Hyper-Kahler versus Calabi-Yau manifolds andtheir Chow groups

Claire Voisin

College de France

Sanya, December 19, 2016

Kahler manifolds

• Complex manifold of dimension n: topological space X with localholomorphic charts, i.e. covering open sets Ui ⊂ X with homeomorphismsci : Ui ∼= Vi, Vi open in Cn, such that the change of coordinates φj φ−1iare holomorphic.

Charts and change of coordinates

• The tangent at each point is endowed with a structure of C-vector space operator I, I2 = −Id, acting on TX,R, of almost complex structure.

• Notion of Hermitian metric on X. In local holomorphic coordinates:h =

∑ij hijdzi ⊗ dzj , with imaginary part ω =

∑ij ωijdzi ∧ dzj ,

ωij = −Imhij .

Definition. The Hermitian metric is Kahler if dω = 0.

Holonomy

• gij = Rehij , Riemannian metric Levi-Civita connection and paralleltransport.

• Other characterization of Kahler metric : the operator I is parallel.Hence characterization by holonomy: I is integrable and the metric isKahler iff the holonomy group is contained in U(n) (preserves g et I).

Example ( Riemann surfaces )

Any metric on an oriented Riemann surface is Kahler for a complexstructure I.

Example (Complex tori)

Torus R2n/Z2n ∼= (S1)2n. Any isomorphism R2n ∼= Cn endows the toruswith a complex structure. A Hermitian metric on Cn induces a flat Kahlermetric on the torus.

Example (projective space)

The Fubini-Study metric on CPn is Kahler, hence any projective complexmanifold is Kahler.

Kahler-Einstein metrics

• Canonical bundle. Holomorphic line bundle KX on X = cplx manifoldgenerated in local holomorphic coordinates by dz1 ∧ . . . ∧ dzn. KX istrivial iff there exists a nowhere zero holomorphic n-form on X.

• Hermitian metric h on X ⇒ Hermitian metric on KX , hence Chernconnection and its curvature. Kahlerian case ⇒ this is the Ricci curvatureof g = Reh.

Theorem (Yau 1978)

Let X be a compact Kahler manifold with trivial canonical bundle. Thereexist Kahler metrics on X which are Ricci flat. There exists exactly onesuch metric for each Kahler class.

• Equivalent conclusions : (a) The holomorphic n-form ηX sur X isparallel for these metrics.(b) The Kahler form ω satisfies ωn = ληX ∧ ηX for some constant λ.

• Viewpoint of holonomy: the holonomy preserves ηX , hence is containedin SU(n). This characterizes Kahler-Einstein metrics (with trivialcanonical bundle).

hyper-Kahler manifolds or symplectic-holomorphic manifolds

Definition: (hyper-Kahler manifold)

X = compact Kahler dimension 2n, X is simply connected and has a unique(up to a multiple) nonzero holomorphic 2-form σX =

∑ij αijdzi∧dzj which

is nowhere degenerate: detαij 6= 0 (that is σnX 6= 0) at any point.

• The 2n-form σnX trivializes the canonical bundle.

Theorem (Beauville 1984)

Let X be a symplectic-holomorphic manifold. Then the holomorphic2-form on X is parallel for the LC connection of any Kahler-Einsteinmetric which thus has holonomy contained in Sp(2n) ∩ U(2n).

Theorem (Quaternionic structure )

X hyper-Kahler, g = Reh= Kahler-Einstein metric.There is an action ofthe field K of quaternions on the tangent space at each point.∀It ∈ K, I2t = −1, the almost complex structure It on X is an integrablecomplex structure on X for which g is Kahler.

• Twistor family parameterized by pure quaternions of modulus 1.

Periods and deformations

• X compact Kahler, I=operator of almost complex structure,Newlander-Nirenberg integrability condition.

• Deformation of X = deformation of I preserving the integrabilitycondition, modulo action of diffeos of X. Small deformations of X remainKahler, the Hodge numbers are constant.

• Kuranishi: existence of universal family (smooth proper map) ofdeformations X → B, X0

∼= X, assuming H0(X,TX) = 0. One hasTB,0 = H1(X,TX) (Kodaira-Spencer). B can be singular.

• The period map. Choose a continuous trivialization X ∼= X ×B overB. Hk(X,C) ∼= Hk(Xt,C) contains Hk,0(Xt), of fixed dimension hk,0, foreach small deformation Xt of X. Hence P : B → Grass(hk,0, Hk(X,C)).

Theorem (Griffiths)

P is holomorphic. The differentialdP : H1(X,TX)→ Hom(Hk,0(X), Hk(X,C)/Hk,0(X)) identifies toy : H1(X,TX)→ Hom(Hk,0(X), H1(X,Ωk−1

X )) via the inclusion

H1(X,Ωk−1X ) = Hk−1,1(X) ⊂ Hk(X,C)/Hk,0(X).

Deformations of hyper-Kahler manifolds

• (Bogomolov-Tian-Todorov unobstructedness theorem.) KX trivial⇒ the deformations of X are parameterized by a smooth analytic space B.

• Griffiths computation ⇒ the local period map P (k = 2) is animmersion with image a analytic hypersurface ImP ⊂ P(H2(Xdiff ,C)).

Theorem (Beauville, Bogomolov)

There exists a quadratic form q on H2(Xdiff ,Q) such that P = openembedding in the quadric Q ⊂ P(H2(Xdiff ,C)) defined par q.

• q is topological: q(γ)2n = µX∫X γ

2n.

• Sketch of proof: f(γ) :=∫X γ

2n, homogeneous degree 2n in γ. As

σn+1Xt

= 0, f vanishes at order ≥ n along ImP, hence f = q′n for somequadratic form q′.

• Projective hyper-Kahler manifolds. An integral cohomology classα ∈ H2(X,Z) has type (1, 1) iff q(σX , α) = 0.

• L = ample line bundle on X, α = c1(L). Polarized defos of X= defospreserving the Hodge class α =hyperplane Dα in D defined by q(α, σ) = 0.

ExamplesBasic example ( K3 surfaces)

Let S ⊂ CP3 be defined by a general degree 4 equation f . Then

σS = ResS∑

i(−1)idX0∧...dXi...∧dX4

f is a nowhere 0 holomorphic 2-form onS. Hence S is symplectic holomorphic.

• Punctual Hilbert schemes. S = smooth complex surface. QuotientS(k) := Sk/Sk=singular variety of dim 2k.• Desingularization for surfaces (Fogarty): Hilbert scheme S[k] of0-dimensional subschemes of S of length k.

S[k] is a desingularization of S(k)

Theorem (Beauville 1984)

If S is a K3 surface, S[k] is hyper-Kahler.

• The holomorphic 2-form comes from the Sk-invariant 2-form∑

i pr∗i σS

on Sk.

O’Grady’s manifolds

• Moduli spaces of stable bundles on K3 surfaces. Mukai : They aresmooth, admit a nonegenerate closed holomorphic 2-form. When they arecompact, they are HK-manifols.

• Yoshioka: they are then deformations of varieties S[k], S a K3.

• When there exist semi-stable non stable sheaves with the giveninvariants, the moduli spaces of semistable sheaves are singular.

• O’Grady’s manifolds. S = K3 surface. Semi-stable sheaves on S derang 2, trivial determinant, c2 = 4.

• non stable examples: Iz1 ⊕ Iz2 , z1, z2 = subschemes of length 2.Thecorresponding moduli space is singular, of dimension 10.

Theorem (O’Grady 1998)

This moduli space admits a HK desingularization. One has b2(Y ) = 24, soY is not a deformation of S[5].

• Similar 6-diml example by O’Grady starting from an abelian surface.

Chow groups

• X projective variety over a field K. CHi(X) = Zi(X)/Zi(X)rat, where:

• Zi(X) = free abelian group on closed irreducible Z ⊂ X, dimZ = i (Zdefined over K, irreducible over K).

• Zi(X)rat = subgroup generated by div φ, φ ∈ K(W )∗, W ⊂ X ofdimension i+ 1.

• X of pure dimension n, CHp(X) = CHn−p(X).

• Fulton. X smooth, then CH∗(X) is a graded ring (intersectionproduct).

• Abel, Lefschetz and Serre. X complex projective. ThenCH1(X) = Pic (X) = Pic (Xan) is an extension of a finitely generatedabelian group (Neron-Severi) by an abelian variety.

• Mumford. If S is a surface over C with h2,0(S) 6= 0, then CH0(S) isinfinite dimensional: the very general fibers of S(n) → CH0(S) arecountable.

Results and conjectures for Chow groups of HK/CY varieties

Theorem (Beauville+Voisin 2004)

Let S be a projective K3 surface. There exists a canonical 0-cycleoS ∈ CH0(S), of degree 1, such that: (i) The intersection D ·D′ of twodivisors is proportional to oS in CH0(S) and (ii) c2(S) = 24oS .

Conjecture. Let X be a simply connected CY-n-fold. Then there exists acanonical 0-cycle oX ∈ CH0(X) s. t. for any D1, . . . , Dn ∈ CH1(X),D1 · . . . ·Dn = µoX in CH0(X).

• Hyper-Kahler case. Stronger conjecture by Beauville: the cycle classmap is injective on the subalgebra of CH(X)Q generated by CH1(X). Ingeneral CY case, we restrict to top degree.

• Evidence provided by work of Beauville, Voisin, Fu, Riess, Bazhov....

• Note that examples by Beauville show that his conjecture does not workin the general Calabi-Yau case, for cycles of intermediate dimensions. Thisis why conjecture above is restricted to 0-cycles.

Speculation

• Yau’s theorem : D ∈ PicX ample ⇒ there exists a Chern formrepresentative ωD with ωnD = λη ∧ η.

View now D ∈ CH1(X),• Conjectural statement. Dn = λoX in CH0(X).

• Obvious formal similarity.

• Can we say more? A cycle provides an integration current. Anequivalence relation Z = div φ, φ a rational function on W provides aLelong type relation between these currents. Is it possible to regularizethese currents canonically and say that η ∧ η represents the canonicalcycle?