ϑ ) < g ( C ) − . Elliptic curves.
In this subsection, we prove Theorem 1.2 as a consequence of thefollowing more general result; compare [Kru18a, Thm. 5.2].
Proposition 4.2.
Let X be a smooth projective variety which has n + 1 automorphisms σ , . . . , σ n ∈ Aut ( X ) with empty pairwise equalizers , which means that σ i ( x ) = σ j ( x ) for all i = j and all x ∈ X . Then, for every n ∈ N and every two Hitchin pairs ( E, ϑ ) , ( F, η ) ∈ Hi V ( X ) , we have ( E, ϑ ) [ n ] ∼ = ( F, η ) [ n ] = ⇒ ( E, ϑ ) ∼ = ( F, η ) . Proof.
Replacing σ i by σ − ◦ σ i , we can assume without loss of generality that σ = id X . Bythe assumption on the automorphisms, the graphs Γ σ i are pairwise disjoint. Hence, for every j ∈ { , . . . , n } , the union G j = G i =0 ,...,ni = j Γ σ i ⊂ X × X is flat and finite of degree n over X . Let f j : X → X [ n ] be the classifying morphism for G j . By Lemma 3.2, we have f ∗ j ( E, ϑ ) [ n ] ∼ = Φ G j O Gj ( E, ϑ ). Since the Γ σ j are disjoint, we have O G j ∼ = ⊕ i = j O Γ σi . Together with Lemma 2.11 and Example 2.9, this gives f ∗ j ( E, ϑ ) [ n ] ∼ = Φ G j O Gj ( E, ϑ ) ∼ = M i = j ϕ j i Φ Γ σi O Γ σi ( E, ϑ ) ∼ = M i = j ϕ j i σ ∗ i ( E, ϑ ) , (12)where ϕ ji : Φ O Gj ( V ) ∼ = ⊕ i = j σ ∗ i V → Φ O Γ σi ( V ) ∼ = σ ∗ i V is the projection, which is induced bythe surjection O G j → O Γ σi . We also set V := ⊕ ni =1 σ ∗ i V , and write ψ j : V → M i =0 ,...,ni = j σ ∗ i V , ϕ k = ϕ jk ◦ ψ k : V → σ ∗ k V for the appropriate projections to the summands. Applying ψ j : Hi ⊕ i = j σ ∗ i V ( X ) → Hi V ( X ) to(12) gives an isomorphism ψ j f ∗ j ( E, ϑ ) [ n ] ∼ = M i = j ϕ i σ ∗ i ( E, ϑ ) ∼ = M i = j σ ∗ i ϕ ( E, ϑ ) . Here, the σ ∗ i on the right-hand side is a shortcut for the autoequivalence α σ i ◦ σ ∗ i of Hi V ( X )where α σ i : V → σ ∗ i V is the canonical isomorphism given by permutation of the summands;compare Remark 2.18.Considering these isomorphisms for varying j = 0 , . . . , n gives( E, ϑ ) [ n ] ∼ = ( F, η ) [ n ] = ⇒ M i =0 ,...,ni = j σ ∗ i ϕ ( E, ϑ ) ∼ = M i =0 ,...,ni = j σ ∗ i ϕ ( F, η ) ∀ j = 0 , . . . , n . (13)Since the category Hi V ( X ) is Krull-Schmidt, [Kru18a, Prop. 5.4] gives M i =0 ,...,ni = j σ ∗ i ϕ ( E, ϑ ) ∼ = M i =0 ,...,ni = j σ ∗ i ϕ ( F, η ) ∀ j = 0 , . . . , n = ⇒ ϕ ( E, ϑ ) ∼ = ϕ ( F, η ) . (14) Combining implication (13) with implication (14) and Lemma 2.3(i) proves the assertion. (cid:3)
Theorem 1.2 is a special case of Proposition 4.2 since, on an elliptic surface, there is aninfinite group of automorphisms with empty pairwise equalizers, namely the transpositions.5.
Stability in the case of curves
In this section, we prove Theorem 1.4. The special case of bundles without the structureof a Hitchin pair was proved in [Kru18b], and here we explain the extra ingredients necessaryfor the same proof to work for Hitchin pairs.Let C be a smooth projective curve, and let ( E, ϑ ) ∈ Hi V ( C ) be a stable (or semi-stable)Hitchin pair with µ ( E ) / ∈ [ − , n −
1] (or µ ( E ) / ∈ ( − , n − π n : C n → C ( n ) be the S n -quotient morphism. As discussed in Subsection 2.10, the pull-back π ∗ n ( E, ϑ ) [ n ] ∈ Hi π ∗ n V [ n ] ( C n )of the tautological Hitchin bundle carries a canonical S n -linearization. By Lemma 2.19, for( E, ϑ ) [ n ] to be stable (or semi-stable), it is sufficient to prove that, for every S n -equivariantHitchin subsheaf A ⊂ π ∗ n E [ n ] with rank A < rank ( π ∗ n E [ n ] ) = n rank ( E ), we have the inequality µ e H n ( A ) < µ e H n ( π ∗ E [ n ] ) (or µ ( A ) ≤ µ ( π ∗ E [ n ] )).The key to the proof of the stability criterion for tautological bundles in [Kru18b] is theexistence of a short exact sequence0 → pr ∗ E ( − δ n (1)) i E −→ π ∗ n E [ n ] p E −−→ pr ∗ π ∗ n − E [ n − → , (15)where pr : C n → C and pr : C n → C n − are the projections to the first factor and to thelast n − δ n (1) = P ni =2 ∆ i where∆ i = { ( x , . . . , x n ) ∈ X n | x = x i } ;see [Kru18b, Prop 1.5]. Let now A ⊂ π ∗ n E [ n ] be an equivariant Hitchin subsheaf. Setting A ′ := i − E A ⊂ pr ∗ E ( − δ n (1)) and A ′′ := p E ( A ), we get a commutative diagram0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / A ′ / / (cid:15) (cid:15) A / / (cid:15) (cid:15) A ′′ / / (cid:15) (cid:15) / / pr ∗ E ( − δ n (1)) i E / / π ∗ n E [ n ] p E / / pr ∗ π ∗ n − E [ n − / / . (16)with exact columns and rows. The proof in [Kru18b] uses the (semi-)stability of E (byassumption) and the (semi-)stability of E [ n − (by induction) to find upper bounds for theslopes of A ′ and A ′′ , which lead to the desired inequality µ e H n ( A ) < µ e H n ( π ∗ E [ n ] ) (or µ ( A ) ≤ µ ( π ∗ E [ n ] )). However, we only know the semi-stability of ( E, ϑ ) as a Hitchin pair, which doesnot imply the stability of E as an ordinary vector bundle; see Example 2.13. The reason that,nevertheless, the proof of loc. cit. works for Hitchin pairs is that we can enhance (15) to ashort exact sequence of π ∗ n V [ n ] -cotwisted Hitchin pairs.For this, we need to have a look at the construction of the short exact sequence (15) in theproof of [Kru18b, Prop 1.5]. We have ( π n × id C ) − Ξ n = D n , where D n = S nk =1 Γ k ⊂ C n × C is the union of the graphs Γ k := Γ pr k of the projections pr k : C n → C to the k -th factor. It OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 15 follows by flat base change that E [ n ] ∼ = Φ O Dn ( E ). Under this isomorphism, (15) is inducedby the canonical short exact sequence of Fourier–Mukai kernels0 → O Γ pr ( − n X k =2 [Γ pr k ∩ Γ pr ]) → O D n → O ∪ nk =2 pr k → O ∪ nk =2 pr k ∼ = ( pr × id C ) ∗ O D n − . In other words, the short exactsequence (15) is isomorphic to the exact sequence0 → Φ O Γ pr ( − P nk =2 [Γ pr k ∩ Γ pr ]) ( E ) → Φ O Dn ( E ) → Φ O ∪ nk =2 pr k ( E ) → . (18)By Lemma 2.10 we get an enhanced version of (18) in the form of a short exact sequence0 → Φ D n O Γ pr ( − P nk =2 [Γ pr k ∩ Γ pr ]) ( E, ϑ ) → Φ D n O Dn ( E, ϑ ) → Φ D n O ∪ nk =2 pr k ( E, ϑ ) → π ∗ n V [ n ] -cotwisted Hitchin pairs. The surjections O D n → O Γ and O D n → O ∪ nk =2 pr k ∼ =( pr i × id C ) ∗ O D n − induce surjections ϕ : π ∗ n V [ n ] → pr ∗ V , ψ : π ∗ n V [ n ] → pr ∗ π ∗ n − V [ n − . Applying Example 2.9 together with Lemma 2.11 to the first term of (19), Lemma 2.12 (re-calling the isomorphism ( π n × id C ) ∗ O Ξ n ∼ = O D n ) to the second term of (19), and Lemma 2.12(recalling the isomorphism O ∪ nk =2 pr k ∼ = ( pr i × id C ) ∗ O D n − ) together with Lemma 2.11 to thethird term of (19), we see that (19) is isomorphic to0 → ϕ pr ∗ ( E, ϑ ) ⊗ O ( − δ n (1)) i E −→ π ∗ n ( E, ϑ ) [ n ] p E −−→ ψ pr ∗ π ∗ n − ( E, ϑ ) [ n − → . (20) Lemma 5.1.
Let, as above, A be an S n -equivariant Hitchin subsheaf of π ∗ n ( E, ϑ ) [ n ] , and let A ′ := i − E ( A ) ⊂ pr ∗ E ( − δ n (1)) and A ′′ := p E ( A ) ⊂ pr ∗ π ∗ n − E [ n − .(i) A ′ is a Hitchin subsheaf of the pr ∗ V -cotwisted Hitchin pair pr ∗ ( E, ϑ ) ⊗ O ( − δ n (1)) .(ii) A ′′ is a Hitchin subsheaf of the pr ∗ π ∗ n − V [ n − -cotwisted Hitchin pair pr ∗ π ∗ n − ( E, ϑ ) [ n − .Proof. Since we now have the short exact sequence of Hitchin pairs (20), this follows directlyfrom Lemma 2.2 and Lemma 2.3(iii). (cid:3)
Now, as in [Kru18b, Subsect. 2.2], we choose the points x , . . . , x n in such a way that allrows and columns of the diagram (16) stay exact after pull-back along the embedding ι : C ֒ → C n , ι ( t ) = ( t, x , . . . , x n ) . Since ι ∗ O C n ( − δ n (1)) ∼ = O C ( − P ni =2 x i ) and pr ◦ ι = id C , we have an isomorphism of ι ∗ pr ∗ V ∼ = V -cotwisted Hitchin pairs ι ∗ pr ∗ (cid:0) ( E, ϑ ) ⊗ O ( − δ n (1)) (cid:1) ∼ = ( E, ϑ ) ⊗ O ( − n X i =2 x i )By Lemma 5.1(i), ι ∗ A ′ is a Hitchin subsheaf of ( E, ϑ ) ⊗ O ( − P ni =2 x i ). Since, by assumption,( E, ϑ ) is stable (or semi-stable), so is (
E, ϑ ) ⊗O ( − P ni =2 x i ); see Remark 2.14. Hence, we have µ ( ι ∗ A ′ ) < µ ( E ( − P ni =2 x i )) (or µ ( ι ∗ A ′ ) ≤ µ ( E ( − P ni =2 x i ))). Now, in the case µ ( E ) > n − µ ( E ) ≥ n − µ e H n ( A ) < µ e H n ( π ∗ n E [ n ] ) (or µ e H n ( A ) ≤ µ e H n ( π ∗ n E [ n ] )) which, as discussed above, by Lemma 2.19implies that E [ n ] is stable (or semi-stable). If µ ( E ) ≤ −
1, as in [Kru18b, Subsect. 2.3], we consider the closed embedding ι : C n − ֒ → C n , ι ( t , . . . , t n ) = ( x, t , . . . , t n ) , where, again, x ∈ C is chosen in such a way that all rows and columns of the diagram(16) stay exact after pull-back along ι ∗ . Since pr ◦ ι = id C n − , we have an isomorphism of ι ∗ pr ∗ π ∗ n − V [ n − ∼ = π ∗ n − V [ n − -cotwisted Hitchin pairs ι ∗ pr ∗ π ∗ n − ( E, ϑ ) [ n − ∼ = π ∗ n − ( E, ϑ ) [ n − . By induction, we may assume that π ∗ n − ( E, ϑ ) [ n − is stable (or semi-stable). By Lemma 5.1(ii), ι ∗ A ′′ is a Hitchin subsheaf of π ∗ n − ( E, ϑ ) [ n − . Hence, we get that µ ( ι ∗ A ′′ ) < µ ( π ∗ n − E [ n − )(or µ ( ι ∗ A ′′ ) ≤ µ ( π ∗ n − E [ n − )). Now, the assertion in the case µ ( E ) < − µ ( E ) ≤ − Higher dimensions
Reconstruction for higher dimensions.
In this section, we proof Theorem 1.3. Aversion of this statement for coherent sheaves without the structure of a Hitchin bundle wasproved in [Kru18a, Thm. 3.2 & Rem. 3.5]. Since the arguments which allow to lift the resultsto Hitchin pairs are very similar to those of the previous two sections, we will restrict ourselvesto a quiet terse explanation this time.Recall that there is the Hilbert–Chow morphism µ : X [ n ] → X ( n ) and the quotient mor-phism π : X n → X ( n ) . We consider the open subsets X n := { ( x , . . . , x n ) ∈ X n | x i = x j for i = j } ⊂ X n ,X ( n )0 := π ( X n ) ⊂ X ( n ) , and X [ n ]0 := µ − ( X ( n )0 ) ⊂ X [ n ] . This gives the following diagram withCartesian squares X [ n ]0 µ / / i (cid:15) (cid:15) X ( n )0 (cid:15) (cid:15) X n π o o j (cid:15) (cid:15) X [ n ] µ / / X ( n ) X nπ o o where all the vertical arrows are open immersions, and µ is an isomorphism.For ( E, ϑ ) ∈ Hi V ( X ), we define the S n -equivariant ( ⊕ ni =1 pr ∗ i V )-cotwisted Hitchin pair C ( E, ϑ ) = ( ⊕ ni =1 pr ∗ i E, ˜ ϑ, λ ) by˜ ϑ : n M i =1 pr ∗ i E ! ⊗ n M i =1 pr ∗ i V ! ∼ = n M i,j =1 pr ∗ i E ⊗ pr ∗ j V → n M i =1 pr ∗ i ( E ⊗ V ) ⊕ pr ∗ i ϑ i −−−−−→ n M i =1 pr ∗ i E , where the middle map is the projection to the appropriate factors, and λ g : ⊕ i pr ∗ i E → g ∗ ( ⊕ i pr ∗ i E ) is the direct sum of the canonical isomorphisms pr ∗ i E ∼ −→ g ∗ pr ∗ g ( i ) E .Note that the open subset X n ⊂ X n is stable under the S n -action on X n , hence themorphism π : X n → X ( n )0 is a S n -quotient. Thus, as explained in Subsection 2.10, the Higgsbundle j ∗ π ∗ µ ∗ i ∗ ( E, ϑ ) [ n ] carries a canonical S n -linearization. Lemma 6.1.
For every ( E, ϑ ) ∈ Hi V ( X ) such that E is reflexive, we have an isomorphismof S n -equivariant ( ⊕ ni =1 pr ∗ i V ) -cotwisted Hitchin pairs j ∗ π ∗ µ ∗ i ∗ ( E, ϑ ) [ n ] ∼ = C ( E, ϑ ) . OGARITHMIC HIGGS BUNDLES ON PUNCTUAL HILBERT SCHEMES 17
Proof.
For vector bundles without the structure of a Hitchin pair, this is [Sta16, Lem. 1.1].For the straightforward generalization to reflexive sheaves, see [Kru18a, Sect. 3.1]. That thetwo structures of Hitchin pairs ( L ni =1 pr ∗ i E ) ⊗ ( L ni =1 pr ∗ i V ) → L ni =1 pr ∗ i E on both sides of thealleged isomorphism agree can be checked on the open dense subset X n ⊂ X n . This can bedone either using the results on Hitchin enhanced Fourier–Mukai transforms of Subsection 2.8,or the concrete description of ϑ { n } of Remark 3.1. (cid:3) Proof of Theorem 1.3.
Let δ : X ֒ → X n , δ ( x ) = ( x, . . . , x ) be the embedding of the small diag-onal. We have δ ∗ C ( E, ϑ ) ∼ = ( E, ϑ ) ⊕ n with the induced S n -linearization given by permutationof the direct summands. Hence, the S n -invariants are given by (cid:0) δ ∗ C ( E, ϑ ) (cid:1) S n ∼ = ( E, ϑ ). ByLemma 6.1, this implies (cid:0) δ ∗ j ∗ π ∗ µ ∗ i ∗ ( E, ϑ ) [ n ] (cid:1) S n ∼ = ( E, ϑ ) , which means that we can reconstruct the isomorphism class of ( E, ϑ ) form (
E, ϑ ) [ n ] . (cid:3) Stability of tautological Hitchin bundles on Hilbert schemes of points onsurfaces.
Let X be a smooth projective surface, and let H be an ample divisor on X . Thereis a unique divisor H ( n ) on the symmetric product X ( n ) with π ∗ n H ( n ) = P ni =1 pr ∗ i H . We set H [ n ] := µ ∗ H ( n ) . This divisor is big and nef, but not ample, as it is trivial along the exceptionaldivisor of µ . The definitions of stable vector bundles and Hitchin bundles still make sensefor non-ample divisors. In [Sta16, Sect. 1], it is shown that, if E is a H -slope stable vectorbundle on X with E = O X , the associated tautological bundle E [ n ] is H [ n ] -slope stable. UsingLemma 6.1 while going through the proof of [Sta16], it is quite easy to see that this result alsogeneralizes to Hitchin pairs: If ( E, ϑ ) is a H -slope stable Hitchin pair on X with E = O X ,then ( E, ϑ ) [ n ] is H [ n ] -slope stable.In [Sta16], it is shown that in a neighborhood of H [ n ] in N ( X ), there is also an ample class I such that E [ n ] is I -slope stable if E = O X is H -slope stable. It seems likely that also thisresult generalizes to Hitchin pairs, but we have not checked the details. Acknowledgements
The first-named author wishes to thank Philipps-Universit¨at Marburg for hospitality whilethis work was carried out. He also acknowledges a partial support of a J. C. Bose Fellowship.
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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai400005, India
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