COMPLETE CURVES IN THE MODULI SPACE OF POLARIZED K3 SURFACES AND HYPER-K ¨AHLER MANIFOLDS

COMPLETE CURVES IN THE MODULI SPACE OF POLARIZED K3 SURFACES AND HYPER-K ¨AHLER MANIFOLDS

OLIVIER DEBARRE AND EMANUELE MACR`I

Abstract. Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the casee= 14), we prove that the moduli space of polarized K3 surfaces of degree 2e contains complete curves for alle≥62 and for some sporadic lower values ofe(starting at 14).

We also construct complete curves in the moduli spaces of polarized hyper-K¨ahler manifolds of K3^[n]-type or Kum_n-type for alln≥1 and polarizations of various degrees and divisibilities.

Contents

1. Introduction. . . 1

2. Ample classes on Kummer surfaces. . . 3

3. Degrees of ample classes on Kummer surfaces. . . 6

4. Ample classes on hyper-K¨ahler manifolds. . . 11

References. . . 19

To the memory of Alberto Collino 1. Introduction

Let F2e⁰ be the (19-dimensional irreducible quasi-projective) moduli space of polarized K3 surfaces of degree 2e. In [BKPS, Theorem 1.3], the authors prove that F₂⁰ is affine, hence contains no complete curves. In [BKPS, Section 3], they construct a complete curve in F₂₈⁰. Their idea is to start from a nonisotrivial family of polarized abelian surfaces, which exists because the moduli space of polarized abelian surfaces has a small boundary in its Satake compactification, and construct a suitable polarization on the associated family of Kummer surfaces (which needs to be ample onallKummer surfaces in the family). Using the same con- struction, we prove the following extension of their results, which partially answers a question asked in [B,§ 6.3.4].

Theorem 1.1. For each integer e≥62 or in the set

{14,18,26,28,29,32,34,36,38,40,42,44,45,46,47,48,49,50,53,54,56,57,58,59,60},

2020Mathematics Subject Classification. 14D20, 14F08, 14J28, 14J42, 14J60.

Key words and phrases. Moduli spaces, stability conditions, K3 surfaces, hyper-K¨ahler manifolds, complete families.

This work was partially supported by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK.

1

there exists a complete curve in F2e⁰ or, equivalently, a nonisotrivial smooth family of K3 surfaces with a relative polarization of degree 2e.

As explained in Remark 3.10, one can make the construction quite explicit and obtain for example, for infinitely many values of e(includinge= 14), complete rational curves in F2e⁰

defined over Q. The Picard numbers of the corresponding K3 surfaces are 19 or 20.

It is plausible that F2e⁰ contain complete curves for all e ≥ 2. The situation is very different for the moduli space F2e of quasi-polarized K3 surfaces of degree 2e:¹ because of the existence of the Baily–Borel projective compactification with one-dimensional boundary, F2e

contains complete subvarieties of dimension 17. This is known to be the maximal possible dimension ([vdGK, Corollary 4.3]).

In the last part of the paper, we construct complete curves in the moduli spaces of polar- ized hyper-K¨ahler manifolds of K3^[n]-type and Kum_n-type by using moduli space techniques as in [M1, M2]. Given a K3 or abelian surface (or more generally a smooth and proper CY2 category) with a Bridgeland stability condition and a Mukai vector, there is a natural polariza- tion on the corresponding moduli space [BM1] of stable objects which behaves well in families.

Hence, the question of finding a suitable polarization on the moduli space translates into the question of understanding stable objects, which can be studied effectively by using techniques from [BM2,MYY, YY, Y2].

The final result is less complete than what can be obtained with the approach described above in the surface case, but it covers more cases. We prove the existence of a complete curve in F2e⁰ for e ∈ {18,32,36,50,54} (see Example 4.14; this cannot be obtained directly with the previous technique), while in higher dimensions, our results imply the following two theorems for Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (for more general statements, see Propositions 4.12 and 4.10).

In what follows, we denote by T a quasi-projective scheme and we let n ≥2.

Theorem 1.2. Let F → T be a smooth family of K3 surfaces with a relatively ample divi- sorH_F. Let H_F,n be the big and nef divisor on Hilbⁿ(F/T)induced by H_F via the symmetric product. Assume that the divisor parameterizing nonreduced subschemes is divisible by 2 and let δ_F be a half. For all positive integers a, b such that a > bp

(n−1)²+ 4(n−1), the divisor a·H_F,n−b·δ_F

is relatively ample on Hilbⁿ(F/T)→T.

In particular, given a complete curve in F2e⁰ and relatively prime positive integers a, b such thata > bp

(n−1)²+ 4(n−1), after a finite base change, we obtain a complete curve in the moduli space of polarized hyper-K¨ahler manifolds of K3^[n]-type of degree 2ea²−2b²(n−1) and divisibility gcd(a,2(n−1)).

Theorem 1.3. Let A → T be a smooth family of abelian surfaces with a relatively ample divisor H_F. For all positive integers a, b such that a > b(n+ 1), the divisor

a·H_A,n−b·δ_A is relatively ample on Kum_n(A/T)→T.

1The moduli spaceF2e⁰ is an open subset ofF2e which is the complement in the period space of a Heegner divisor; this Heegner divisor has one or two irreducible components depending on whether e6≡1 (mod 4) or e≡1 (mod 4).

The notationH_A,n andδ_A means, as before, the restrictions of the corresponding divisors in the Hilbert scheme (we suppose again that the global halfδ_A exists).

In particular, given a complete curve of primitively polarized abelian surfaces of degree 2d and positive integers a, b such that gcd(a, b) = 1 and a > b(n+ 1), we obtain, after a finite base change, a complete curve in the moduli space of polarized hyper-K¨ahler manifolds of Kum[n]-type of degree 2da²−2b²(n+ 1) and divisibility gcd(a,2(n+ 1)).

Acknowledgements. We would like to thank Ignacio Barros, Daniel Huybrechts, Kieran O’Grady, and Claire Voisin for useful discussions and suggestions.

2. Ample classes on Kummer surfaces

Let A be an abelian surface and let ε: Ab → A be the blow up of the sixteen 2-torsion points of A. Let Kum(A) =A/b ±1 be the Kummer surface of A, with quotient map π: Ab→ Kum(A), and let E₁, . . . , E₁₆ be the images by π of the exceptional curves of ε; each E_i is a rational curve with self-intersection −2. By [BHPV, Propositions VIII.(5.1) and VIII.(5.2)], there is an injective morphism of Hodge structures

α :=π∗ε^∗:H²(A,Z)−→H²(Kum(A),Z) and it satisfies, for allx, y ∈H²(A,Z),

α(x)·α(y) = 2x·y.

In particular, α(x)² ∈4Z. Moreover, Im(α) is the orthogonal inH²(Kum(A),Z) of the sublat- tice generated by (the classes of) E₁, . . . , E₁₆ ([BHPV, Corollary VIII.(5.6)]).

Any integral divisor class on Kum(A) can therefore be written as

(1) L=aα(N)−

16

X

i=1

a_iE_i,

where N is primitive in NS(A) and a, a₁, . . . , a₁₆ ∈ Q. By taking the product with E_i, we see that a_i ∈ ¹₂Z. By taking the product with α(x), where x ∈ H²(A,Z) is such that N ·x = 1, we see that a ∈ ¹₂Z. The a_i are not necessarily integers: for example, the class ¹₂P16

i=1E_i is integral. We will from now on drop α in (1).

Let H_A be an ample divisor class on A of degree 2d, for some d ∈ Z_>0. The divisor class H :=α(HA) on Kum(A) is nef and big, but not ample; one has H² = 4d.

We look for conditions for a classLas in (1) to be ample. AssumeL² >0 and 0< H·L= aH·N (so for example, a >0 andH·N >0), so that L is in the positive cone; we have then ([H, Proposition 2.1.4])

L ample ⇐⇒ L·C >0 for all irreducible curvesC ⊂Kum(A) with C² =−2.

The next proposition gives sufficient conditions for a class L as in (1) to be ample. Its proof follows that of [GS2, Proposition 4.4] (itself identical to that of [GS1, Proposition 3.4]), where similar results are proven under the additional assumption Pic(A) = ZH_A.

Proposition 2.1. On any Kummer surface, any rational class L=aH −

16

X

i=1

a_iE_i,

where a₁ ≥ · · · ≥a₁₆>0 and such that a > a₁+a₂+a₃+a₄, is ample.

Proof. We have

(2) a² >(a1 +a2+a3+a4)² = X

1≤i,j≤4

aiaj ≥

16

X

i=1

a²_i,

hence

L² = 4a²d−2

16

X

i=1

a²_i >(4d−2)a² >0,

and L·H = 4ad > 0, hence L is in the positive cone. Consider now an irreducible curve C with self-intersection −2; we must prove L·C > 0. One can write

C =bM −

16

X

i=1

b_iE_i,

with b, b₁, . . . , b₁₆ ∈ ¹₂Z. Since L·E_i = 2a_i > 0, we may assume that C is not one of the E_i. We then have b_i = ¹₂C ·E_i ≥ 0 and 0 < H · C = bH ·M, so we may assume b > 0 and H·M > 0. Finally, the integerM² is divisible by 4 and nonnegative, because 2bM is the class of the effective divisor ε∗π^∗C onA.²

Assume by way of contradictionL·C ≤0, that is,

(3) abH·M −2

16

X

i=1

a_ib_i ≤0.

Following [GS1], we use the inequalities X¹⁶

i=1

b_i2

≤X¹⁶

i=1

a²_iX¹⁶

i=1

b²_i

(Cauchy–Schwarz) , (H·M)² ≥H²M² (Hodge).

They imply

2 = −C² =−b²M²+ 2

16

X

i=1

b²_i

≥ −b²M²+ 2

P16

i=1a_ib_i2

P16

i=1a²_i (use Cauchy–Schwarz)

≥ −b²M²+a²b²(H·M)² 2P16

i=1a²_i (use (3))

≥ −b²M²+a²b²H²M² 2P16

i=1a²_i (use Hodge)

=b²M²

2a²d P16

i=1a²_i −1

.

If b²M² ≥2, we obtain a²d≤P16

i=1a²_i, which contradicts (2).

2This is an essential point, and the only one where we use the geometry of the situation. The rest is just formal lattice-theoretic manipulations.

Assume now b²M² < 2. Since M² is divisible by 4 and nonnegative and b ∈ ¹₂Z_>0, this leaves only two cases to be considered

• b= ¹₂ and M² = 4, so that

(4) 2 =−C² =−1 + 2

16

X

i=1

b²_i.

Sinceb_i ∈ ¹₂Z≥0, we get (after reordering) eitherb₁ = 1, b₂ =b₃ = ¹₂, or b₁ =· · ·=b₆ =

1

2, and the others vanish. Plugging that back into (3), we get that abH ·M is at most 2a1+a2+a3 in the first case, and at mosta1+· · ·+a6 in the second case. The Hodge inequality also gives bH ·M ≥ √

dM² ≥ 2. Therefore, we get a ≤ ¹₂(2a₁ +a₂) in the first case anda≤ ¹₂(a₁+· · ·+a₅) in the second case, which contradicts our hypothesis ona.

• M² = 0 (butM 6= 0). In that case, we get from (4) (after reordering), eitherb₁ = 1 or b₁ =· · ·=b₄ = ¹₂, and the others vanish. As in the first case, we get thatabH·M is at most 2a1 in the first case, and at mosta1+· · ·+a4 in the second case, which contradicts our hypothesis on a. In the first case, we remark that since the linear system |H| is base-point-free and only contracts the curves E₁, . . . , E₁₆, the linear system |H−E₁| has no fixed divisor, hence is nef; in particular, bH ·M −2 = (H −E₁)·C ≥0. This impliesa≤a₁ and contradicts our hypothesis ona.

We have therefore obtained contradictions in all cases. This proves L·C > 0 for all (−2)-

curves L, henceL is ample.

Corollary 2.2. On any Kummer surface, the class aH−P16

i=1E_i, with a ∈ Q, is ample for all a >4 and nef for a= 4.

This seems to have been known, at least whena is an integer, to the authors of [BKPS]

(see their Section 3) but we could not find a published proof.

Remark 2.3. LetA₁andA₂be elliptic curves. The ample classH_A:= (A₁×{0})+d({0}×A₂) on the abelian surfaceA :=A₁×A₂ induces a polarizationH of degree 4don the Kummer surface Kum(A). The image in Kum(A) of the proper transform ε^∗({0} ×A₂)−P4

i=1E_i in Ab of the elliptic curve {0} ×A₂ is a smooth rational curve with self-intersection −2 whose intersection number with 4H−P16

i=1Ei is 0. The nef class 4H−P16

i=1Ei is therefore not ample on Kum(A).

Remark 2.4. If we make the additional generality assumption Pic(A) = ZH_A, we can take M =H in the proof of Proposition 2.1. In particular, the case M² = 0 needs not be consid- ered and the hypotheses can be relaxed. One checks for example that when d = 1, the class aH−P16

i=1E_i is ample for all a >3.³ It is classical that when (A, H_A) is an indecomposable principally polarized abelian surface (that is, the Jacobian of a smooth curve), the integral degree-8 class 2H− ¹₂P16

i=1E_i is very ample ([GH, p. 773–787]). See also [GS2, Section 4] for other results under the assumption Pic(A) = ZH_A.

3Assume a >3. Ifb≥2, we get the contradiction 2≥b²M² P^2a16^2d i=1a²_i −1

≥2a²−16>2. Ifb= ¹₂, we get P16

i=1b²_i = ³₂, hence P16

i=1bi ≤3, hencea≤ _2b¹ P16

i=1bi ≤3. If b= 1, we getP16

i=1b²_i = 3, henceP16

i=1bi ≤6, hencea≤_2b¹ P16

i=1bi ≤3. Ifb= ³₂, we getP16

i=1b²_i = ¹¹₂, henceP16

i=1bi≤9, hencea≤ _2b¹ P16

i=1bi≤3.

3. Degrees of ample classes on Kummer surfaces

Our aim in this section is to prove that all large enough even numbers can be realized as degrees of primitive ample classes on Kummer surfaces.

Proposition 3.1. Let A be a principally polarized surface. For every integer e ≥ 160, there exists a primitive ample class of degree 2e on Kum(A).

3.1. Primitive ample classes of large degrees not divisible by 8. We keep the same notation as before. In this section, we consider integral classes on Kum(A) of the form

L=aH−

15

X

i=1

aiEi−E16,

wherea, a₁, . . . , a₁₅are integers,a₁ ≥ · · · ≥a₁₅>0, anda > a₁+a₂+a₃+a₄. By Proposition2.1, these classes are ample. We assume H² = 4, so that

L² = 4a²−2

15

X

i=1

a²_i −2.

Proposition 3.2. Any integer n ≥164, be written as

(5) n= 2a²−

15

X

i=1

a²_i,

where a, a₁, . . . , a₁₅ are integers, a₁ ≥ · · · ≥a₁₅≥1, and a > a₁+a₂+a₃+a₄.

Proof. We will first prove the result for all n ≥1218. The remaining numbers can be checked by computer.

Lemma 3.3. Every integer m≥34 is the sum of five positive perfect squares.

Proof. This is classical and easy if we use Lagrange’s theorem that every positive integer is the sum of four squares. Assume first m >169; then one can write m−169 = a²₁+· · ·+a²₄, with a₁ ≥a₂ ≥a₃ ≥a₄ ≥0.

If a₄ > 0, writing m = 13² +a²₁ +· · ·+a²₄, we are done. Assume a₄ = 0; if a₃ > 0, writing m = 12² + 5² +a²₁ +a²₂ +a²₃, we are done. Assume a₃ = a₄ = 0; if a₂ > 0, writing m= 12²+ 4²+ 3²+a²₁+a²₂, we are done. Assumea₂ =a₃ =a₄ = 0; then a₁ >0 and, writing m= 12²+ 7²+ 4²+ 2²+a²₁, we are done.

The remaining cases 34≤m≤169 can be checked by hand.

Lemma 3.4. Every integer m ≥ 36 is the sum of the squares of fifteen positive integers that are all ≤p_m

3 −3.

Proof. Write m = 3n +r, with r ∈ {0,1,2}. If m ≥ 102, we have n ≥ 34. By Lemma 3.3, we can write both n and n+ 1 as the sum of the squares of five positive integers that are all

≤√

n−3≤p_m

3 −3. Adding up these decompositions, we obtain the required decomposition of m.

The remaining cases 36≤m≤101 can be checked by computer.

A consequence of Lemma 3.4 is that all integers between 2a² −36 and 2a² − b^3a2₁₆⁺¹⁴³c can be written as in (5), where the ai are integers such that

1≤a_i ≤ r1

3

3a²+ 143 16

−3< a 4. In order to have no gap when we go from a toa+ 1, we need

2(a+ 1)²−j3(a+ 1)²+ 143 16

k≤2a²−35 or, equivalently,

3(a+ 1)²+ 143

16 ≥4a+ 37, that is, a≥26. This means that all integers

n ≥2·26²−j3·26²+ 143 16

k

= 1217 can be written as in (5) with a >4a₁, which is more than we need.

The remaining cases 164≤n≤1216 can be checked by hand.

Corollary 3.5. Let A be a principally polarized surface. For all integers e≥163 not divisible by 4, there exists a primitive ample class of degree2e on Kum(A).

Proof. Writee+ 1 = 2a²−P15

i=1a²_i as in Proposition 3.2. By Proposition 2.1, the class L=aH−

15

X

i=1

a_iE_i−E₁₆

is ample on Kum(A), of degree 2e. Since L·E₁₆ = 2, either L is primitive and we have what we want, or it is divisible by 2. In the latter case, L² = 2e must be divisible by 8 and this

contradicts our hypothesis one.

3.2. Primitive ample classes of large degrees divisible by 4. We keep the same notation as before. Recall that the class P16

i=1E_i is (uniquely) divisible by 2 in Pic(Kum(A)) (because it is the class of the branch locus of the double cover π: Ab→ Kum(A)). In this section, we consider integral classes of the form

L=aH−

15

X

i=1

a_i− 1

2

E_i− 1 2E₁₆

where a, a₁, . . . , a₁₅ are integers, a₁ ≥ · · · ≥ a₁₅ ≥ 1, and a > a₁ + a₂ +a₃ +a₄ −2. By Proposition 2.1, these classes are ample. They are also primitive, because L·E₁₆ = 1. We assume H² = 4, so that

L² = 4a²−8−4

15

X

i=1

a_i 2

(always divisible by 4). Recall that a triangular number is an integer of the form ^r₂ . Proposition 3.6. Every integer n ≥30can be written as

(6) n=a²−

15

X

i=1

a_i 2

,

where a, a₁, . . . , a₁₅ are integers, a₁ ≥ · · · ≥a₁₅≥1, and a > a₁+a₂+a₃+a₄−2.

Proof. We will first prove the result for all n ≥ 757. The remaining numbers can be checked by computer.

Lemma 3.7. Every nonnegative integer is the sum of three nonnegative triangular numbers.

Proof. This is classical. Every positive integer of the form 8m+ 3 can be written as the sum of three squares. These squares must be odd, so we may write

8m+ 3 = (2a₁−1)²+ (2a₂ −1)²+ (2a₃−1)², with a₁ ≥a₂ ≥a₃ ≥1, or

m= a₁

2

+ a₂

2

+ a₃

2

.

This proves the lemma.

Lemma 3.8. Every integer m ≥ 24 is the sum of fifteen nonnegative triangular numbers of the form ^a₂

with 1≤a≤ ¹₂ + q2m

5 +³³₄.

Proof. Write m = 5n + r, with r ∈ {0, . . . ,4} and n ≥ 0. By Lemma 3.7, we can write n, . . . , n+ 4 each as the sum of the squares of three nonnegative triangular numbers ^a₂

. We have ^a₂

≤ n + 4 ≤ ^m₅ + 4, hence a ≤ ¹₂ +q

2m

5 + ³³₄. Adding up these decompositions, we

obtain the required decomposition of m.

A consequence of Lemma 3.8 is that all integers between a² and a² − b^5a2₃₂⁻⁶⁶¹c can be written as in (6), where the a_i are integers such that

1≤a_i ≤ 1 2 +

r2 5

5a²−661 32

+33

4 < a+ 2 4 . In order to have no gap when we go from a toa+ 1, we need

(a+ 1)²−j5(a+ 1)² −661 32

k≤a²+ 1 or, equivalently,

5(a+ 1)²−661

16 ≥2a,

that is, a≥14. This means that every integer

n ≥14²−j5·14²−661 32

k

= 187

can be written as in (6) with a >4a₁−2, which is more than we need.

The remaining cases 30≤n ≤186 can be checked by computer.

Corollary 3.9. Let A be a principally polarized surface. For every even integer e ≥56, there exists a primitive ample class of degree 2e on Kum(A).

Proof. Write ¹₂e+ 2 =a² −P15 i=1

ai

2

as in Proposition 3.6. By Proposition 2.1, the primitive integral class

L=aH−

15

X

i=1

a_i− 1

2

E_i− 1 2E₁₆

is ample on Kum(A), of degree

4a²−8−4

15

X

i=1

ai

2

= 2e.

Because of Corollary3.9, we may assume, for the proof of Proposition3.1, thate is odd.

Corollary3.5 applies whene≥163, but one sees by a direct computer check that it also applies for e= 34 and for the following (odd) values of e:

53,79,97,101,103,107,109,113,119,125,131, 135,137,139,143,145,149,151,155,157,161.

This proves Proposition 3.1. For the record, one also sees by a direct computer check that Corollary 3.9 also applies for the following (even) values of e:

14,26,28,40,42,44,46,48.

3.3. Ample classes of intermediate degrees. We keep the same notation as before. Not only is the class P16

i=1E_i divisible by 2 in Pic(Kum(A)), but this is also true for some sums of eight classes among the Ei.⁴

(1) We may consider primitive integral classes of the form aH−

15

X

i=1

a_iE_i− 1 2E₁₆

where a is an integer, a1, . . . , a15 ∈ ¹₂Z>0 and exactly seven of them are not integers, a1 ≥

· · · ≥a15>0, and a > a1+a2+a3+a4 ([GS2, Remark 2.3]). By Proposition2.1, these classes are ample.

For possible degrees 2e, we get the following new values for e:

38,57,59,71,73,75,77,79,81,83,85, and all numbers between 95 and 159.

(2) Whend is odd, there are integral classes of the type ([GS2, Theorem 2.7])⁵ 1

2(H+E_i1 +· · ·+E_i6),

4This fact can be explained geometrically as follows. Write A=A⁰/{0, α}, whereA⁰ is an abelian surface andα∈A⁰ has order 2. The translation byαonA⁰ induces an involution on Kum(A⁰) whose fixed points are the eight images in Kum(A⁰) of the sixteen pointsx∈Asuch that 2x=α. Blowing up these points, we get a double cover of Kum(A) branched along the union of the eight (−2)-curves corresponding to the images inA of the pointsx. The sum of these eight (−2)-curves is therefore divisible by 2.

5When d = 1 and the principally polarized surface (A, H_A) is indecomposable, the image of Kum(A) by the morphism associated with the linear system|H|is a quartic surface inP³ with sixteen nodes and sixteen everywhere tangent planes (classically called “tropes”), each containing six nodes. Each trope gives rise to a conic with class of the type ¹₂(H−Ei₁ − · · · −Ei₆). When the surface (A, HA) is the product of principally polarized elliptic curves A1 and A2, these sixteen conics become unions of copies of A1 and A2 meeting transversely at one point.

so we may also consider primitive integral classes of the form

a+1 2

H−

15

X

i=1

a_iE_i− 1 2E₁₆

where a is an integer, a₁, . . . , a₁₅∈ ¹₂Z_>0 and exactly five of them are not integers, a₁ ≥ · · · ≥ a₁₅ > 0, and a ≥ a₁ +a₂ +a₃ +a₄. By Proposition 2.1, these classes are ample. Still taking d= 1, we get possible degrees 2e for

e ∈ {29,45,46,47,49,63,65,67,69,71,73}

and all integerse between 85 and 159.

All in all, in our list of possible degrees 2e (all obtained with d = 1), we get for e the values

14,26,28,29,34,38,40,42,44,45,46,47,48,49,53,56,57,58,59,60

and all integerse≥62. The casese∈ {18,32,36,50,54}will be covered later in Example4.14.

Proof of Theorem 1.1. LetT be an irreducible complete curve contained in the (3-dimensional) coarse moduli spaceA2 of principally polarized abelian surfaces; such curves exist because the boundary of the (3-dimensional) projective Satake compactifications has dimension 1.

We fixm even,m ≥4, and we consider the pullback T⁰ of T in the moduli spaceA2(m) of principally polarized abelian surfaces with full level-m structures. Since m≥3, this moduli space is fine hence the inclusion T⁰ ⊂ A2(m) defines a family A → T⁰ of abelian surfaces with a relative principal polarization H_A on A; since m is even, there are sixteen sections corresponding to the 2-torsion points in the fibers.

Let ε: Ac→ A be the blow up of the images of these sixteen sections. Multiplication by−1 onA lifts toAc. Letπ: Ac→K :=Ac/±1 be the quotient map and letE ⊂Acbe the image byπof the exceptional divisorε⁻¹(A[2]). The divisorE is the branch locus ofπhence its class is divisible by 2 in Pic(K ); it splits as the sum of sixteen irreducible divisors E1, . . . ,E16. The relative polarization HA lifts to a relative quasi-polarization onAc→T⁰ which induces a relative quasi-polarization H onK →T⁰.

By Proposition 3.1, there are relative polarizations on K of all even degrees ≥ 320.

Method (1) of Section 3.3 also applies: using the T⁰-isomorphism A →^∼ Pic⁰(A/T⁰) given by the principal polarizationHA and a section of A →T⁰ of order 2, one can construct a double

´

etale covering A⁰ →A and proceed as in footnote 4. We obtain relative polarizations on K of all even degrees ≥190.

Finally, to use method (2), one needs global halves of classes of the typeH −E1−· · ·−E6. To achieve this, one needs to take a suitable ramified double cover of the base curve T⁰, as

explained in [BKPS, Section 3].

Remark 3.10. One can be more precise concerning the nature of the complete curves that we constructed in F2e⁰. For example, in the proof above, we can take for T a complete Shimura curve. More precisely, let Q be an indefinite quaternion algebra over Q of reduced discrimi- nantD and consider principally polarized abelian surfaces whose endomorphism rings contain a maximal order ofQ. Their Picard number is at least 3 and their locus in the moduli spaceA2

of principally polarized abelian surfaces only depends onD and is the finite union of images of

projective Shimura curves (defined overQ; see [R, Proposition 4.3]). WhenD∈ {6,10}, these loci are irreducible and the Shimura curves are isomorphic toP¹overQ([R, Proposition 7.1]).⁶ During the proof above, we needed to take a ramified cover T⁰ → T in order to work with a family A →T⁰ of abelian surfaces. When t⁰ ∈T⁰, the Kummer variety Kum(At⁰) only depends on the image t of t⁰ in T, but the polarization that we construct depends in most cases on an ordering of the (−2)-curves E1, . . . , E16. However, if we consider primitive ample line bundles of the type

L=aH− c 2

16

X

i=1

E_i,

the image of the pair (Kum(At⁰), L) will only depend on the point t and the modular map T⁰ →F2e⁰ will factor throughT.

ForL to be primitive, we need gcd(a, c) = 1 and for ampleness, we need a >2c (Corol- lary 2.2). It follows that in degrees 2e = 4a²−8c², for all positive integers a and c subject to these two conditions, we obtain rational curves in F2e⁰.

4. Ample classes on hyper-K¨ahler manifolds

In this section, we use moduli spaces of stable sheaves/complexes on twisted K3 or abelian surfaces, more generally on CY2 categories, to obtain complete families of polarized hyper-K¨ahler manifolds in any dimension, once we start from a complete family of polar- ized CY2 categories. Our tool is a minor generalization of [M2], together with techniques from [BM1, BM2]. We start by first giving in Section 4.1 a short review of twisted K3 and abelian surfaces, which provide our main set of examples. The main result, Theorem 4.8, is stated in full generality in Section 4.2and is a direct rewriting of the results in [BL+]. Finally, we discuss various examples in Section4.3.

4.1. Twisted K3 and abelian surfaces. Following [HS], we give a short review of twisted K3 and abelian surfaces. Let S be a complex K3 surface. We define its Brauer group as

Br(S) := H²(S,OS^∗)_tors.

It parameterizes equivalence classes of Azumaya algebras over S.

Definition 4.1. Atwisted K3 surface is a pair (S, α), whereS is a K3 surface andα∈Br(S). We denote by coh(S, α) the abelian category ofα-twisted coherent sheaves on S and by D^b(S, α) its bounded derived category.

Using the exponential sequence onS, we see that for anyα∈Br(S), there exists aB-field B ∈H²(S,Q) such that

exp(B^0,2) = α.

Such a B-field is unique modulo H²(S,Z) and NS(S)⊗Q.

Definition 4.2. Let (S, α) be a twisted K3 surface and letB ∈H²(S,Q) be a B-field. We define a weight-2 polarized Hodge structure H(S, B,e Z) on the cohomology H^∗(S,Z), together with the Mukai pairing, by setting

He^2,0(S, B) :=C·exp(B)[η] =C·([η] +B ∧[η]),

6The abelian surfaces that they parametrize were described in [HM, Theorem 1.3] (see also [R, Section 7])) as the Jacobians of explicit genus-2 curves.

where [η] ∈ H²(S,C) is the class of a nondegenerate holomorphic 2-form η on S. We denote byH_alg^∗ (S, B,Z) its (1,1)-part, given by

H_alg^∗ (S, B,Z) := He^1,1(S, B)∩H^∗(S,Z).

This Hodge structure does not depend on the choice of B, up to noncanonical isomor- phism. FixingB, by [HS, Proposition 1.2], we also have a well-defined notion of Mukai vector

v^B =p

td_S·ch^B: K(S, α)−→H_alg^∗ (S, B,Z),

where K(S, α) denotes the Grothendieck group of the abelian category coh(S, α).

Example 4.3. Ifαis trivial, we can chooseB = 0 and the Hodge structureH(S, B,e Z) coincides with the usual Mukai Hodge structure onH^∗(S,Z) and the Mukai vector with the usual Mukai vector.

Mukai’s moduli theory on twisted K3 surfaces works as in the untwisted case. We sum- marize the main results as follows.

Let (S, α) be a twisted K3 surface. We denote by Stab^†(D^b(S, α)) the distinguished connected component of the space of Bridgeland stability conditions on D^b(S, α), with respect to the algebraic Mukai latticeH_alg^∗ (S, B,Z), described in [Bri2] (see also [HMS] for the twisted case). The original definition is in [Bri1]; here we use [BL+, Definition 21.15], with the addition that moduli spaces exist.

Forσ ∈Stab^†(D^b(S, α)) andv∈H_alg^∗ (S, B,Z) a Mukai vector, we denote by M_(S,α),σ(v) be the moduli space of σ-semistable objects on D^b(S, α) with B-twisted Mukai vector v (see [HS, Y1,T, BM1, BL+]).

Theorem 4.4(Yoshioka). Let(S, α)be a twisted K3 surface and letB ∈H²(S,Q)be a B-field.

Let v ∈ H_alg^∗ (S, B,Z) be a primitive Mukai vector and let σ ∈ Stab^†(D^b(S, α)) be v-generic.

Then,

(a) The moduli space M_(S,α),σ(v) is nonempty if and only if v² + 2 ≥ 0. In that case, M_(S,α),σ(v)is a smooth projective hyper-K¨ahler manifold of K3^[n]-type, where n = ^v2₂⁺². (b) The Mukai isomorphism gives isometries of Hodge structures

ϑ: v^⊥/v−→^∼ H²(M_(S,α),σ(v),Z) if v² = 0, ϑ: v^⊥−→^∼ H²(M_(S,α),σ(v),Z) if v² ≥2,

where v^⊥ ⊂ H^∗(S, B,Z) is endowed with the induced sub-Hodge structure, while H²(M_(S,α),H(v),Z) is endowed with the standard Hodge structure together with the Beauville–Bogomolov–Fujiki form q.

Thev-genericity of the stability condition σ means M_(S,α),σ(v) =M_(S,α),σ^st (v), namely all σ-semistable objects are σ-stable.

Given an ample divisor H on S, we denote the classical moduli spaces (particular cases of the above theorem) of H-Gieseker semistable (resp. stable) α-twisted sheaves by M_(S,α),H(v) (resp.M_(S,α),H^st (v)) and of slope-semistable (resp. slope-stable) torsion-free sheaves byM_(S,α),H^µ (v) (resp. M_(S,α),H^µ−st (v)).

Remark 4.5. Analogously, one can define twisted abelian surfaces and their moduli spaces. Let (A, α) be a twisted abelian surface. Given σ ∈Stab^†(D^b(A, α))⁷, an analogue of Theorem 4.4 holds for the generalized Kummer moduli space K_(A,α),σ(v), namely the fiber at 0 of the Al- banese morphism alb : M_(A,α),σ(v)→A×A^∨. More precisely, let v∈H_alg^∗ (A, B,Z) be a primi- tive Mukai vector. Whenv² ≥6, the moduli spaceK_(A,α),σ(v) is nonempty of dimensionv²−2 and the Mukai isomorphism gives an isomorphism ϑ: v^⊥⊂H^∗(A, B,Z)→^∼ H²(K_(A,α),σ(v),Z).

The casev² = 4 is slightly degenerate: the generalized Kummer moduli space is isomor- phic to a K3 surface. We still have the Mukai morphism, but it is not an isomorphism: we can identify v^⊥ with the part of the cohomology of K_(A,α),H(v) coming from the abelian surface plus an extra class. This identification is not an isometry in this case: it satisfies

q(ϑ(a), ϑ(b)) = 2·(a,b),

for all a,b ∈v^⊥. When v= (1,0,−2) (and so B = 0) and σ is v-generic, thenK_(A,α),H(v)' Kum(A), the class −(0, D_A,0) gets identified with the cohomology class D:=α(D_A) induced byD_A, while the class −(1,0,2) gets identified with ¹₂P16

i=1E_i.

As in the K3 case, given an ample divisor H onA, we use the notationK_(A,α),H(v), and so on, for the particular case of H-Gieseker semistable sheaves.

4.2. Polarized families of moduli spaces. We follow [P, Sections 2–5] for the basic notions used in this section (see also [MS, Section 2]). The goal is to give a short review of [BL+, Part IV]: a polarized family of CY2 categories gives rise to a polarized family of moduli spaces of stable objects. We will apply our results only in geometric situations arising from CY2 categories of twisted K3 of abelian surfaces, as in Section 4.1.

Let D be a smooth proper CY2 category over the complex numbers, as in [P, Defi- nition 6.1]. We denote by H(e D,Z) its topological K-theory, together with the Mukai Hodge structure. We also have a Mukai vectorv: K(D)→H_alg(D,Z) with the same formal properties as in the twisted K3 or abelian surface case.

Example 4.6. The main example we will consider is when (S, α) is a twisted K3 or abelian surface and D = D^b(S, α). In such a case, H(e D,Z) is isometric to H(S, B,e Z) as Hodge structures, once a B-field lift is fixed.

As in the twisted setting, for σ = (Z,P) ∈ Stab(D), with respect to the algebraic Mukai lattice Halg(D,Z), and v∈H_alg^∗ (D) a Mukai vector, we denote by M_D,σ(v) the moduli space ofσ-semistable objects on D with Mukai vectorv. It is a proper algebraic space over C ([AP,AHLH]). The stable locus M_D^st_,σ(v) is open, smooth, and symplectic. The main result of [BM1] shows that there exists a real numerical Cartier divisor class `σ(v) on M_D,σ(v) which is strictly nef (it is ample in geometric situations, as we will see below).

We now consider the relative situation and introduce the notion of polarized family of CY2 categories. Let T be a quasi-projective scheme over C. Let D/T be a CY2 category over T (still in the sense of [P, Definition 6.1]). We denote by H(e D/T,Z) the Mukai local system, as in [P, Definition 6.4]. The uniformly numerical relative Grothendieck group of D over T (see [BL+, Proposition and Definition 21.5]) is denoted byN (D/T); it is a free abelian group of finite rank which can be thought of as the numerical Grothendieck group of Dt, for a very

7For twisted abelian surfaces, the space of stability conditions is actually connected.

general closed point t ∈ T, or equivalently the sections of H(e D/T,Z) which are algebraic on each fiber.

The notion of stability condition on D over T was introduced in [BL+]. We denote by Stab_N(D/T) the space of stability conditions σ ={σ_t}t∈T on D over T whose central charge factors throughN (D/T) (see [BL+, Theorem 22.2]⁸). It leads to the following definition.

Definition 4.7. We say that a CY2 category D/T over T is a polarized family if there exists a stability condition σ = {σ_t}t∈T ∈ Stab_N(D/T) on D over T whose central charge factors through N (D/T).

One of the main results in [BL+], together with the generalization of [M1] given in [P], implies that a family of polarized CY2 categories gives a family of quasi-polarized moduli spaces.

Theorem 4.8. Let (D/T, σ) be a polarized CY2 category over a complex quasi-projective scheme T. Let v ∈ N (D/T) be a Mukai vector for which the relative moduli M_σ(v) con- sists only of stable objects. Then M_σ(v) →T is a smooth and proper algebraic space over T, all the fibers are projective and symplectic, and there exists a relative real numerical Cartier divisor class`σ ∈N¹(Mσ(v)/T) that is relatively strictly nef. IfT is smooth, then Mσ(v)→T is also projective.

Proof. The fact that M_σ(v) → T is proper and the existence and positivity property of the divisor class`_σ is exactly [BL+, Theorem 21.24 & 21.25]. Since the relative moduli spaceM_σ(v) consists only of stable objects, the smoothness over T follows from [P, Theorem 1.4]. The projectivity of the fibers, or the more general statement if T is smooth, follows then from [V-P, Corollary 3.4]. Finally, the symplectic form comes from relative Serre duality: it is

nondegenerate by assumption and skew-symmetric by [vdB].

The relative divisor class is ample when one of the fibers Dt over a closed point t∈T is geometric.

Proposition 4.9. In addition to the assumptions made above, let us further assume that there exists a closed pointt0 ∈T such that Dt0 'D^b(S, α), for a twisted K3 or abelian surface (S, α), and that σt0 ∈Stab^†(D^b(S, α)). Then `σ is relatively ample.

Proof. The argument is based on [BL+, Section 33] and was found by Giulia Sacc`a. We repeat it here for completeness.

We can slightly deformσ and, upon taking multiples of the divisor class `<sub>σ</sub>, assume that this class is integral. By [BM1, Corollary 7.5], the class `_σt

0 =`_σ|_Mσt

0(v) is ample on M_σt

0(v).

Since ampleness is an open property, `σt is ample for allt in a Zariski open subset U ⊂T. Let us fix a line bundleLσ whose numerical class is`σ. By relative Serre vanishing, we can assume that L_σt has no higher cohomology for all t∈U. Hence

h⁰(M_σt(v), L^⊗m_σt ) =χ(M_σt(v), L^⊗m_σt ) for all t ∈U and m >0, and thus this number is independent of t.

By semicontinuity, this shows that h⁰(M_σt(v), L^⊗m_σ

t ) has maximal growth for all t ∈ T. Therefore it is big on M_σt(v) for all t ∈ T. Since M_σt(v) is a smooth, projective, symplectic

8In the notation of loc. cit., this means that the finite-rank free abelian group Λ is N (D/T) and the morphismv:Knum(D/T)→Λ is given by the Mukai vector.

variety, it has trivial canonical bundle. Hence, by the Base Point Free Theorem, since the line bundle L_σt is also strictly nef, it must be ample for all t∈T, as we wanted.

4.3. Examples. We use Theorem 4.8 to construct families of polarized HK manifolds. We separate the case of twisted abelian and K3 surfaces: the ideas are similar but slightly more involved in the K3 case.

Generalized Kummer varieties. Let r, d ≥ 1. Let (A, α, H_A) be a polarized twisted abelian surface, with H_A² = 2d and α^r =id. We let B ∈ H²(A,Q) be a B-field associated with α and setB0 :=r·B ∈H²(A,Z). We set a:=HA·B0 ∈Z and b := ^(B₂⁰⁾² ∈Z.

Let us further fix integersc and s and consider the Mukai vector v:= (r, c·H_A+r·B, s)∈H_alg^∗ (A, B,Z).

We assume

v² ≥4 and gcd(r,2cd+a) = 1.

We consider the Mukai vectors `,δ ∈v^⊥⊂H_alg^∗ (A, B,Z) given by

`:=−(0, r·H_A,2cd+a) and δ :=− r·v+v²·w , where w:= (0,0,1).

Finally, we let K := K_(A,α),HA(v) be the generalized Kummer variety arising from the moduli space of H-Gieseker stable sheaves (see Remark4.5). On K, the divisor

D_u :=ϑ(u·`−δ)∈NS(K)_R

is ample for allusufficiently large ([BM1, Corollary 9.14]). The following result gives an explicit lower bound.

Proposition 4.10. In the above notation, the divisor D_u on K is ample for all u > r^v₂². Moreover, if v² ≤2r−2, then D∞:=ϑ(`) is also ample.

Proof. We are looking at Mukai vectorsa= (α, D, β)∈H_alg^∗ (A, B,Z) satisfying a² ≥0 and (a,v)∈n

1, . . . ,v² 2

o .

To estimate whenD_u is ample, for u6=∞, we can further consider those Mukai vectors asuch that

(a,`)·(a,δ)≥1.

By using [MYY, YY, Y2],⁹ to prove that D_u is ample for allu > r^v₂², we have to show

(7) (a,δ)

(a,`)

≤? rv² 2 . Explicitly, we have

(a,`) = H_A·D,e De :=r·D−α(c·H_A+r·B) (a,δ)² =v²De²+r² (a,v)²−v²a²

.

9This is not explicitly stated inloc. cit. It is the version for abelian surfaces of [BM2, Theorem 12.1], with a similar proof.

Hence, (7) can be written as

(8) v²De²+r² (a,v)² −v²a^{2 ?}

≤r²(v²)²

4 (H_A·D)e ². If De² ≤0, (8) follows immediately from the inequalities

(9) 1≤(a,v)≤ (v²)

2 and a² ≥0.

Hence, we assumeDe² ≥2. By the Hodge Index Theorem (HA·D)e ² ≥H_A²De², it is enough to show the inequality

(10) v²De²+r² (a,v)²−v²a^{2 ?}

≤r²(v²)²

4 H_A²De².

By using again (9) and dividing by v² ≥4, to prove (10), we have to show the inequality

(11) De²

r²v²

4 H_A² −1 _?

≥r²v² 4 . Since De² ≥2, checking (11) amounts to showing the inequality

r²v²

4 2H_A² −1 ^?

≥2, which holds true since r ≥1,v² ≥4, and H_A² ≥2.

To show that D∞ is ample, we only have to show that there are no Mukai vectors a as above such that (a,`) = 0. This can be done by a direct computation. We use instead a quicker, more geometric, argument. The divisor class D_∞ is associated with the Donaldson–

Ulhenbeck–Yau compactification: it induces the birational morphism K_(A,α),HA(v)−→K_(A,α),H^µ

A(v).

To show that D∞ is ample, it is enough to show that all slope-semistable torsion-free sheaves are actually slope-stable vector bundles.

To this end, we first note the equality

H_A·(c·H_A+r·B) = 2cd+a.

Hence, since gcd(r,2cd+a) = 1, there are no properly slope-semistable torsion-free sheaves.

The assumption v² ≤2r−2 then guarantees that all torsion-free sheaves are actually vector bundles: indeed, the Mukai vector v⁰ of the double dual would satisfy (v⁰)² < 0, which is

impossible.

In the special case r = 1, we obtain the following generalization of Corollary 2.2. Let n ≥ 1 and v = (1,0,−n−1), and consider the generalized Kummer variety Kum_n(A) which is isomorphic to K in this case. Explicitly, we have

`=−(0, H_A,0) and δ :=−(1,0, n+ 1).

ThenH_n :=D_∞=ϑ(`) is big and nef, but not ample. We letδ :=ϑ(δ); it is half the restriction of the divisor on the Hilbert scheme parameterizing nonreduced subschemes.

Corollary 4.11. On any generalized Kummer variety Kum_n(A), the class aH_n−δ is ample for all real numbers a > n+ 1.

Theorem 1.3 follows immediately from Corollary 4.11.

Recall from Remark4.5that ifn= 1, then Kum₁(A) = Kum(A), the notation forH =H₁ is coherent with above, andδ = ¹₂P16

i=1E_i. Hence, the statement is exactly Corollary 2.2. For n≥2, it is optimal since NS(Kum_n(A))'NS(A)⊕Z·δ. Moreover, if there exists a divisor D withD² = 0 andD·H_A= 1, we get a Mukai vectora= (1, D,0) witha² = 0 and (a,v) = n+1.

Thus the class (n+ 1)H_n−δ is nef but not ample on Kum_n(A) in that case.

K3-type. Let r, e ≥ 1. Let (S, α, H) be a polarized twisted K3 surface, with H² = 2e and α^r =id. We let B ∈ H²(S,Q) be a B-field associated with α, B₀ :=r·B ∈ H²(S,Z). We let a:=H·B₀ ∈Z and b := ^(B₂⁰⁾² ∈Z.

Let us further fix integersc and s and consider the Mukai vector v:= (r, c·H+r·B, s)∈H_alg^∗ (S, B,Z).

We assume

v² ≥0 and gcd(r,2ce+a) = 1.

We setw:= (0,0,1) and consider the Mukai vectors`,δ ∈v^⊥ ⊂H_alg^∗ (S, B,Z) given by

` :=−(0, r·H,2ce+a) and δ :=− r·v+v²·w .

Finally, we letM :=M_(S,α),H(v) be the moduli space of H-Gieseker stable sheaves on S with Mukai vector v. On M, the divisor

D_u :=ϑ(u·`−δ)∈NS(M)_R

is ample for all u sufficiently large. The following result is the analog of Proposition 4.10 for K3 surfaces.

Proposition 4.12. In the above notation, the divisor D_u on M is ample for all u > r

r(v²)²

4 + 2v². Moreover, if v² ≤2r−4, then D_∞:=ϑ(`) is also ample.

Proof. The proof goes along the same lines as the proof of Proposition4.10. The only difference is that, to show thatD_u is ample, we use [BM2, Theorem 12.1]. Thus, we are looking at Mukai vectors a= (α, D, β)∈H_alg^∗ (S, B,Z) satisfying

a² ≥ −2 and (a,v)∈n

1, . . . ,v² 2

o .

This is the reason why the estimate on u is more complicated, but the rest of the proof is

analogous.

As in the generalized Kummer case, by considering the case r = 1 (and so B = 0) and c= 0 in Proposition 4.12, we obtain immediately Theorem 1.2.

Explicit polarized families. To apply Theorem4.8, we need to make sure that starting from a polarized family of K3 or abelian surfaces, we can take a twist along the whole family, after taking a finite cover. This is the content of the following lemma.

Lemma 4.13. LetT be an integral quasi-projective scheme and let S →T be a smooth family of K3 or abelian surfaces. Let 0 ∈T be a closed point and let α0 ∈ Br(St) be a Brauer class on S0. Then there exists an integral quasi-projective scheme T⁰, a generically finite projective surjective morphism ϕ: T⁰ T, a flat family of Azumaya algebras AT⁰ on the base change ST⁰ → T⁰, and a closed point 0⁰ ∈ T⁰ with ϕ(0⁰) = 0, such that the class of the Azumaya algebra AT⁰|₀⁰ is exactly α₀.

Proof. Let us briefly sketch a proof of this well-known fact.¹⁰ We consider the relative moduli space f: T → T of Azumaya algebras over T. On the special fiber at 0 ∈ T, we can choose an Azumaya algebraA0 of minimal rank with class α0. Hence,A0 is a slope-polystable vector bundle with trivial first Chern class, so it deforms along over any complex deformation of the K3 or abelian surface. This shows thatf is dominant in a neighborhood of A0. The scheme T⁰

is then given by a generic multisection of f through A0.

Example 4.14. To show that F2e⁰ contains a complete curve for e∈ {18,32,36,50,54}, we can apply Proposition 4.10 to the divisor D∞, together with Lemma 4.13. Indeed, when v² = 4, thenK is a K3 surface and D∞ is ample with self-intersectionD²_∞= 4dr². We then take c= 0, s= 0, a= 1, b= 2. For e= 18, we further taked= 1, r= 3. For e= 32, we taked= 1, r= 4.

Fore= 36, we take d= 2, r = 3. Fore= 50, we take d= 1, r= 5. For e= 54, we taked= 3, r= 3.

Example 4.15 (Mukai, O’Grady). Consider a polarized K3 surface (S, H) of degree 2e and a Brauer classα ∈Br(S) of order r≥2. We further assume that there is a B-fieldB ∈H²(S,Q), B₀ := r·B ∈ H²(S,Z), associated with α such that a = H·B₀ = 1 and b = ^B₂⁰² = r. If we consider the Mukai vector

v= (r, r·B,1),

the moduli space M := M(S,α),H(v) is a smooth projective K3 surface and the divisor D∞ is ample on M of degree 2r²e.

Given a polarized K3 surface of degree 2e, we can always find a Brauer class α ∈Br(S) satisfying the above conditions. Hence, by applying Lemma4.13, we obtain a diagram of finite morphisms

Fe,r⁰ g

""

f

}}F⁰2e F⁰_2r²_e

wheref is surjective andg is dominant. In particular, given a complete subvariety inF_2e⁰, this gives us a complete subvariety inF_2r⁰²_e, for allr≥2. The morphismgis not surjective: it misses some irreducible components of the Heegner divisor associated with a class of square −2r².

These morphisms and their degrees can be better studied by using lattice theory and period domains. Indeed, as proved in [O2, Appendix], the morphism g is an open embedding and the morphism f can be extended to quasi-polarized K3 surfaces to give a finite surjective morphism

f:F2r²e −→F2e

10The general theory of moduli spaces of polarized twisted K3 surfaces has recently been studied in [Br].