The nilpotent cone in the Mukai system of rank two and genus two

We study the nilpotent cone in the Mukai system of rank two and genus two. We compute the degrees and multiplicities of its irreducible components and describe their cohomology classes.


Introduction
Let (S, H ) be a polarized K3 surface of genus g and fix two coprime integers n ≥ 1 and s. The moduli space M = M H (v) of H -Gieseker stable coherent sheaves with Mukai vector v = (0, n H, s) is a smooth Hyperkähler variety of dimension 2(n 2 (g − 1) + 1). A point in M corresponds to a stable sheaf E on S such that E is pure of dimension one with support in the linear system |n H|. Taking the (Fitting) support defines a Lagrangian fibration f : M −→ |n H| ∼ = P n 2 (g−1)+1 , [E] → Supp(E) known as the Mukai system [5,22]. Over a general point in |n H| which corresponds to a smooth curve D ⊂ S the fibers of f are abelian varieties isomorphic to Pic δ (D), where δ = s −n 2 (1− g). So, M can also be viewed as a relative compactified Jacobian associated to the universal curve C → |n H|.
The Mukai system is of special interest because of its relation to the classical and widely studied Hitchin system, see [14] for a survey. Let C be a smooth curve of B Isabell Hellmann igb@math.uni-bonn.de 1 Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany genus g. A Higgs bundle on C is a pair (E, φ) consisting of a vector bundle E on C and a morphism φ : E → E ⊗ ω C , called Higgs field. The moduli space M Higgs (n, d) of stable Higgs bundles of rank n and degree d is a smooth and quasi-projective symplectic variety. Sending (E, φ) to the coefficients of its characteristic polynomial χ(φ) defines a proper Lagrangian fibration It is equivariant with respect to the C * -action that is given by scaling the Higgs field on M Higgs (n, d) and by multiplication with t i in the corresponding summand on the base. As a corollary the topology of M Higgs (n, d) is controlled by the fiber over the origin. This fiber is called the nilpotent cone. In the late '80s Beauville, Narasimhan, and Ramanan discovered a beautiful interpretation of the space of Higgs bundles [4]. They showed that a Higgs bundle (E, φ) with characteristic polynomial s corresponds to a pure sheaf of rank one on a so called spectral curve C s ⊂ T * C inside the cotangent bundle of C. The curve C s is defined in terms of s = χ(φ) and is linearly equivalent to nC, the n-th order thickening of the zero section C ⊂ T * C. This idea was taken up by Donagi, Ein, and Lazarsfeld in [9]: The space M Higgs (n, d) appears as a moduli space of stable sheaves on T * C that are supported on curves in the linear system |nC|. Consequently, M Higgs (n, d) has a natural compactification M Higgs (n, d) given by a moduli space of sheaves on the projective surface S 0 = P(ω C ⊕ O C ) with respect to the polarization H 0 = O S 0 (C). The Hitchin map extends to M Higgs (n, d) → |n H 0 | ∼ = P(⊕ n i=0 H 0 (ω i C )) and is nothing but the support map; the nilpotent cone is the fiber over the point nC ∈ |n H 0 |. However, M Higgs (n, d) cannot admit a symplectic structure as it is covered by rational curves. At this point the Mukai system enters the picture. If S is a K3 surface that contains the curve C as a hyperplane section, one can degenerate (S, H ) to (S 0 , H 0 ) and consequently the Mukai system M H (v) → |n H| with v = (0, n H, d + n(1 − g)) degenerates to the compactified Hitchin system [9, §1]. From our perspective, this is a powerful approach to studying the Hitchin system. For instance, in a recent paper [7], de Cataldo, Maulik and Shen prove the P=W conjecture for g = 2 by means of the corresponding specialization map on cohomology.
In this note, we study the geometry of the nilpotent cone in the Mukai system, which is defined in parallel to the Hitchin system for some curve C ∈ |H |. Alternatively, one could say that we study the most singular fiber type, see (3.1). We will fix the invariants n = 2 and g = 2 and the Mukai vector v = (0, 2H , −1). In this case and if C is irreducible, the nilpotent cone has two irreducible components where the first component is isomorphic to the moduli space M C (2, 1) of stable vector bundles of rank two and degree one on C and the second component is the closure of N C \ N 0 . Both components are Lagrangian subvarieties of M = M H (v). If C is smooth, then N 0 is smooth and the singularities of N 1 are contained in N 0 ∩ N 1 (each understood with their reduced structure). However, both components occur with multiplicities.
Our first result is the computation of the multiplicities of the components as well as their degrees. Here, the degree is meant with respect to a naturally defined distinguished ample class u 1 ∈ H 2 (M, Z), see Definition 4.7. Theorem 1.1 Let C ∈ |H | be an irreducible curve. The degrees of the two components of the nilpotent cone N C are given by deg u 1 N 0 = 5 · 2 9 and deg u 1 N 1 = 5 2 · 2 11 and their multiplicities are Moreover, any fiber F of the Mukai system has degree 5 · 3 · 2 13 .
As the smooth locus of every component with its reduced structure deforms from the Mukai to the Hitchin system, the multiplicities and degrees must coincide. Here, indeed, the same multiplicities can be found in [23,Propositions 34 and 35] and [15,Proposition 6], whereas, up to our knowledge, the degrees have not been determined in the literature. In our case, the degrees determine the multiplicities.
Our second result is a description of the classes [N 0 ] and [N 1 ] ∈ H 10 (M, Z). The projective moduli spaces of stable sheaves on K3 surfaces are known to be deformation equivalent to Hilbert schemes of points. In our case, M is actually birational to S [5] [6, Lemma 3.2.7]. In particular, there is an isomorphism H * (M, Z) ∼ = H * (S [5] , Z). The cohomology ring of S [5] is well understood, e.g. [19, §4] and the references therein. Recall that for any Hyperkähler variety X of dimension 2n there is an embedding where [F] is the class of a general fiber of the Mukai system and 0 = β ∈ (S 5 H 2 (M, Q)) ⊥ satisfies β 2 = 0. As deg u 1 β = 0, the class β is not effective.

Outline
In Sect. 2 we introduce the Mukai system. In Sect. 3 we reduce the study of N C to the case of a smooth curve C. We describe the irreducible components of the nilpotent cone following [9, §3], where it is shown that any point [E] ∈ N C \ N 0 fits into an extension of the form where L ∈ Pic 1 (C) is a line bundle and x ∈ C a point. We specify a space W → Pic 1 (C) × C parameterizing such extensions, and a compactification W of W that comes with a birational map ν : W → N 1 . In the Hitchin case, this idea originates from [23]. In Sect. 4 we prove Theorem 1.

Notation
All schemes are of finite type over k = C. In the entire paper, S is a K3 surface polarized by a primitive, ample class H ∈ NS(S) with H 2 = 2g − 2.

The Mukai system
In this section, we give a brief recollection on moduli spaces of sheaves on K3 surfaces and define the Mukai system. First recall that the Mukai vector induces an isomorphism v : K(S) num Here,g = n 2 (g − 1) + 1. Moreover, M H (v) is irreducible holomorphic symplectic of dimension n 2 H 2 + 2 = 2g and hence, by Matsushita's result [21,Corollary 1] this morphism is a Lagrangian fibration (for an explicit proof see [9, Lemma 1.3]), called the Mukai system (of rank n and genus g).
In particular, a stable vector bundle of rank two and degree one on a smooth curve C ∈ |H | defines a point in M := M H (v). We have dim M = 8g − 6 = 10 and M is birational to the Hilbert scheme S [5] of five points on S.

Taking (Fitting) supports defines a Lagrangian fibration
We have a natural morphism |H | × |H | → |2H |. We define ⊂ |2H | as its image and ⊂ as the image of the diagonal. Then ∼ = Sym 2 |H | and ∼ = |H |. If every curve in |H | is irreducible (e.g. if Pic(S) = Z · H ) then and are exactly the loci of non-reduced and non-integral curves, respectively. And in this case we can distinguish three cases following [6, Proposition 3.7.1]: is reduced and irreducible if x ∈ |2H | \ is reduced and has two irreducible components if x ∈ \ has two irreducible components with multiplicities if x ∈ .
(3.1) In the general case, the list is still valid for for the geometric generic point in the respective subvariety. However, over points that correspond to curves with more irreducible components, one also finds more irreducible components in the fiber [6, Proof of Lemma 3.3.2].
We will study fibers of the third type, namely where C ∈ |H | is irreducible. In analogy with the Hitchin system, we call N C nilpotent cone.
For the rest of the paper, we fix a smooth curve C ∈ |H | and write N instead of N C . We will now identify the irreducible components of N following the ideas of [9].

Pointwise description of the nilpotent cone N = N C
Let [E] ∈ N and consider its restriction E| C to C. There are two cases, either E| C is a stable rank two vector bundle on C or E| C has rank one. By dimension reasons, the sheaves of the first kind contribute an irreducible component N 0 of N isomorphic to the moduli space M C (2, 1) of stable rank two and degree one vector bundles on C. In the second case, E| C ∼ = L ⊕ O D , where the first factor L := E| C /torsion is a line bundle on C and D ⊂ C is an effective divisor. We set Proof Noting that ω −1 C is the conormal bundle of C in S, it is straightforward to obtain the sequence (3.2). Let us prove the numerical restrictions. From (3.2) we have On the other hand, E is stable and therefore the reduced Hilbert polynomials [16, Definition 1.2.3] of E and L satisfy p(E, t) < p(L, t), which amounts to We conclude that every point in E 1 defines a class in Ext 1 ) defines a point in E 1 if and only if its middle term is stable and has the point x as support of its torsion part when restricted to C, i.e. if it is not pushed forward from C. It turns out that all such extensions are stable.

Lemma 3.3 Consider a coherent sheaf E on S that is given as an extension
where L and L are line bundles on C of degree k and 1 − k, respectively, with k ≥ 1. Moreover, assume that E itself does not admit the structure of an O C -module. Then E is H -Gieseker stable.

Corollary 3.4 The closed points of E 1 are in bijection with the following set
) such that v is not pushed forward from C. Here, L varies over all line bundles on C with deg L = 1, and x varies over all points in C. The bijection is established by Lemma 3.1.
In Proposition 3.5 below, we will see that there is a short exact sequence where ρ L,x has the following interpretation modulo a scalar factor. If E is the middle term of a representing sequence of v ∈ Ext 1 Hence, another way to phrase Corollary 3.4 is by fixing for every x ∈ C a defining section

Extension spaces
So far, we have given a pointwise description of the nilpotent cone. Next, we will identify its irreducible components and their scheme structures. This subsection is a technical parenthesis in this direction. The reader may like to skip it. Let S be a smooth projective surface and i : C → S a smooth curve with normal bundle N C/S ∼ = O C (C). Let T be any scheme and let F and F be two vector bundles on T × C considered as families of vector bundles on C. Denote by π : T × S → T and π C : T × C → T the projections. For a morphism f : X → Y , we write Ext f instead of R f * R Hom.

4) as well as for every t ∈ T a short exact sequence of vector spaces
Note that the fibers of (3.4) must, in general, not coincide with (3.5), see Lemma 3.6.
Proof Apply Rπ C * R Hom( , F) or R Hom( , F t ), to the exact triangle [17,Corollary 11.4]) or its counterpart in D b (C), respectively, and consider the induced cohomology sequence.
We can explicitly describe the morphism ρ t in the sequence (3.5).
. This gives a well-defined, linear map As im ξ = ker δ, it follows by dimension reasons, that δ has to be surjective. So, ρ t = δ up to post-composition with an isomorphism of Ext 0

Lemma 3.6 For every t ∈ T there is a commutative diagram of short exact sequences
where the first vertical arrow is an isomorphism.
has constant dimension for all t ∈ T all vertical arrows are isomorphisms.
Proof The vertical morphisms are the usual functorial base change morphisms. The lower line is (3.5) and hence also exact on the left. The first vertical arrow is an isomorphism because Ext 2 C (F t , F t ) = 0. Consequently, also the upper line is exact on the left.

Irreducible components of N
In this section, we show that E 1 is irreducible and has the same dimension as N . Therefore its closure with reduced structure is an irreducible component of N . For the proof, we need some more notation. Let P 1 be a Poincaré line bundle on Pic 1 (C) × C and ⊂ C × C the diagonal. Set T := Pic 1 (C) × C and on T define the following sheaves where p i j are the appropriate projections from Pic 1 (C) × C × C. Considering the fiber dimensions, we see that V and U are vector bundles of rank 2 and 1, respectively. In fact, s induces an isomorphism s : O T ∼ − −→ U = p 12 * p * 23 O( ). Moreover, by Proposition 3.5 they fit into a short exact sequence Consequently, also W is a vector bundle and ρ induces a map of geometric vector bundles We set with the projection τ : W → T . We retain some immediate consequences of the construction.
(i) W is a principal homogeneous space under Spec T (Sym • V ∨ ). In particular, it is an affine bundle over T . (ii) Let t = (L, x) ∈ T . Then by Lemma 3.6 we have (iii) dim W = 5.
(vi) W is compactified by the projective bundle W := P(W) with boundary isomorphic to P(V), i.e. W = W ∪ P(V).
Next, we relate E 1 and W . Recall that N 1 := E 1 ⊂ N red . We keep all the notations from the previous section, and

Proposition 3.8 There exists a 'universal' extension represented by
In particular, N red = N 0 ∪ N 1 is a decomposition into irreducible components.
Proof We set F := P 1 O( ) ω −1 C and F := p * 13 P 1 . We are looking for a 'universal' extension, i.e. for such that for w ∈ W ⊂ W the restriction of v univ to {w} × S is the extension corre- We get where the indicated isomorphism comes from the Leray spectral sequence. It is an isomorphism, because where π C : T × C → T . The last equality follows from the long exact sequence Finally, we consider the universal surjection as an element in H 0 (W , τ * W ⊗ O τ (1))) and take its image under (3.7). This produces the desired extension. By construction, G univ ∈ Coh(W × S) defines a morphism ν : W → N 1 ⊂ M which restricts to a bijection W → E 1 (see Corollary 3.4 and (3.3)). By degree reasons an extension on C of the form 0 → L → E → L → 0, where deg L = 1 and deg L = 0 is stable or split. However, the split extensions do not occur in P(V). Hence, ν is everywhere defined. Moreover, the boundary W \ W = P(V) maps to

Remark 3.9
One can show that ν : W → E 1 is actually an isomorphism of schemes. Moreover, ν : W → N 1 is finite and hence a normalization map. Its tangent map is analyzed in [11,Proposition 7.5] and provides a characterization of the singularities of N 1 .

Proof of Theorem 1.1
We will now prove Theorem 1.1.
Theorem Let C ∈ |H | be an irreducible curve. The degrees of the two components of the nilpotent cone N C = N 0 ∪ N 1 are given by deg u 1 N 0 = 5 · 2 9 and deg u 1 N 1 = 5 2 · 2 11 and their multiplicities are Moreover, any fiber F of the Mukai system has degree 5 · 3 · 2 13 . All degrees will be computed with respect to a naturally defined distinguished ample class u 1 ∈ H 2 (M, Z), which we construct in Sect. 4.1. We set The multiplicity is defined as follows. Let η i be the generic point of N i . Then as stated in the theorem. Luckily, it turns out that the multiplicities are small in comparison with the degrees so that it is possible to determine the multiplicities from the knowledge of the degrees but not vice versa.  Hence, N 1 is not reduced and the first solution is ruled out.

Remark 4.1
We will prove Theorem 1.1 for a fixed smooth curve C ∈ |H |, which implies the case of an irreducible and possibly singular curve by a deformation argument as follows. According to the careful analysis in [6, Section 3.7, in particular Propositions 3.7.23 & 3.7.19] the above description of the irreducible components of f −1 (2C) is valid for every irreducible curve C ∈ |H |. Hence, if one deforms from a smooth to a singular, irreducible curve in |H |, the irreducible components of the fiber with their reduced structure deform as well. Consequently, degrees and multiplicities remain constant.

Construction of the ample class u 1
We use the determinant line bundle construction [16, Lemma 8.1.2] in order to produce an ample class on the moduli space M. Let X and T be two projective varieties and assume that X is smooth. Let p : T × X → T and q : T × X → X denote the two projections. For any W ∈ Coh(X × T ) flat over T , we define λ W : K(X ) num → H 2 (T , Z) to be the following composition We will take advantage of the functorial properties of this definition. These are The construction is especially interesting if X = M T (c) is a fine moduli space, that parametrizes coherent sheaves of class c on T . Let E univ be a universal sheaf on M T (c) × T , then is well-defined and does not depend on the choice of universal sheaf. Here, Fix n ≥ 1 and d ∈ Z coprime and let c = (n, d) ∈ K(C) num . Then M C (c) = M C (n, d) is the moduli space of stable vector bundles of rank n and degree d on C and we find c ⊥,χ = (−n, d + n (1 − g) . The generalized Theta divisor can be defined by see [12,Théorème D]. A special case is M C (1, k) = Pic k (C). In this case, we find c ⊥,χ = (−1, k + 1 − g) and is the class of the canonical Theta divisor in Pic k (C).

Lemma 4.4 Let C be a smooth curve of genus g and P a Poincaré line bundle on
Pic k (C) × C. Then is given by By tensoring with a suitable line bundle on Pic k (C), we can assume c 2,0 1 (P) = 0.
Thus the first arrow is given by (r , c , s ) → (−r , c , −s − r ). The value of s does not have any relevance for our computations. However, with the results of [3], it can be proven that l s is ample for s 0 and one can even compute the precise boundary of the ample cone.

Definition 4.7
For everything what follows, we fix s 0 0 such that l s 0 is ample and set u 1 := l s 0 .

Degree of a general fiber
We compute the degree of a general fiber.

Degree of the vector bundle component N 0
Next, we deal with the component N 0 , which is isomorphic to M C (2, 1). u = (2c.H , c, s) ∈ v ⊥,χ . Then where ∈ H 2 (N 0 , Z) is the the generalized Theta divisor. In particular, Proof Let i : C → S be the inclusion. The inclusion N 0 → M is defined by (id ×i) * E univ , where E univ is the universal vector bundle on N 0 × C. Hence, maps (r , c, s) to (r , c.H ). In particular, Li * u = c.H (2, 1), whereas by definition θ = λ N 0 (−2, −1).
The value SM C (2,1) 3 SM = 4 is given by the leading term of the Verlinde formula [27].

Degree of the other component N 1
We complete the proof of Theorem 1.1 by dealing with the remaining component N 1 . Recall from Proposition 3.8 that there is a birational map ν : In particular, deg u 1 N 1 = 5 2 · 2 11 .
Note that the first equality in (4.3) is immediate, because ν : W → N 1 is birational. For the proof of the proposition, we need to introduce some more notation. We abbreviate Pic 1 (C) to Pic 1 and in the following all cohomology groups have Z coefficients. We set for the pullback of the class of a point on C. If no confusion is likely, we suppress pullbacks from our notation, e.g. we will write 1 ∈ H 2 (Pic 1 ×C) and also 1 ∈ H 2 (P(W)) instead of p * 1 1 and τ * p * 1 1 , respectively. Moreover, we define where P is a Poincaré line bundle. Note that π is independent of the choice of P.
Proof of Proposition 4. 13 We will split the proof into the following three steps. 2c.H , c, s). Then (ii) We have (iii) The top cohomology group H 10 (W ) generated by 1 2 ζ 2 ρ 2 1 and we have P(W) Proof of (i) In Proposition 3.8, we defined the morphism ν : W → N 1 by means of G univ ∈ Coh(W × S), which sits in the (universal) extension , is determined in Lemmas 4.14 and 4.4. Note that each summand depends on the choice of a Poincaré line bundle, whereas the sum does not. Together, Lemma 4.14 Let F ∈ Coh(X × C). Then for all x = (r , d) ∈ K(C) num . In particular, where i : C → C × C is the diagonal and thus for all x ∈ K(C) num . Now, To prove the remaining steps, we need to understand the cohomology ring H * (W ).

Lemma 4.15
We have In particular, Proof By definition, W = P(W). Hence, We use the short exact sequence 0 → V → W → O T → 0 from (3.6) to compute the Chern classes of W. Note that So the Chern classes of V can be computed by the push forward along the first projection of the following the short exact sequence We find
Finally, we want to show that W ζ 2 ρ 2 = 2. Indeed, This concludes the proof of the proposition.

Proof of Theorem 1.2
In this section, we prove Theorem 1. From now on, all cohomology groups have Q-coefficients. Before coming to the proof, we want to point out, that the irreducible components over points in \ (see (3.1)) are of different cohomological nature. Let D ∈ \ be a reducible curve with two smooth components C 1 and C 2 meeting transversally. Then the two components N 1 and N 2 of f −1 (D) contain an open sublocus parametrizing line bundles on D of bi-degree (2, 1) and (1, 2), respectively [6, Proposition 3.7.1 and Lemma 3.3.2]. The monodromy around \ exchanges C 1 and C 2 and consequently the classes of the irreducible components. We find In particular, the two components are linearly dependent. This is not true over . Moreover, our computation will use the following two inputs. Let α ∈ H 2 (SM C (2, 1)) be the degree two Künneth component of (c 2 1 − c 2 )(V univ ) with V univ being a universal bundle on SM C (2, 1) × C. It is known, e.g. [26, §5A], that c 1 (T SM C (2,1) ) = 2α, c 2 (T SM C (2,1) ) = 3α 2 and SM C (2,1) α 3 = 4.
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