Rigid stable vector bundles on hyperk\"ahler varieties of type $K3^{[n]}$

We prove existence and unicity of slope stable vector bundles on a general polarized hyperk\"ahler (HK) variety of type $K3^{[n]}$ with certain discrete invariants, provided the rank and the first two Chern classes of the vector bundle satisfy certain equalities. The latter hypotheses at first glance appear to be quite restrictive, but in fact we might have listed almost all slope stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type $K3^{[n]}$ with $20$ moduli.

1. Introduction 1.1.Background.A prominent rôle in the theory of K3 surfaces is played by spherical (i.e.rigid and simple) vector bundles.In [O'G22] we have proved existence and uniqueness results for stable vector bundles on general polarized hyperkähler (HK) variety of type K3 r2s with certain discrete invariants (of the polarization and of the vector bundle).In the present paper we show that the main result in [O'G22] extends to HK varieties of type K3 rns of arbitrary (even) dimension.More precisely, we prove that for certain choices of rank and first two Chern classes on a polarized HK variety pX, hq of type K3 rns , there exists one and only one stable vector bundle with the assigned rank and first two Chern classes provided the moduli point of pX, hq is a general point of a certain irreducible component of the relevant moduli space of polarized HK varietes.
We like to think of this result as evidence in favour of the following slogan: stable vector bundles on higher dimensional HK manifolds behave as well as stable sheaves on K3 surfaces, provided one restricts to (stable) vector bundles whose projectivization extends to all small deformations of the base HK manifold (i.e.projectively hyperhomolomorphic vector bundles).
1.2.The main result.Let K d e p2nq be the moduli space of polarized HK varieties of type K3 rns of degree e and divisibility d.Thus K d e p2nq parametrizes isomorphism classes of couples pX, hq where X is a HK manifold of type K3 rns , h P NSpXq is the class of an ample divisor class (we assume that h is primitive) such that q X phq " e and divphq " d, where q X is the Beauville-Bogomolov-Fujiki (BBF) quadratic form of X and divphq is the divisibility of h, i.e. the positive generator of q X ph, H 2 pX; Zqq.Note that divphq divides 2pn ´1q.It is known under which hypotheses K d e p2nq is not empty.If that is the case, then it is a quasi-projective variety (not necessarily irreducible) of pure dimension 20.
We recall that if F is a (coherent) sheaf on a complex smooth variety the discriminant of F is defined to be the Betti cohomology class ∆pF q :" 2rc 2 pF q ´pr ´1qc 1 pF q 2 " ´2r ch 2 pF q `ch 1 pF q 2 . (1.2.1) Date: 17th October 2023.Partially supported by PRIN 2017YRA3LK.
1 Theorem 1.1.Let n, r 0 , g, l, e P N `, with n ě 2, and let e :" # e if r 0 is even, 4e if r 0 is odd.

Rank and Chern classes.
The choice of rank and first two Chern classes in Theorem 1.1 is not as special as one would think.Let us first consider a rigid stable vector bundle E on a polarized K3 surface pS, hq of rank r with c 1 pE q " ah.Since χpS, End E q " 2, we have 2rc 2 pE q ´pr ´1qa 2 h 2 " ∆pE q " 2pr 2 ´1q. (1.3.2) It follows that gcdtr, au " 1.The following result is a (weak) extension to higher dimensions.
Proposition 1.2.Let pX, hq be a polarized HK variety of type K3 rns , and let F be a slope stable vector bundle on X. Suppose that c 1 pF q " ah and that the natural morphism DefpX, F q Ñ DefpX, hq is surjective.Then rpF q " r n 0 m, a ¨divphq " r n´1 0 mb 1 0 , (1.3.3)where r 0 , m, b 1 0 are integers, and gcdtr 0 , b 1 0 u " 1.
Note that in Proposition 1.2 we do not assume that F is rigid.There are examples of slope stable projectively hyperholomorphic vector bundles for which m ą 1 which are not rigid see [Mar21,Bot22] and [Fat23].The tangent vector bundle of a HK manifold of type K3 r2s is an example of rigid slope stable projectively hyperholomorphic vector bundle with m ą 1 (in fact m " 4), see [Gav21].There are no other examples of the latter type that I am aware of.
The stable vector bundles in Theorem 1.1, together with those obtained by tensoring with powers of the polarization, cover many of the choices of rank and first Chern class with m " 1 which are a priori possible according to Proposition 1.2.
Regarding the discriminant of the vector bundle(s) E in Theorem 1.1 we note the following.First, the formula for the discriminant in (1.2.6) for n " 1 is exactly the formula in (1.3.2).Next, since ´∆pE q is the second Chern class of the pushforward to X of the relative tangent bundle of PpE q Ñ X, and PpE q extends to all small deformations of X (because H 2 pX, End 0 pE qq " 0), the cohomology class ∆pE q remains of type p2, 2q for all deformations of X.It follows that ∆pE q is a linear combination of c 2 pXq and q _ X , see the main result in [Zha15].If n P t2, 3u then c 2 pXq and q _ X are linearly dependent, and hence it follows (without doing any computation) that ∆pE q is a multiple of c 2 pXq.If n ą 3 then c 2 pXq and q _ X are linearly independent, hence there is no "a priori" reason why ∆pE q should be a multiple of c 2 pXq.In fact I know of no examples of stable projectively hyperholomorphic vector bundles on HK varieties of type K3 rns whose discriminant is not a multiple of the second Chern class.
The vector bundles E in Theorem 1.1 are atomic, in fact they are in the O Xorbit (see [Mar21]) and hence the Beckmann-Markman extended Mukai vector r vpE q (see [Mar21, Theorem 6.13], [Bec23, Definition 4.16], [Bec22, Definition 1.1]) is determined uniquely (if we require that its first entry equals rpE q), in particular r qpr vpE qq " ´pn `3qr 2n´2 0 {2 by [Bec23,Lemma 4.8].By the formula relating the projection of the discriminant ∆pE q on the Verbitsky subalgebra and the square r qpr vpE qq in [Bot22, Proposition 3.11], we get that if ∆pE q is a multiple of c 2 pXq, then it is given by the formula in (1.2.6).
The natural question to ask is the following: are we close to having listed all slope stable rigid vector bundles on a polarized HK variety of type K3 rns with 20 moduli?1.3.3.Projective bundles.Let X and E be as in Theorem 1.1.The projectivization PpE q extends (uniquely) to a projective bundle on all (small) deformations of X because H 2 pX, End 0 pE qq " 0. Actually Markman [Mar21,Theorem 1.4] shows that PpE q extends to a projective bundle on all deformations of X (it is projectively hyperholomorphic).In fact the (possibly twisted) locally free sheaf E on S rns appearing in Markman loc.cit. is obtained by deforming the vector bundle F rns àssociated to a spherical vector bundle F on S, see Definition 3.1 (or Definition 5.1 in [O'G22]), and likewise the vector bundles in Theorem 1.1 are obtained by deforming F rns `.Theorem 1.1 should provide a uniqueness result for stable projective bundles P of dimension pr n 0 ´1q with characteristic class given by the third equality in (1.2.6) (i.e.´c2 pΘ P{X q, where Θ P{X is the relative tangent bundle of P Ñ X) on a general HK manifold of type K3 rns .In order to turn this into a precise statement one would need to specify with respect to which Kähler classes the projective bundle is supposed to be stable.The zoo of conditions in Theorem 1.1 would then correspond to the cases in which the projective bundle is the projectivization of a vector bundle, i.e. to the vanishing of the relevant Brauer class.1.3.4.Franchetta property.Let U il e p2nq Ă K il e p2nq be an open non empty subset with the property that there exists one and only one stable vector bundle E on rpX, hqs P U il e p2nq such that the equations in (1.2.6) hold, and let X Ñ U il e p2nq be the tautological family of HK (polarized) varieties (we might need to pass to the moduli stack).By Theorem A.5 in [Muk87], there exists a quasi tautological vector bundle E on X , i.e. a vector bundle whose restriction to a fiber pX, hq of X Ñ U il e p2nq is isomorphic to E 'd for some d ą 0, where E is the vector bundle of Theorem 1.1.If rpX, hqs P U il e p2nq the Generalized Franchetta conjecture, see [FLV19], predicts that the restriction to CH 2 pXq Q of ch 2 pEq P CHpX q Q is equal to ´d r 2n´2 0 pr 2 0 ´1q 12 c 2 pXq.In other words it predicts that the third equality in (1.2.6) holds at the level of (rational) Chow groups.In general it is not easy to give a rationally defined algebraic cycle class on a non empty open subset of the moduli stack of polarized HK varieties.Theorem 1.1 produces such a cycle, and hence it provides a good test for the generalized Franchetta conjecture.
1.4.Basic ideas.The key elements in the proof of the main result are the following.First there is the extension to modular sheaves (defined in [O'G22]) on higher dimensional HK manifolds of the decomposition of the (real) ample cone of a smooth projective surface into open chambers for which slope stability of sheaves with fixed numerical characters does not change, see The second element is the behaviour of modular vector bundles on a Lagrangian HK manifold.If the polarization is very close to the pull-back of an ample line bundle from the base, then the restriction of a slope stable vector bundle to a general Lagrangian fiber is slope semistable, and if it is slope stable then it is a semi-homogeneous vector bundle, in particular it has no non trivial infinitesimal deformations keeping the determinant fixed.In the reverse direction, if the restriction of a vector bundle to a general Lagrangian fiber is slope stable, then the vector bundle is slope stable (provided the polarization is very close to the pull-back of an ample line bundle from the base).
The key element in the proof of existence is a construction discussed in [O'G22] (and in [Mar21]) which associates to a vector bundle F on a K3 surface S a sheaf F rns `on S rns .The sheaf F rns `is locally free by Haiman's highly non trivial results in [Hai01].If F is a spherical vector bundle then End 0 pF rns `q has no non zero cohomology by Bridgeland-King-Reid's derived version of the McKay correspondence.This gives that F rns `extends to all (small) deformations of pS rns , det F rns `q, and that the projectivization PpF rns `q extends to all (small) deformations of S rns (the last result follows from a classical result of Horikawa).We prove slope stability of F rns `in the case of an elliptic K3 surface S, by using our results on vector bundles on Lagrangian HK manifolds.In fact if S is an elliptic K3 surface, then there is a Lagrangian fibration S rns Ñ pP 1 q pnq -P n , whose general fiber is the product of n fibers of the elliptic fibration.If F is a slope stable rigid vector bundle on S, then the restriction to an elliptic fiber is slope stable.It follows that the restriction of F rns `to a general fiber of the Lagrangian fibration S rns Ñ P n is slope stable.From this one gets that the (unique) extension of F rns `to a general Lagrangian deformation of pS rns , det F rns `q is slope stable with respect to det F rns `(provided we move in a Noether-Lefschetz locus with high enough discriminant).
Uniqueness of a general slope stable vector bundle with the given numerical invariants is obtained by proving uniqueness for vector bundles on (polarized) HK varieties with Lagrangian fibrations (with discriminant high enough and almost coprime to the rank).The main points in the proof of the latter result are the following.Let F be a spherical vector bundle on an elliptic K3 surface S: the vector bundle E X on a (small) Lagrangian deformation X of S rns obtained by extension of F rns `restricts to slope stable semi-homogeneous vector bundles on Lagrangian fibers parametrized by a large open subset of the base (the complement has codimension at least 2).Any slope stable vector bundle E on X with the same rank, c 1 and c 2 as E X restricts to a slope stable semi-homogeneous vector bundle on a general Lagrangian fiber.Any two simple semi-homogeneous vector bundles on an abelian variety with the same rank and determinant are obtained one from the other via tensorization with a (torsion) line bundle.This, together with a monodromy argument, gives that E X and E restrict to isomorphic vector bundles on a general Lagrangian fiber.Since the set of Lagrangian fibers for which the restriction of E X is slope stable has complement of codimension at least 2, one concludes that E X and E are isomorphic.
1.5.Outline of the paper.Sections 2 and 3 are devoted to the computation of the discriminant of the vector bundle F rns `on S rns , provided F is a spherical vector bundle on the K3 surface S. Since PpF rns `q extends to all (small) deformations of S rns , one knows a priori that the discriminant is a linear combination of c 2 pS rns q and the inverse q _ S rns of the BBF quadratic form.From this it follows that one can work on the open subset of S rns parametrizing subschemes whose support has cardinality at least n ´1, and then a straightforward computation gives that the discriminant is as in (1.2.6).
In Section 4 we show that by starting from slope stable spherical vector bundles F on an elliptic surface S Ñ P 1 we can produce vector bundles F rns `on S rns with rank and first two Chern classes covering all the cases in Theorem 1.1.Moreover we study the restriction of such an F rns `to fibers of the Lagrangian fibration S rns Ñ pP 1 q pnq -P n .
Section 5 is the most demanding part of the paper.The key ideas, outlined in Section 1.4, are combined together in order to give the proof of Theorem 1.1 (and of Proposition 1.2).1.6.Acknowledgments.Thanks to Marco Manetti for explaining Horikawa's classical result [Hor74, Theorem 6.1], and to Emanuele Macrì for helping me out by introducing me to the powerful [Sim92, Theorem 2].
2. The isospectral Hilbert scheme 2.1.Summary of results.We start by introducing notation and recalling known results.Let S be a K3 surface.The isospectral Hilbert scheme of n points on S, denoted by X n " X n pSq, was introduced and studied by Haiman, see Definition 3.2.4 in [Hai01].We have a commutative diagram In fact X n pSq is the reduced scheme associated to the fiber product of S n and S rns over S pnq .Moreover the map τ is identified with the blow up of S n with center the big diagonal, see Corollary 3.8.3 in [Hai01].Let pr i : S n Ñ S be the i-th projection, and let τ i : X n pSq Ñ S be the composition τ i :" pr i ˝τ .Let pS n q ˚:" tx " px 1 , . . ., x n q P S n | at most two entries of x are equalu, (2.1.2) and let X n pSq ˚:" τ ´1ppS n q ˚q.Let E n Ă X n pSq ˚be the exceptional divisor of X n pSq ˚Ñ pS n q ˚.Then E n is smooth because the restriction of the big diagonal to pS n q ˚is smooth.We let e n :" clpE n q P H 2 pX n pSq ˚; Qq.
(2.1.3)Let η P H 4 pS; Qq be the fundamental class.If X is a HK manifold, the non degenerate BBF symmetric blinear form H 2 pXqĤ 2 pXq Ñ C defines a symmetric bilinear form H 2 pXq _ ˆH2 pXq _ Ñ C, i.e. a symmetric element of H 2 pXq b H 2 pXq, whose image in H 4 pXq via the cup product map is a rational Hodge class q _ X .Below are the results obtained in the present section.
Proposition 2.1.Let n ě 2. We have the following equalities in the rational cohomology of X n pSq ˚: (2.1.4) (2.1.5) Before stating the next result we note that while q _ X is a rational cohomology class, p2n ´2qq _ X lifts to an integral class (uniquely because the group H ˚pS rns ; Zq is torsion free by the main result in [Mar07]).
Proposition 2.2.Let n ě 4, and let T n Ă H 4 pS rns ; Zq be the saturation of the subgroup spanned by c 2 pS rns q and p2n ´2qq _ X .The map The proof of Propositions 2.1 and 2.2 are respectively in Subsections 2.2 and 2.3.

2.2.
Proof of Proposition 2.1.We start by recalling a couple of formulae.First suppose that j : D ãÑ W is the embedding of a smooth divisor in a smooth ambient variety, and that F is a sheaf on D. Then by Grothendieck-Riemann-Roch and the push-pull formula we have chpj ˚pF qq " j ˚pchpF qq ¨TdpO W pDqq ´1 " j ˚pchpF qq ¨p1 ´clpDq 2 `clpDq 2 6 `. ..q.
(2.2.1)Next we recall how one computes the Chern classes of a blow up.Let Z be a smooth variety, and let Y Ă Z be a smooth subvariety of pure codimension c.Let f : r Z Ñ Z be the blow of Y .Let j : E ãÑ r Z be the inclusion of the exceptional divisor of f , and let e P H 2 p r Z; Qq be the class of E. If N Y {Z is the normal bundle of Y in Z, then E -PpN Y {Z q, and the restriction of O r Z pEq to E is isomorphic to the tautological sub line bundle O E p´1q.Let Q be the quotient bundle on E, i.e. the vector bundle fitting into the exact sequence where (2.2.3) Taking Chern characters, and applying the formula in (2.2.1) to the inclusion j : E ãÑ r Z and the sheaf Q we get the formula chp r Zq " f ˚pchpZ qq ´chpj ˚pQqq " f ˚pchpZ qq ´j˚p chppQqq ¨´1 ´e 2 `. . .

¯"
" f ˚pchpZ qq ´j˚`p c ´1q `f E pc 1 pN Y {Z q ´j˚p eq ˘¨´1 ´e 2 `. . .

¯"
" Proof of the equality in (2.1.4).Since X n pSq ˚is the blow up of pS n q ˚with center the smooth locus of the big diagonal, we can relate the Chern characters of X n pSq ånd pS n q ˚via the equality in (2.2.4).Since ch 2 pS n q " ´24 ř n i"1 τ i pηq, and the normal bundle of the big diagonal in S n has trivial first Chern class, the equation in (2.2.4) gives that (2.2.5) The differential of the map ρ : X n pSq Ñ S rns gives the exact sequence where ι : E n ãÑ X n pSq ˚is the inclusion map.Taking Chern characters we get that ch 2 pX n pSq ˚q " ρ ˚ch 2 ppS rns q ˚q ´3 2 e 2 n . (2.2.7) The equality in (2.1.4)follows from the equalities in (2.2.5) and (2.2.7).
Proof of the equality in (2.1.5).Let tα 1 , . . ., α 22 u be an orthonormal basis of H 2 pS; Cq.Then and hence Let D n Ă S n be the big diagonal.Then clpD n q " pn ´1q (2.2.3) Moreover it follows from the "Key Formula"(see for example Proposition 6.7 in [Ful84]) that we have the relation The equality in (2.1.5)follows at once from the equalities in (2.2.3) and (2.2.4).
Proof.We prove the lemma by integrating α n and e 2 n over algebraic 2 cycles on X n pSq ˚defined as follows.Let p 1 , . . ., p n´1 Ă S be n ´1 distinct points, and let Γ :" ρ ´1ptp 1 , . . ., p n´1 , xq | x P Suq. (2.3.1) Clearly Γ Ă X n pSq ˚, and it is isomorphic to the blow up of S at p 1 , . . ., p n´1 .In order to define the second 2 cycle we assume (as we may) that S contains two smooth curves C 1 , C 2 intersecting with transverse intersection of cardinality d ą 0. Let q 1 , . . ., q n´2 Ă pSzC 1 zC 2 q be n ´2 distinct points, and let Clearly Ω Ă X n pSq ˚, and it is isomorphic to the blow up of C 1 ˆC2 at the d points px, xq for It makes sense to integrate α n and e 2 n over Γ, Ω because the latter are compact (complex) surfaces contained in X n pSq ˚.One checks easily that the 2 ˆ2 "Gram matrix" of the integrals of α n and e 2 n over Γ and Ω is given by Table 1.It follows that α n and e 2 n are linearly independent.Now we can prove Proposition 2.2.Proposition 2.1 expresses the restriction to X n pSq ˚of ρ ˚pc 2 pS rns q and q _ as linear combinations of α n and e n .The determinant of the 2 ˆ2 matrix with entries the corresponding coefficients is non singular if and only if n R t2, 3u, hence Proposition 2.2 follows from Lemma 2.3.
Remark 2.4.Let n P t2, 3u.By Proposition 2.1 the classes ρ ˚`ch 2 pS rns ˘|XnpSqå nd ρ ˚pq _ q |XnpSq˚a re linearly dependent.This agrees with known results.In fact if X is a HK of type K3 r2s then c 2 pXq and q _ X are linearly dependent because Sym 2 H 2 pX; Qq " H 4 pX; Qq, and if X is a HK of type K3 r3s then c 2 pXq and q _ X are linearly dependent although Sym 2 H 2 pX; Qq is strictly contained in H 4 pX; Qq, see Example 14 in [Mar02], or Remark 3.3 in [GKLR22].

Basic modular vector bundles on S rns
3.1.Summary of results.Let S be a K3 surface.We maintain the notation introduced in Subsection 2.1.Let F be a locally free sheaf on S. Then X n pF q :" τ 1 pF q b . . .b τ n pF q (3.1.1)is a locally free sheaf on X n pSq.The map ρ in (2.1.1)is finite, and moreover it is flat because X n pSq is CM by Theorem 3.1 in [Hai01].It follows that the pushforward ρ ˚pX n pF qq is also locally free.The symmetric group S n acts on X n pSq compatibly with its permutation action on S n , and hence the action lifts to an action µ ǹ on X n pF q.Since µ n maps to itself any fiber of ρ : X n pSq Ñ S rns , we get an action µ ǹ : S n Ñ Autpρ ˚Xn pF qq.
Definition 3.1.Let F rns `Ă ρ ˚Xn pF q be the sheaf of S n -invariants for µ ǹ .
Since ρ ˚Xn pF q is locally free, so is F rns Let r 0 be the rank of F .Below is the main result of the present section.
Proposition 3.2.Suppose that F is spherical, i.e. h p pS, End 0 pF qq " 0 for all p.
If F is spherical, then the vector bundle F rns `is modular by Proposition 3.2, and we refer to it as a basic modular vector bundle.The proof of Proposition 3.2 is in Subsection 3.4.
Remark 3.4.The equalities in Proposition 3.2 should hold with the weaker hypothesis χpS, End 0 pF qq " 0. To prove this it would suffice to show that such a vector bundle is the limit of spherical vector bundles.

Chern classes of ρ
Q pS rns q be given by In the present subsection we prove the following result.
Proposition 3.5.Let S be a K3 surface, and let F be a vector bundle on S such that χpS, End 0 pF qq " 2. Let r 0 be the rank of S, and let h `P H 1,1 Q pS r2s q be as in (3.2.1).Then the following equalities hold:

3.2.4)
Proof.Let D n Ă pS n q ˚be the (intersection of pS n q ˚with the) big diagonal.For 1 ď j ă k ď n let D n pj, kq Ă D n be the set of points px 1 , . . ., x n q such that x j " x k .
We have the open embedding x j , x j`1 , . . ., x k´1 , x x k , x k`1 , . . ., x n q (3.2.5)Let τ En : E n Ñ D n be the restriction of τ to E n , and let E n pj, kq :" τ ´1 En pD n pj, kqq.Then E n " š E n pj, kq.Let τ j,k : E n pj, kq Ñ D n pj, kq be defined by the restriction of τ En , and let Let Q j,k be the locally free sheaf on E n pj, kq defined by Let ι j,k : E n pj, kq ãÑ X n pSq ˚be the inclusion map.We have an exact sequence It follows that ρ ˚chpF rns `q " τ 1 chpF q ¨. . .¨τ n chpF q ´ÿ 1ďjăkďn chpι j,k,˚p Q j,k qq. (3.2.9) Since χpS, End 0 pF qq " 2, the Hirzebruch-Riemann-Roch Theorem gives that 2r 0 ch 2 pF q " ch 1 pF q 2 ´2pr 2 0 ´1qη. (3.2.10) Using the above equality, one gets that modulo H 6 pX n pSq ˚; Qq we have The equalities in (3.2.2), (3.2.3) and (3.2.4) follow at once from the equalities in (3.2.9), (3.2.11) and the above equality.
3.3.Deformations of pS rns , PpF rns `qq.Let F be a vector bundle on S, and let f : PpF rns `q Ñ S rns be the structure map.We let DefpPpF rns `q, f, S rns q be the deformation functor of the map f , see Definition 8.2.7 in [Man22].
Proposition 3.6.Suppose that the K3 surface S is projective and that F is a spherical vector bundle on S. Then the natural map DefpPpF rns `q, f, S rns q Ñ DefpS rns q is smooth.
Proof.The result follows from a Theorem of Horikawa.In fact let X :" PpF rns `q, Y :" S rns , and consider the exact sequence of locally free sheaves on X By [Hor74, Theorem 6.1] (see also Corollary 8.2.14 in [Man22]) it suffices to prove that the map H 1 pX, Θ X q Ñ H 1 pX, f ˚ΘY q is surjective and the map H 2 pX, Θ X q Ñ H 2 pX, f ˚ΘY q is injective.By the exact sequence in (3.3.1) it suffices to show that H 2 pX, Θ X{Y q " 0. By the local-to-global spectral sequence abutting to H 2 pX, Θ X{Y q we are done if we prove that We have H p pS rns , End 0 F rns `q " 0 @p, (3.3.4) and this finishes the proof.
Corollary 3.7.Suppose that the K3 surface S is projective and that F is a spherical vector bundle on S. If n ď 3 then ∆pF rns `q belongs to the saturation of c 2 pS rns q, if n ě 4 then ∆pF rns `q belongs to the saturation of the span of c 2 pS rns q and q _ .Proof.Let X F ÝÑ Y G ÝÑ T be representative of DefpPpF rns `q, f, S rns q.Thus both F and G are proper holomorphic maps of analytic spaces, there exists 0 P T such that F ´1pG ´1p0qq Ñ G ´1p0q is identified with f : PpF rns `q Ñ S rns , and every (small) deformation of f is identified with F ´1pG ´1ptqq Ñ G ´1ptq for some t P T (close to 0).For t P T (close to 0) the map F ´1pG ´1ptqq Ñ G ´1ptq is a P r´1 fibration, where r " rkpF rns `q, and hence the push-forward F ˚pΘ X {Y q is a vector bundle on Y (of rank r 2 ´1).By Proposition 3.6 the family G : Y Ñ T is versal at t " 0, and hence the characteristic class c 2 pF ˚pΘ X0{Y0 q (here X 0 " F ´1pG ´1p0qq and Y 0 " G ´1p0q) remains of type p2, 2q for all small deformation of Y 0 " S rns .If n ď 3 it follows that c 2 pF ˚pΘ X0{Y0 q belongs to the saturation of c 2 pS rns q, and if n ě 4 it follows that c 2 pF ˚pΘ X0{Y0 q belongs to the saturation of the span of c 2 pS rns q and q _ , see [Zha15].We are done because c 2 pF ˚pΘ X0{Y0 qq " c 2 pEnd 0 F rns `q " ´∆pF rns `q.
Remark 3.8.Let DefpS rns , det F rns `qq be the deformation functor of the couple pS rns , det F rns `qq.The natural map DefpF rns `q Ñ DefpS rns , c 1 pF rns `qq is an isomorphism, by the Artamkin-Mukai Theorem [Muk84, Art88] (see also [IM19]) and by the vanishing in (3.3.4).Hence F rns `extends (uniquely) to a vector bundle on any small deformation of S rns keeping c 1 pF rns `q of type p1, 1q.
3.4.Proof of Proposition 3.2.The equality in (3.1.2) follows at once from the equality in (3.2.2).Similarly, the equality in (3.1.3)follows at once from the equality in (3.2.3), because the restriction map H 2 pS rns ; Zq Ñ H 2 pρpX n pSq ˚qq is an isomorphism (the complement of ρpX n pSq ˚q in S rns has codimension greater than one).Lastly we prove the equality in (3.1.4).Proposition 3.5 and a straightforward computation give that ρ ˚∆pF rns `q|XnpSq˚" ρ ˚ˆr 2n´2 0 pr 2 0 ´1q 12 c 2 ´Srns ¯˙|XnpSq˚.
If n ď 3 then by Proposition 3.7 (note: we may assume that S is projective) ∆pF rns `q is a (possibly rational) multiple of c 2 `Srns ˘.Since the restriction of ρ ˚c2 `Srns ˘to X n pSq ˚is non zero (by the equality in (2.1.4)and Lemma 2.3), the equality in (3.1.4)follows.If n ě 4 then by Proposition 3.7 ∆pF rns `q is a linear combination (possibly with rational coefficients) of c 2 `Srns ˘and q _ , and the equality follows from Proposition 2.2.
4. Basic modular vector bundles on S rns for S an elliptic K3 surface 4.1.Contents of the section.We show that by starting from slope stable spherical vector bundles F on an elliptic surface S we can produce vector bundles F rns òn S rns with rank and first two Chern classes covering all the cases in Theorem 1.1.We also study the restriction of such an F rns `to fibers of the Lagrangian fibration S rns Ñ P n .4.2.Basic modular sheaves with the required topology.The present section contains analogues of the results in Sections 6.2 and 6.3 of [O'G22].Let S be a K3 surface with an elliptic fibration S Ñ P 1 ; we let C be a fiber of the elliptic fibration.The claim below follows from surjectivity of the period map for K3 surfaces.
Claim 4.1.Let m 0 , d 0 be positive natural numbers.There exist K3 surfaces S with an elliptic fibration S Ñ P 1 such that NSpSq " ZrDs ' ZrCs, D ¨D " 2m 0 , D ¨C " d 0 . (4.2.1) The following result is a (slight) extension of Proposition 6.2 in [O'G22] (and is more or less well known by experts).
Proposition 4.2.Let m 0 , r 0 , s 0 P N `and let t, d 0 P Z. Suppose that (a) t 2 m 0 " r 0 s 0 ´1, (b) d 0 is coprime to r 0 , (c) we have Let S be an elliptic K3 surface as in Claim 4.1.Then there exists a vector bundle F on S such that the following hold: (1) vpF q " pr 0 , tD, s 0 q, (2) χpS, EndpF qq " 2, (3) F is L slope-stable for any polarization L of S, (4) and the restriction of F to every fiber of the elliptic fibration S Ñ P 1 is slope-stable.
(Notice that every fiber is irreducible by our assumptions on NSpSq, hence slopestability of a sheaf on a fiber is well defined, i.e. independent of the choice of a polarization.) Proof.One proceeds literally as in the proof of Proposition 6.2 in [O'G22].
Claim 4.3.Keep notation and hypotheses as above, in particular F is the vector bundle on S such that the equation in (4.2.5) and Items (2)-( 4) of Proposition 4.2 hold.Let E :" F rns `.Then we have rpE q " r n 0 , c 1 pE q " g ¨rn´1 0 i h, ∆pE q " r 2n´2 0 pr 2 0 ´1q 12 c 2 pS rns q, (4.2.6) where h P NSpS rns q is primitive, qphq " e and divphq " il.Definition 4.4.Let S Ñ P 1 be an elliptically fibered K3 surface.If x P P 1 we let C x be the (scheme theoretic) elliptic fiber over x.Let B " tb 1 , . . ., b m u Ă P 1 be the set of x such that C x is singular.Then B is not empty (generically m " 24).The Lagrangian fibration associated to S Ñ P 1 is the map π : S rns Ñ P n given by the composition S rns Ñ S pnq Ñ pP 1 q pnq -P n .(4.3.1)

Proof
We record a few facts regarding the (scheme theoretic) fibers of π.Let x 1 , . . ., x n P P 1 be pairwise distinct: then Next we describe the discriminant locus of π : S rns Ñ pP 1 q pnq , i.e. the subset D Ă pP 1 q pnq parametrizing cycles x 1 `. . .`xn such that π ´1px 1 `. . .`xn q is singular.For b j P B let Dpb j q Ă pP 1 q pnq be the irreducible divisor parametrizing cycles x 1 `. . .`xn such that x i " b j for some i P t1, . . ., nu.Let T :" t ÿ i m i x i P pP 1 q pnq | m i ě 2 for some i P t1, . . ., nuu. (4.3.3) Note that the fibers of π over points of T are reducible and non reduced.
Proposition 4.5.The irreducible decomposition of the discriminant locus D of π : S rns Ñ P n is given by Below is the main result of the present subsection.
Proposition 4.6.Let S be a K3 surface with an elliptic fibration S Ñ P 1 as in Claim 4.1, and let π : S rns Ñ P n be the associated Lagrangian fibration.Let F be a vector bundle on S as in Proposition 4.2.Then the following hold: (a) If x 1 , . . ., x n P P 1 are pairwise distinct, then the restriction of F rns `to π ´1px 1 `. . .`xn q is slope stable for any product polarization (this makes sense by the isomorphism in (4.3.2)).(b) Let U Ă pP 1 q pnq be the open subset parametrizing cycles x 1 `. . .`xn such that the restriction of F rns `to π ´1px 1 `. . .`xn q is a simple sheaf.The complement of U has codimension at least two.
Before proving Proposition 4.6, we notice that Proposition 6.10 in [O'G22] holds for products of projective varieties of arbitrary dimension.
Lemma 4.7.For i P t1, 2u let pX i , L i q be an irreducible polarized projective variety of dimension d i , and let V i be a slope stable vector bundle on Proof.Suppose that there exists an injection α : The open subset U Ă X 1 ˆX2 of points p at which α is an injection of vector bundles (i.e. the stalk of E at p is free and α defines an injection of the fiber of E at p to the fiber of V 1 b V 2 at p) has complement of codimension at least 2. Let p " px 1 , x 2 q P U .The restrictions of α to tx 1 uˆX 2 and to X 1 ˆtx 2 u are generically injective maps of vector bundles.Let We have µ L pE q " A 1 µ L1 pE |X1ˆtx2u q `A2 µ L2 pE |tx1uˆX2 q, (4.3.6) and µ L pV 1 b V 2 q " A 1 µ L1 pV 1 q `A2 µ L2 pV 2 q. (4.3.7)Since the restrictions of V 1 b V 2 to X 1 ˆtx 2 u and to tx 1 u ˆX2 are isomorphic to the polystable vector bundles V 1 b C C rpV2q and V 1 b C C rpV1q respectively, it follows from (4.3.5),(4.3.6) and (4.3.7) that µpE |X1ˆtx2u q " µpV 1 q and µpE |tx1uˆX2 q " µpV 2 q.In turn, these equalities give that there exist vector subspaces 0 This is a contradiction.
Proof of Proposition 4.6.(a): Follows from the stability of the restriction of F to any elliptic fiber of S Ñ P 1 (Proposition 4.2), and Lemma 4.7.(b): Let V Ă pP 1 q pnq be the open subset parametrizing cycles Γ :" d 1 x 1 `. ..`d m x m such that d j ď 2 for all j P t1, . . ., mu, and C xj is smooth if d j " 2. If Γ is such a cycle then the restriction of F rns `to π ´1pΓq is a simple sheaf.In fact π ´1pΓq is a product of schemes C 1 ˆ. . .ˆCm , where C j " C xj if d j " 1, while if d j " 2 then C j is identified with the scheme theoretic fiber over 2x j P pP 1 q p2q of the Lagrangian fibration S r2s Ñ pP 1 q p2q .Moreover simple.Since the complement of V in pP 1 q pnq is a (closed) subset of codimension at least two, this proves Item (b).
Remark 4.8.By Item (a) of Proposition 4.6 the restriction of F rns `to a (singular) Lagrangian fiber X t parametrized by a general point t P Dpb j q (notation as in (4.3.4)) is slope stable for any product polarization.
The following remarks place Item (a) of Proposition 4.6 in the context of known results.
Remark 4.9.Let X Ñ P n be a Lagrangian fibration of a HK manifold.For t P P n let X t :" π ´1ptq be the schematic fiber of X Ñ P n over t.If X t is smooth there exists an ample primitive class θ t P H 1,1 Z pX t q such that the image of the restriction map H 2 pX; Zq Ñ H 2 pX t ; Zq is contained in Zθ t , see [Wie16].If F is a sheaf on X t slope-(semi)stability of F will always mean θ t slope-(semi)stability.
Remark 4.10.Let X Ñ P n be a Lagrangian fibration of a HK manifold of type K3 rns , and let X t be a smooth Lagrangian fiber.Then the primitive ample class θ t P H 1,1 Z pX t q is a principal polarization of X t , see [Wie16].If S rns Ñ P n is the Lagrangian fibration in (4.3.1), and π ´1px 1 `. . .`xn q -C x1 ˆ. . .C xn is a smooth Lagrangian fiber, then θ x1`...`xn is the product principal polarization.5. Proof of Theorem 1.1 and Proposition 1.2 5.1.Contents of the section.In the present section we prove the following two statements.
Proposition 5.1.Let n, r 0 , g, l, e, i be as in Theorem 1.1.There exists an irreducible component K il e p2nq good of K il e p2nq such that for a general polarized HK variety pX, hq parametrized by K il e p2nq good there exists an h slope-stable vector bundle E on X such that the equalities in (1.2.6) hold and moreover H p pX, End 0 pE qq " 0 for all p. Proposition 5.2.Let n, r 0 , g, l, e, i be as in Theorem 1.1.If rpX, hqs P K il e p2nq good is a general point, then there exists a unique h slope-stable vector bundle E on X such that the equalities in (1.2.6) hold.
The same exact argument given in the "Proof of Theorem 1.4" on p. 30 of [O'G22] shows that Theorem 1.1 follows from Propositions 5.1 and 5.2.
In Subsections 5.2 and 5.3 we prove results that are used in the proof of Propositions 5.1 and 5.2.Proposition 5.1 is proved in Subsection 5.4.Subsections 5.5 and 5.6 contain results that are used in the proof of Proposition 5.2.Propositions 5.2 and 1.2 are proved in Subsections 5.7 and 5.8 respectively.

The relevant component of K il
e p2nq, and Noether-Lefschetz divisors.Let S be an elliptic K3 surface as in Claim 4.3, and let C, D be divisor classes generating NSpSq as in loc.cit.Let X 0 " S rns , and let X 0 π0 ÝÑ pP 1 q pnq " P n (5.2.1) be the Lagrangian fibration associated to the elliptic fibration of S, see (4.3.1).Let (5.2.2) Definition 5.3.Let K il e p2nq good Ă K il e p2nq be the irreducible component containing rpS rns , h 0 qs.
(5.2.3) (The last equality follows from (4.2.3).)Let L 0 , F 0 be the line bundles on X 0 such that c 1 pL 0 q " h 0 and c 1 pF 0 q " f 0 .
Definition 5.4.Let ϕ : X Ñ B be an (analytic) contractible representative of the functor DefpX 0 , L 0 , F 0 q.Let 0 P B be the base point, so that X 0 " ϕ ´1p0q -S rns .For b P B we let X b :" ϕ ´1pbq, and we let L b , F b be the line bundles on X b which are deformations of L 0 , F 0 respectively.We let h b :" c 1 pL b q and f b :" c 1 pF b q.
Our first observation is that if d 0 is large enough, then h b is ample for a general b P B. Before proving this, we recall the following elementary result.
(5.2.4) Proposition 5.6.Keep notation as above, and assume that ild 0 ą pn ´1q 2 pn `3qpe `1q. (5.2.5) Proof.Let b P B be a very general point, in the sense that NSpX b q " xh b , f b y.By the inequality in (5.2.5) and Lemma 5.5 there are no ξ P NSpX b q such that ´2pn ´1q 2 pn `3q ď qpξq ă 0. By [Mon15, Corollary 2.7] it follows that the ample cone of X b is equal to the intersection of NSpX b q and the positive cone (if R is the integral generator of an extremal ray then, viewed by duality as an element of H 2 pK3 rns , Qq, the multiple 2pn ´1qR is integral because the divisibility of any element of H 2 pX b , Zq is a divisor of 2n ´2).Hence either h b or ´hb is ample.By considering the limit case b " 0 we get that h b is ample.This proves that h b is ample for b very general.Since h b is ample for b in the complement of an anaytic subset of B, we are done.
Assume that the inequality in (5.2.5) holds.By Proposition 5.6 we have the moduli map Note that the image of the above period map is contained in a unique (irreducible) Noether-Lefschetz divisor in K il e p2nq good .Definition 5.7.If the inequality in (5.2.5) holds, we let NLpd 0 q Ă K il e p2nq good be the unique irreducible Noether-Lefschetz divisor containing the image of the moduli map in (5.2.6).
Remark 5.8.Let rpX, hqs P NLpd 0 q be a general point.Then there exists a well defined rank two subspace V Ă NSpXq such that V " xh, f y, where qph, f q " ild 0 , qpf q " 0.
(5.2.7) (For rpX, hqs in a proper Zariski closed subset of NLpd 0 q there might be more than one such rank two subspace).For almost all choices (of n, r 0 , g, l, e, i and d 0 ) there is a unique class f P V such that V " xh, f y and the equalities in (5.2.7) hold, while for special choices there are two such classes.If monodromy exchanges these two isotropic classes there is no intrinsic way of distinguishing them.Abusing language we will speak of "the class f ".If pX, hq " pX 0 , h 0 q (recall that X 0 " S rns where S is our elliptic K3 surface) then f 0 " c 1 pπ 0 O P n p1qq, where π 0 is the Lagrangian fibration given by in (5.2.1).By [Mat17, Theorem 1.2] it follows that if rpX, hqs P NLpd 0 q is a general point then there exists a Lagrangian fibration π X : X Ñ P n such that f " c 1 pπ X O P n p1qq.
Proposition 5.9.Keep notation as above, and assume that the inequality in (5.2.5) holds, and that d 0 is coprime to r 0 .Let NLpd 0 q Ă K il e p2nq good be the Noether-Lefschetz divisor of Definition 5.7.There exist an open dense NLpd 0 q ˚Ă NLpd 0 q and for each rpX, hqs P NLpd 0 q ˚a vector bundle E X on X such that H p pX, End 0 pE X qq " 0 for all p, and the restriction of E X to a general smooth fiber of the Lagrangian fibration X Ñ P n (see Remark 5.8) is slope stable.
Proof.Let E 0 :" F rns `be the vector bundle on X 0 of Claim 4.3.Note that c 1 pE 0 q " h 0 by loc.cit.If B is small enough, then by Remark 3.8 the vector bundle E 0 on X 0 deforms uniquely to a vector bundle E b on X b , and hence we get a vector bundle E X on X for rpX, hqs in a dense open subset U Ă NLpd 0 q.The equations in (5.2.8) hold by the equations in (4.2.6).Since H p pX, End 0 pE 0 qq " 0 for all p (see (3.3.4)) it follows from upper semicontinuity of the dimension of cohomology sheaves that for rpX, hqs in a smaller dense open subset U 1 Ă U we have H p pX, End 0 pE X qq " 0 for all p. Lastly, it follows from Item (a) of Proposition and Remark 4.10 that for rpX, hqs in a smaller dense open subset U 2 Ă U 1 the restriction of E X to a general fiber of the Lagrangian fibration X Ñ P n (see Remark 5.8) is slope stable.We set NLpd 0 q ˚:" U 2 .5.3.Suitable polarizations.We recall that if h is a-suitable (see [O'G22, Definition 3.5]) and E is a vector bundle on X with apE q ď a (see (3.1.1)loc.cit.for the definition of apE q), then slope stability of the restriction of E to a general Lagrangian fiber (there is a canonical choice of polarization of any smooth Lagrangian fiber, see Remark 4.9) implies slope stability of E , and the following weak converse holds: slope stability of E implies that the restriction of E to a general Lagrangian fiber is slope semistable.
Lemma 5.10.Keep assumptions and notation as above, and let a ą 0. Suppose that ild 0 ą ape `1q. (5.3.1) Let rpX, hqs P NLpd 0 q be a general point, and let f P V Ă NSpXq be as in Remark 5.8.Then there does not exist ξ P NSpXq such that ´a ď q X pξq ă 0, q X pξ, hq ą 0, q X pξ, f q ą 0. (5.3.2) Proof.Let xh, f y " V Ă NSpXq be as in Remark 5.8.Applying Lemma 5.5 to Λ :" V , α " f and β " h one gets that there are no ξ P V such that ´a ď q X pξq ă 0.
In particular if NSpXq " xh, f y (as is the case for very general rpX, hqs P NLpd 0 q), then there is no ξ P V such that the inequalities in (5.3.2) hold.It follows that the set of rpX, hqs P NLpd 0 q for which there exists ξ P NSpXq such that the inequalities in (5.3.2) hold is the intersection of NLpd 0 q with a finite union of Noether-Lefschetz divisors in K il e p2nq, each of which does not contain NLpd 0 q.In fact suppose that the inequalities in (5.3.2) hold.Let D be the (finite) index of xh, f y ' pxh, f y K X NSpXqq in NSpXq.It is crucial to note that D has an upper bound which only depends on the discriminant of the restrictions of q X to V (i.e.´pild 0 q 2 ) and to V K , and the latter has an upper bound depending only on the discriminant of V and the discriminant of q X (i.e.2n ´2).Then Moreover we have just proved that ξ 2 is non zero.Since the restriction of q X to h K X NSpXq is negative definite we get that qpξ 2 q ă 0. We also have qpξ 1 q ă 0 by the last two inequalities in (5.3.2).
Hence by the first inequality in (5.3.2) there exists a positive M independent of pX, hq such that ´M ď qpξ 2 q ă 0 (here it is crucial that D has an upper bound which only depends on pild 0 q 2 ) and 2n´2).Hence the moduli point of pX, hq belongs to the intersection of NLpd 0 q with a finite union of Noether-Lefschetz divisors in K il e p2nq, and none of them contains NLpd 0 q because if ρpXq " 2 then rpX, hqs is not contained in any of these Noether-Lefschetz divisors.
Proposition 5.11.Keep notation as above, and assume that ild 0 ą r 4n´2 0 pr 2 0 ´1qpn `3qpe `1q 8 . (5.3.4) Let rpX, hqs P NLpd 0 q (the inequality in (5.3.4) implies that the inequality in (5.2.5) holds, and hence the Noether-Lefschetz divisor NLpd 0 q Ă K il e p2nq good is defined).If E is a vector bundle on X such that the equalities in (1.2.6) hold, then h is apE q-suitable (relative to the associated Lagrangian fibration π : X Ñ P n ).
The inequality in (5.3.4) and Lemma 5.10 give that h is apE q-suitable.
5.4.Proof of Proposition 5.1.Keeping notation and assumptions as above.Assume in addition that d 0 is coprime to r 0 and that the inequality in (5.3.4) holds.(Note that the set of such d 0 is infinite.)Let rpX, hqs P NLpd 0 q ˚, and let E X be a vector bundle on X as in Proposition 5.9.By Proposition 5.11 the polarization h is apE X q-suitable relative to the Lagrangian fibration π X : X Ñ P n .By Proposition 5.9 the restriction of E X to a general fiber of π X : X Ñ P n is slope stable.Since h is apE X q-suitable, the vector bundle E X is h slope stable by [O'G22, Proposition 3.6].We have H p pX, End 0 pE X qq " 0 for all p, and hence E X extends (uniquely) to all small deformations of pX, det E X q by Remark 3.8.Since c 1 pE X q is a multiple of h, we get that E X extends to a vector bundle E 1 on a general deformation pX 1 , h 1 q of pX, hq.By openness of slope stability E 1 is slope stable and by upper semicontinuity of cohomology dimension H p pX, End 0 pE 1 qq " 0 for all p.
5.5.Tate-Shafarevich twists.A basic example of Lagrangian fibration is obtained as follows.Let pS, h S q be a polarized K3 surface of genus n.Let J pSq be the moduli space of rank 0 pure O S p1q semistable sheaves ξ with χpξq " 1 ´n, i.e. sheaves with Mukai vector p0, h S , 1 ´nq.The generic point of J pSq is represented by i ˚L , where i : C ãÑ S is the inclusion of a smooth C P O S p1q, and L is a line bundle of degree 0. Suppose that all divisors in the complete linear system |O S p1q| are irreducible and reduced.Then every semistable sheaf parametrized by J pSq is stable, and J pSq is a HK projective variety of Type K3 rns .Moreover the support map J pSq Ñ |O S p1q| -P n is a Lagrangian fibration.Let NLpd 0 q Ă K il e p2nq good be the Noether-Lefschetz divisor of Definition 5.7, and let rpX, hqs P NLpd 0 q be a general point.Then the associated Lagrangian fibration π : X Ñ P n is related to a (general) moduli space J pSq via a Tate-Shafarevich twist.In order to be more precise, we recall a result of Markman.First, if rpX, hqs P NLpd 0 q is a general point, then there is an associated polarized K3 surface pS, Dq of genus n, and moreover pS, Dq is a general such polarized surface -see [Mar14, Subsection 4.1].
Proposition 5.12.Keep notation as above, and assume that the inequality in (5.2.5) holds.Let rpX, hqs P NLpd 0 q be a general point, let X Ñ P n be the associated Lagrangian fibration, and let pS, Dq be the associated polarized K3 surface (which is a general polarized K3 surface of genus n).Then X Ñ P n is isomorphic to a Tate-Shafarevich twist of J pSq Ñ |D| via an identification P n " ÝÑ |D|.
Proof.Suppose first that ρpXq " 2.Then, as shown in the proof of Proposition 5.6, the ample cone of X is equal to the positive cone (because of the inequality in (5.2.5)), and hence every bimeromorphic map X X 1 , where X 1 is a HK, is actually an isomorphism.It follows that X is isomorphic to a Tate-Shafarevich twist of J pSq Ñ |D| by Theorem 7.13 in [Mar14].The result follows from this because the locus in NLpd 0 q parametrizing pX, hq such that ρpXq " 2 is dense.Let Pic 0 pX{P n q be the relative Picard scheme of the Lagrangian fibration X Ñ P n (notice that all fibers of X Ñ P n are irreducible by Proposition 5.12).Let U Ă P n be the open dense set of regular values of X Ñ P n .If t P U , the fiber of Pic 0 pX{P n q Ñ P n over t is an abelian variety A t (of dimension n) and the fundamental group π 1 pU, tq acts by monodromy on the subgroup A t,tors of torsion points.
Corollary 5.13.Keep hypotheses and notation as above, and suppose that V Ă A t rr n 0 s is a coset (of a subgroup of A t rr n 0 s) of cardinality r 2n 0 invariant under the action of monodromy.Then V " A t rr 0 s.
Proof.Let pS, Dq be the polarized K3 surface of genus n associated to X following Markman.Let J pSq 0 Ă J pSq be the open dense subset of smooth points with G slope-stable.Arguing as in the proof of [O'G22, Proposition 4.4] one proves that c 1 pG q " aθ, c 1 pH q " bθ, ∆pG q ¨θn´2 " ∆pH q ¨θn´2 " 0. (5.6.3) Let us prove that G and H are locally free.Let r :" rpF q and let m r : A Ñ A be the multiplication by r map.Let L be a line bundle on A such that c 1 pL q " ´raθ, and let E :" m r pF q b L .Then E is slope semistable (because F is) and c 1 pE q " 0, ∆pE q ¨θn´2 " 0.
By [Sim92, Theorem 2] it follows that every quotient of the (slope) Jordan-Hölder filtration of E is locally free.Since m r pG q b L is a polystable subsheaf of E we get that it is locally free.Thus m r pG q is locally free and hence also G is locally free.By the equalities in (5.6.3)we may iterate this argument to show that also H is locally free.
is a (slope) Jordan-Hölder filtration of F , then each quotient Q i :" G i {G i´1 is a slope stable locally free sheaf with c 1 pQ i q rkpQ i q " c 1 pF q rpF q , ∆pQ i q ¨θn´2 " 0. (5.6.4) Hence each Q i is simple semi-homogeneous by [O'G22, Proposition A.2], and therefore by [O'G22, Proposition A.3] (see also [Muk78,Remark 7.13]) there exist coprime integers r i , b i , with r i ą 0, such that rpQ i q " r n i and c 1 pQ i q " r n´1 i b i θ.Let i, j P t1, . . ., mu; since the slopes of Q i and Q j are equal, we get that b i r j " b j r i .It follows that r i " r j and b i " b j because gcdtr i , b i u " gcdtr j , b j u " 1.Thus rpF q " mr n 0 and c 1 pF q " mr n´1 0 b 0 θ where r 0 " r i and b 0 " b i for all i P t1, . . ., mu.We have m ě 2 because we assumed that F is strictly slope semistable.
Corollary 5.16.Let pA, θq be a principally polarized abelian variety of dimension n, and let F be a θ slope-semistable vector bundle on A such that ∆pF q ¨θn´2 " 0. If rpF q " r n 0 and c 1 pF q " r n´1 0 b 0 θ, where r 0 , b 0 are coprime integers, then F is θ slope-stable.
Proof.By contradiction.Suppose that F is not θ slope-stable.By Proposition 5.15 we may write rpF q " s n 0 m, c 1 pF q " s n´1 0 c 0 mθ where s 0 , m, c 0 are integers (with s 0 , m ą 0), s 0 , c 0 are coprime and m ą 1.It follows that s 0 b 0 " c 0 r 0 .Since gcdtr 0 , b 0 u " 1 and gcdts 0 , c 0 u " 1, we get that r 0 " s 0 and hence m " 1.This is a contradiction.
Proof of Proposition 5.14.Let ϕ : X Ñ B be as in Definition 5.4.Recall that X 0 " ϕ ´1p0q -S rns , where S is an elliptic K3 surface as in Claim 4.3.Let E 0 :" F rns `be the vector bundle on X 0 of Claim 4.3.If B is small enough, the vector bundle E 0 on X 0 deforms uniquely to a vector bundle E b on X b , hence we get a vector bundle E X on X for rpX, hqs in a dense open subset U Ă NLpd 0 q.Moreover E X is h slope stable and Items (1) and (2) of Proposition 5.14 hold, see the proof of Proposition 5.1.For rpX, hqs P NLpd 0 q ˚, where NLpd 0 q ˚Ă U is an open dense subset, the restriction of E X to a general smooth fiber of the Lagrangian fibration X Ñ P n is slope stable, see Proposition 5.1.
Let V 0 Ă P n be the set of t such that the restriction of E 0 to the fiber over t of the Lagrangian fibration S rns Ñ P n is simple.By Item (b) of Proposition 4.6, V 0 is an open subset whose complement has codimension (in P n ) at least 2. It follows that there exists an open dense subset NLpd 0 q s Ă NLpd 0 q ˚with the following property: if rpX, hqs P NLpd 0 q s the subset V X Ă P n parametrizing fibers X t of the Lagrangian fibration X Ñ P n such that the restriction of E X to X t is simple has complement of codimension (in P n ) at least 2.
Let rpX, hqs P NLpd 0 q s .We claim that if t P V X and X t is smooth, then the restriction E X|Xt is slope stable.In order to prove this, we start by noting that for any Lagrangian (scheme-theoretic) fiber X t we have ż rXts ∆pE X|Xt q ¨`h |Xt ˘n´2 " 0. (5.6.5) In fact the above equality is an easy consequence of the modularity of E X , see [O'G22, Lemma 2.5].Let X t be a general smooth Lagrangian fiber.By Proposition 5.9 the restriction of E X to X t is slope stable, hence E X|Xt is semi-homogeneous because of the equality in (5.6.5),see [O'G22, Lemma 2.5].It follows that if t 0 P V X and X t0 is smooth, then E X|Xt 0 is (simple) semi-homogeneous.To prove this, we introduce some notation.For t P P n such that X t is smooth let Φ 0 pE X|Xt q :" tpx, rξsq P X t ˆp X t | D T x pE X|Xt q " ÝÑ pE X|Xt q b ξu, Ψ 0 pE X|Xt q :" tpx, rξsq P X t ˆp X t | HompT x pE X|Xt q, pE X|Xt q b ξq " 0u, where T x : X t Ñ X t is translation by x P X t (locally in t we may assume that X t is a family of abelian varieties rather than torsors over abelian varieties).Recall that E X|Xt is semi-homogeneous if and only if Φ 0 pE X|Xt q has dimension at least n, and that if that is the case then the group Φ 0 pE X|Xt q has pure dimension n.Let X t be a general smooth Lagrangian fiber, so that E X|Xt is slope stable and semi-homogeneous.Then Φ 0 pE X|Xt q has pure dimension n, and moreover (by slope stability) Φ 0 pE X|Xt q " Ψ 0 pE X|Xt q.By upper semicontinuity of cohomology dimension it follows that every irreducible component of Ψ 0 pE X|Xt 0 q has dimension at least n.Now p0, rO Xt 0 sq P Ψ 0 pE X|Xt 0 q and, since E X|Xt 0 is simple, every non zero homomorphism E X|Xt 0 Ñ E X|Xt 0 is an isomorphism.It follows that Φ 0 pE X|Xt 0 q has dimension at least n, and hence E X|Xt 0 is semi-homogeneous.By [Muk78, Proposition 6.13] we get that E X|Xt 0 is slope semistable (actually it is Gieseker stable, see Proposition 6.16 loc.cit.).Lastly we prove that E X|Xt 0 is slope-stable.Let θ t0 be the principal polarization of X t0 , see Remark 4.9.Since q Xt 0 ph, f q " ild 0 (see (5.2.7)), we have h |Xt 0 " ild 0 θ t0 .Hence rpE X|Xt 0 q " r n 0 , c 1 pE X|Xt 0 q " r n´1 0 gld 0 θ t0 .
(5.6.6)By hypothesis g, l and d 0 are coprime to r 0 .By Corollary 5.16 we get that E X|Xt 0 is slope-stable.We finish the proof by showing that if rpX, hqs P NLpd 0 q s is general then the restriction of E X to a general singular Lagrangian fiber is slope-stable.Since rpX, hqs P NLpd 0 q s is general the discriminant divisor D X Ă P n parametrizing singular Lagrangian fibers of X Ñ P n is the dual of an embedded K3 surface S Ă pP n q _ .In fact this holds by Proposition 5.12.Hence D X is an irreducible divisor.Thus it suffices to prove that there exist t P D X such that E X|Xt is slope stable (for the restriction of h to X t ).This follows from Remark 4.8 and openness of slope stability.5.7.Proof of Proposition 5.2.The key result is the following.
Proposition 5.17.Keep notation as in Subsection 5.2.Assume in addition that d 0 is coprime to r 0 , and that the inequality in (5.3.4) holds.Let rpX, hqs P NLpd 0 q ˚be a general point.Then (up to isomorphism) there exists one and only one h slope stable vector bundle E on X such that the equalities in (1.2.6) hold.the complement of U : X in P n has codimension at least 2).Then Y :" π ´1pT q is a smooth projective (integral) variety of dimension n `1 and the sheaves F :" E X|Y and G :" E |Y satisfy the hypotheses of [O'G22, Lemma 7.5], and hence the isomorphism in (5.7.4) holds by the quoted lemma.
Since E X|Xt and E |Xt are simple for all t P U : X , and since c 1 pE X q " c 1 pE q, it follows that the restrictions of E X and E to π ´1pU : X q are isomorphic, see the proof of Proposition 7.4 in [O'G22], in particular the beginning of the proof of Lemma 7.5.The complement of π ´1pU : X q in X has codimension at least 2 because π is equidimensional, and hence the isomorphism E X|π ´1pU : X q " ÝÑ E |π ´1pU : X q extends to an isomorphism E X " ÝÑ E .
We are ready to prove Proposition 5.2.Let n, r 0 , g, l, e be as in Theorem 1.1.Since the result is trivially true fo r 0 " 1 we assume that r 0 ě 2. Let X Ñ T il e p2nq be a complete family of polarized HK varieties of Type K3 rns parametrized by K il e p2nq good .Since K il e p2nq good is irreducible we may, and will, assume that T il e p2nq is irreducible.For t P T il e p2nq we let pX t , h t q be the corresponding polarized HK of Type K3 rns .Let m : T il e p2nq Ñ K il e p2nq good be the moduli map sending t to rpX t , h t qs.
By Gieseker and Maruyama there exists a relative moduli space M pr 0 , gq f ÝÑ T il e p2nq, (5.7.5) such that for every t P T il e p2nq the (scheme theoretic) fiber f ´1ptq is isomorphic to the (coarse) moduli space of h t slope-stable vector bundles E on X t such that (1.2.6) holds.Moreover the morphism f is of finite type by Maruyama [Mar81], and hence f pM pr 0 , gqq is a constructible subset of T il e p2nq.Let dpr 0 , e, lq be the right hand side of the inequality in (5.3.4).For t in a dense subset of Ť dądpr0,e,lq m ´1pNLpdq good q the preimage f ´1ptq is a singleton by Proposition 5.17.Since Ť dądpr0,e,lq m ´1pNLpdq good q is Zariski dense in T il e p2nq (it is the union of an infinite collection of pairwsie distinct divisors), and since f pM pr 0 , gqq is a constructible subset of T il e p2nq, it follows that for general t P T il e p2nq the fiber f ´1ptq is a singleton.
Let rE s be the unique point of f ´1ptq for t a generic point of m ´1pNLpdq good q, where d ą dpr 0 , e, lq.Then H p pX t , End 0 pE qq " 0 by Proposition 5.14.Hence the last sentence of Theorem 1.1 follows from upper semicontinuity of cohomology.5.8.Proof of Proposition 1.2.Since the natural morphism DefpX, F q Ñ DefpX, hq is surjective, for d 0 " 0 there exist extensions of F to polarized HK varieties pY, hq of type K3 rns with a Lagrangian fibration π : Y Ñ P n such that q Y ph, f q " d 0 ¨divphq.
(5.8.1) (As usual f :" c 1 pπ ˚OP n p1qq.)Let X t be a smooth (Lagrangian) fiber of π, and let θ t be the principal polarization of X t induced by the Lagrangian fibration (see Remark 4.10).We claim that h |Yt " d 0 ¨divphq θ t . (5.8.2) In fact the above equality follows from the equalities ż Yt ph |Yt q n " ż X h n ¨f n " n! q X ph, f q n " n! pd 0 ¨divphqq n .
By the equality in (5.8.2) we get that c 1 pE |Y t q " a ¨d0 ¨divphq θ t .