Mukai lattice of a generalised Kummer variety Claudio Onorati

Mukai lattice of a generalised Kummer variety Claudio Onorati

Mukai lattice of a generalised Kummer variety
Claudio Onorati
Department of Mathematical Sciences,
University of Bath
Advisor: Prof. Gregory Sankaran
c.onorati@bath.ac.uk
Abstract
We will give a different and intrinsic characterisation of the Mukai lattice for a generalised Kummer variety. It was shown by Markman ([Mar2]) that for a moduli space
of sheaves M on a K3 surface, there exists an isometry of lattices between the Mukai lattice and a quotient of the fourth cohomology group H 4 (M, Z). This fact
realizes the Mukai lattice as an intrinsic object in the geometry of an irreducible holomorphic symplectic manifold. This new lattice is invariant under the monodromy
group and provides a preferred class of primitive embeddings of H 2 (M, Z) into the Mukai lattice. Currently, we are generalising the same computations in the case of
a generalised Kummer variety. The subject of this poster is the description of the isometry between these two lattices.
Let A be an abelian surface and H an ample class on A. Let K(A) = K0(A) be the
Grothendieck group of A, which we identify
with the even cohomology ring H ev (A, Z),
endowed with the Mukai pairing (u, v) :=
−χ(u∨ ⊗ v). Then M = MH (v) is the moduli space of (Gieseker) H-stable sheaves
on A with invariants fixed by v ∈ K(A). We
suppose that H and v satisfy hypotheses
such that M is a smooth projective variety
of dimension v 2 + 2 and there exists a universal sheaf E˜ on the product M ×A. By an
important result of Yoshioka ([Yos]), there
exists an Albanese map (Aˆ is the abelian
surface dual to A)
a : M −→ A × Aˆ
such that the fibre K = KH (v) := a−1(0, 0)
is an irreducible holomorphic symplectic
manifold of dimension m = v 2 − 2. This
construction is independent of the choice
ˆ We will refer to K
of the point (0, 0) ∈ A×A.
as a generalised Kummer variety. The restriction of the universal sheaf E˜ on K×A is
still an important object and we will refer to
˜ K×A. With this notation we get
it as E := E|
a natural homomorphism of Grothendieck
groups (here pi is the projection from K ×A
to the ith factor)
u : K(A) −→ K(K)
sending an element
x to u(x)
=
!
∨
Endowing the secp1,! p2(x ) ⊗ [E] .
ond cohomology group H 2(K, Z) with the
Bogomolov-Beauville pairing, the morphism
θ : v ⊥ → H 2(K, Z),
sending x to c1(u(x)), is an isometry of
Hodge structures ([Yos]).
In the same way, we can try to define a
map from K(A) to H 4(K, Z) by sending x
to c2(u(x)). Nevertheless, in this way the
resulting morphism is not a group homomorphism. Let Q4(K, Z) be the quotient of
H 4(K, Z) by the image of the cup product
map from H 2(K, Z). The composition of
the above map with the projection to the
quotient yields a morphism
φ : K(A) −→ Q4(K, Z).
(1)
A straightforward computation shows that
this map is a homomorphism of groups
(and it is independent of the choice of a
universal sheaf). Our goal is to show that
this map is an isomorphism. Later we will
also need the first K-theory groups K1(A)
and K1(K).
Generators for the cohomology ring.
Following the method of the diagonal used
by Ellingsrud and Strømme in their paper ”Towards the Chow ring of the Hilbert
scheme of P2”, we can give explicit generators for the cohomology ring H ∗(K, Z)
once we can express the (Poincare dual δ
of the) class of the diagonal ∆ ⊂ K × K as
a linear combination of polynomials with integral coefficients in the Chern classes of
the Kunneth
factors of the sheaf E. More
¨
explicitly, under the Kunneth
isomorphism
¨
the class of E in K0(K ×A) decomposes as
[E] =
n1
X
i=1
xi ⊗ ei +
n2
X
yj ⊗ f j
(2)
j=1
where {x1, · · · , xn1 } (resp. {y1, · · · , yn2 })
is a basis of K0(A) (resp. K1(A)) and ei
(resp. fj ) are elements in K0(K) (resp.
K1(K)). Following an argument of Markman ([Mar1], Theorem 1), we can compute
"
#!
L
O
!
!
δ = cm −π13,! π12
(E)∨
π23
(E)
(3)
where πij are the projections from K × A ×
K to the product of the ith and j th factor.
Substituting the decomposition (2) in the
expression above and expanding the factors, we can state
X
δ=
p∗2 (αk ) ∪ p∗1 (βk ),
(4)
k
where αk (resp. βk ) are polynomials with
integral coefficients in the Chern classes of
the xi and yj (resp. ei and fj ). Finally, it is a
general fact that every class x ∈ H ∗(K, Z)
can be written as
x = q1∗ (δ ∪ q2∗(x)) ,
(5)
where qi is the projection from K ×K. Substituting the expression (4) in the (5) above,
we get that the cohomology ring is generated by the Chern classes of the ei and fj .
With some easy computations, we can further show that
H ev (K, Z) is generated by cj (u(xi)).
(6)
In particular, H 4(K, Z) is generated by
c2(u(xi)). Moreover, these classes do not
vanish in the quotient Q4(K, Z) and so the
map φ defined in (1) is surjective.
Observe that, thanks to a result of Bogomolov, the cup product induces an injective homomorphism Sym2(H 2(K, Q)) →
H 4(K, Q), whose quotient is exactly
Q4(K, Q). Using the Goettsche formula for
a generalised Kummer variety, we can easily see that for m ≥ 6 the dimension of
Q4(K, Q) (and hence the rank of Q4(K, Z))
is equal to
dim H 4(K, Q) − dim Sym2(H 2(K, Q)) = 8.
Since the rank of the Mukai lattice of a generalised Kummer variety is known to be 8,
the map φ must be an isomorphism.
Conclusions.
As a consequence,
Q4(K, Z) is a free abelian group of rank 8.
Pushing forward the Mukai pairing on K(A)
through φ, we can define an even unimodular integral nondegenerate symmetric
bi
linear form B(ξ, η) := φ−1(ξ), φ−1(η) on
Q4(K, Z). With this definition, φ extends to
an isometry of lattices. The composition
e = φ ◦ θ−1 : H 2(K, Z) ,→ Q4(K, Z)
is a primitive embedding of the generalised
Kummer lattice into the Mukai lattice.
Coming next. In the K3[n]-case the pairing B is invariant under the monodromy
group and e generates a rank 1 Mon(M )-
submodule of Hom H 2(K, Z), Q4(K, Z)
which is a Hodge-sub-structure of type
(1, 1). We claim that this still holds true in
the generalised Kummer case.
References
[Mar1] E. Markman, Generators of the cohomology ring of moduli spaces of
sheaves on symplectic surfaces, 2001.
[Mar2] E. Markman, Integral constrains
on the monodromy group of the Hy¨
perkahler
resolution of a symmetric
product of a K3 surface, 2008.
[Yos] K. Yoshioka, Moduli spaces of stable
sheaves on abelian surfaces, 2001.
´
´
´
´ Leuven, 1-5 June 2015
GAeL XXIII, Geom
etrie
Algebrique
en Liberte,