K3S - Rice Universitysf31/WAGS_poster.pdf · 2019. 8. 25. · SHEAVES ON A K3SURFACE AND GALOIS...
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S HEAVES ON A K3S URFACE AND G ALOIS R EPRESENTATIONS S ARAH F REI · U NIVERSITY OF O REGON · [email protected] M OTIVATING G OAL Let S be a K3 surface defined over a finite field, and let M 1 and M 2 be fine moduli spaces of sta- ble sheaves on S of equal dimension. We want to establish that the zeta function of M 1 is equal to that of M 2 : Z (M 1 ,t)= Z (M 2 ,t). O VERVIEW • Moduli spaces M of stable sheaves on a K3 sur- face S are deformation equivalent to the Hilbert scheme S [n] parameterizing dimension zero sub- schemes of length n = 1 2 dim M in S . • While two such moduli spaces M 1 and M 2 are deformation equivalent, they need not be bira- tional, so it is surprising that these two varieties should have the same zeta function. • The proof of the main result below is straight- forward for i =1 [1]. We transport the result to i> 1 by showing: The Galois action on H 2 (M ¯ k , Q ‘ ) and the ring structure on H * (M ¯ k , Q ‘ ) determine the Galois action on all of H * (M ¯ k , Q ‘ ). • This result parallels a result of Looinjenga and Lunts [2] on Hodge structures: For a compact hyperkähler manifold X , the algebra structure on H * (X, Q) and the Hodge structure on H 2 (X ) determine the Hodge structure on all of H * (X ). T HE MODULI SPACE • The moduli space M := M H (S, v ) parameterizes Gieseker H -semistable sheaves on S with Mukai vector v . • When v is primitive and H is generic, M is a smooth projective variety over k [1, Theorem 2.4]. R EFERENCES [1] François Charles. Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for K 3 surfaces, and the Tate conjecture. Ann. of Math. (2), 184(2):487–526, 2016. [2] Eduard Looijenga and Valery A. Lunts. A Lie algebra attached to a projective variety. Invent. Math., 129(2):361–412, 1997. [3] Eyal Markman. On the monodromy of moduli spaces of sheaves on K 3 surfaces. J. Algebraic Geom., 17(1):29–99, 2008. [4] Mikhail Verbitsky. Cohomology of compact hyperkähler manifolds and its applications. Geometric and Functional Analysis, 6(4):601–611, 1996. S KETCH OF THE PROOF Step 1: Given two Mukai vectors v 1 ,v 2 ∈ N (S ), we show that v ⊥ 1 ∼ = v ⊥ 2 as Galois representations. Step 2: Markman [3] constructs a graded ring R(v ) generated by v ⊥ in degree 2 and by M 2i ∼ = e H * (S ¯ k , Q ‘ )= v ⊥ ⊕hv i in degree 2i, and a ring homomorphism h : R(v ) → e H * (M ¯ k , Q ‘ ) which sends v ⊥ in degree 2 isomorphically onto H 2 (M, Q ‘ (1)) via θ v,‘ , and for α ∈ M 2i , h(α)=[π 2* (u v · π * 1 (α)] 2i , where u v , defined in [3, Eq. (27)], is a normalization of v (U ) which Markman shows is invariant under the action of a finite-index subgroup of the monodromy group. The maps π 1 and π 2 are the projections from S ×M onto S and M, respectively, and U is the universal sheaf on S ×M. Extending the results of Markman to étale cohomology, we show that h is surjective. This requires lifting S to characteristic zero and applying the smooth base change theorem to the relative moduli space. Step 3: The Beauville-Bogomolov form q on H 2 (M ¯ k , Q ‘ ) gives rise to a natural action of SO(q ) on e H * (S ¯ k , Q ‘ ) via the isomorphism v ⊥ ∼ = H 2 (M ¯ k , Q ‘ (1)) and by extending the action trivially to hv i. Looi- jenga and Lunts [2], following Verbitsky [4], construct a natural action of so(q ) on the entire cohomology ring e H * (M ¯ k , Q ‘ ) which integrates and descends to an action of SO(q ). Remark: Over C, SO(q ) is the connected component of the Zariski closure of the monodromy group of M. Markman [3] shows that these two SO(q ) actions agree. Step 4: Decompose R(v 1 ) into irreducible O(q ) representations. For each 1 ≤ i ≤ dim M 1 = dim M 2 , there exists an O(q )-invariant subspace V such that: V R(v 1 ) H 2i (M 1, ¯ k , Q ‘ (i)) R(v 2 ) H 2i (M 2, ¯ k , Q ‘ (i)) ∼ ∼ h 1 ∼ ˜ h 2 h 2 Since Gal( ¯ k/k ) ⊂ O(q ), V is also Galois invariant, and hence the Gal( ¯ k/k )-actions on H 2i (M 1, ¯ k , Q ‘ (i)) and on H 2i (M 2, ¯ k , Q ‘ (i)) are equal to the action on V . F UTURE R ESEARCH We aim to prove the corresponding result for moduli spaces of sheaves on an abelian surface. In this case, there is an added difficulty because the Galois action on the second cohomology no longer determines the Galois action on the higher cohomology groups. M AIN R ESULT We deduce the motivating goal from the follow- ing more general result. Theorem. Suppose that S is a K3 surface de- fined over an arbitrary field k , and let ‘ be a prime number which is invertible in k . Then for all i, H 2i (M 1, ¯ k , Q ‘ ) ∼ = H 2i (M 2, ¯ k , Q ‘ ) as Gal( k/k )-representations. M UKAI LATTICE AND VECTORS • The ‘-adic Mukai lattice of S is the Gal( ¯ k/k )- module e H * (S ¯ k , Z ‘ ) := H 0 (S ¯ k ,Z ‘ ) ⊕ H 2 (S ¯ k , Z ‘ (1)) ⊕ H 4 (S ¯ k , Z ‘ (2)) endowed with the Mukai pairing. • Let ω be the numerical equivalence class of a point on S ¯ k .A Mukai vector on S is an element v ∈ N (S ) := Z ⊕ NS(S ) ⊕ Zω. We consider this as an element of e H * (S ¯ k , Z ‘ ) un- der the natural inclusion. • The Mukai vector of a coherent sheaf F on S is v (F ) := ch(F ) p td(S ) ∈ e H * (S ¯ k , Z ‘ ). • Charles [1] shows θ v,‘ : v ⊥ → H 2 (M ¯ k , Q ‘ (1)), given by θ v,‘ (α)= π 2* (v (U ) · π * 1 (α)) is a Gal( ¯ k/k )- equivariant isomorphism.
Transcript of K3S - Rice Universitysf31/WAGS_poster.pdf · 2019. 8. 25. · SHEAVES ON A K3SURFACE AND GALOIS...
SHEAVES ON A K3 SURFACE AND GALOIS REPRESENTATIONSSARAH FREI · UNIVERSITY OF OREGON · [email protected]
MOTIVATING GOALLet S be a K3 surface defined over a finite field,and letM1 andM2 be fine moduli spaces of sta-ble sheaves on S of equal dimension. We want toestablish that the zeta function ofM1 is equal tothat ofM2:
Z(M1, t) = Z(M2, t).
OVERVIEW•Moduli spacesM of stable sheaves on a K3 sur-face S are deformation equivalent to the Hilbertscheme S[n] parameterizing dimension zero sub-schemes of length n = 1
2 dimM in S.
•While two such moduli spacesM1 andM2 aredeformation equivalent, they need not be bira-tional, so it is surprising that these two varietiesshould have the same zeta function.
• The proof of the main result below is straight-forward for i = 1 [1]. We transport the result toi > 1 by showing:
The Galois action on H2(Mk̄,Q`) and thering structure on H∗(Mk̄,Q`) determinethe Galois action on all of H∗(Mk̄,Q`).
• This result parallels a result of Looinjenga andLunts [2] on Hodge structures:
For a compact hyperkähler manifold X ,the algebra structure on H∗(X,Q) and theHodge structure on H2(X) determine theHodge structure on all of H∗(X).
THE MODULI SPACE• The moduli space
M :=MH(S, v)
parameterizes Gieseker H-semistable sheaves onS with Mukai vector v.
• When v is primitive and H is generic, M is asmooth projective variety over k [1, Theorem 2.4].
REFERENCES
[1] François Charles. Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for K3 surfaces, and the Tate conjecture. Ann. ofMath. (2), 184(2):487–526, 2016.
[2] Eduard Looijenga and Valery A. Lunts. A Lie algebra attached to a projective variety. Invent. Math., 129(2):361–412, 1997.
[3] Eyal Markman. On the monodromy of moduli spaces of sheaves on K3 surfaces. J. Algebraic Geom., 17(1):29–99, 2008.
[4] Mikhail Verbitsky. Cohomology of compact hyperkähler manifolds and its applications. Geometric and Functional Analysis, 6(4):601–611, 1996.
SKETCH OF THE PROOF
Step 1: Given two Mukai vectors v1, v2 ∈ N(S), we show that v⊥1 ∼= v⊥2 as Galois representations.
Step 2: Markman [3] constructs a graded ring R(v) generated by v⊥ in degree 2 and byM2i∼= H̃∗(Sk̄,Q`) = v⊥ ⊕ 〈v〉 in degree 2i, and a ring homomorphism
h : R(v)→ H̃∗(Mk̄,Q`)
which sends v⊥ in degree 2 isomorphically onto H2(M,Q`(1)) via θv,`, and for α ∈M2i,
h(α) = [π2∗(uv · π∗1(α)]2i,
where uv , defined in [3, Eq. (27)], is a normalization of v(U) which Markman shows is invariant underthe action of a finite-index subgroup of the monodromy group. The maps π1 and π2 are the projectionsfrom S ×M onto S andM, respectively, and U is the universal sheaf on S ×M.Extending the results of Markman to étale cohomology, we show that h is surjective. This requires liftingS to characteristic zero and applying the smooth base change theorem to the relative moduli space.
Step 3: The Beauville-Bogomolov form q on H2(Mk̄,Q`) gives rise to a natural action of SO(q) onH̃∗(Sk̄,Q`) via the isomorphism v⊥ ∼= H2(Mk̄,Q`(1)) and by extending the action trivially to 〈v〉. Looi-jenga and Lunts [2], following Verbitsky [4], construct a natural action of so(q) on the entire cohomologyring H̃∗(Mk̄,Q`) which integrates and descends to an action of SO(q).
Remark: Over C, SO(q) is the connected component of the Zariski closure of the monodromy group ofM. Markman [3] shows that these two SO(q) actions agree.
Step 4: Decompose R(v1) into irreducible O(q) representations. For each 1 ≤ i ≤ dimM1 = dimM2,there exists an O(q)-invariant subspace V such that:
V
R(v1) H2i(M1,k̄,Q`(i))
R(v2) H2i(M2,k̄,Q`(i))
∼
∼
h1
∼h̃2
h2
Since Gal(k̄/k) ⊂ O(q), V is also Galois invariant, and hence the Gal(k̄/k)-actions on H2i(M1,k̄,Q`(i))
and on H2i(M2,k̄,Q`(i)) are equal to the action on V .
FUTURE RESEARCHWe aim to prove the corresponding result formoduli spaces of sheaves on an abelian surface.In this case, there is an added difficulty becausethe Galois action on the second cohomology nolonger determines the Galois action on the highercohomology groups.
MAIN RESULTWe deduce the motivating goal from the follow-ing more general result.
Theorem. Suppose that S is a K3 surface de-fined over an arbitrary field k, and let ` be aprime number which is invertible in k. Thenfor all i,
H2i(M1,k̄,Q`) ∼= H2i(M2,k̄,Q`)
as Gal(k/k)-representations.
MUKAI LATTICE AND VECTORS
• The `-adic Mukai lattice of S is the Gal(k̄/k)-module
H̃∗(Sk̄,Z`) := H0(Sk̄,Z`)⊕H2(Sk̄,Z`(1))
⊕H4(Sk̄,Z`(2))
endowed with the Mukai pairing.
• Let ω be the numerical equivalence class of apoint on Sk̄. A Mukai vector on S is an element
v ∈ N(S) := Z⊕NS(S)⊕ Zω.
We consider this as an element of H̃∗(Sk̄,Z`) un-der the natural inclusion.
• The Mukai vector of a coherent sheaf F on S is
v(F ) := ch(F )√
td(S) ∈ H̃∗(Sk̄,Z`).
• Charles [1] shows
θv,` : v⊥ → H2(Mk̄,Q`(1)),
given by θv,`(α) = π2∗(v(U) ·π∗1(α)) is a Gal(k̄/k)-equivariant isomorphism.
