10 Variations of Hodge Structure
this means that for every curve X and every universal family of deformations
X -> В of X, the set of points b e В such that Xb is hyperelliptic is a proper
analytic subset of В if g > 2, and the set of points b e В such that Xb is trigonal
or isomorphic to a smooth plane curve of degree 5 is a proper analytic subset
of В if g > 5.
Theorem 10.24 (See Arbarello et al. 1985)
(a) (Noether) Let X be a non-hyper elliptic curve. Then the map given by the
x : Я°(Х, Kxf1 -> H°(X, Kf)
(b) (Petri) Let X be a curve which is non-hype relliptic, non-trigonal and not
isomorphic to a planar quintic. Then X is determined by
Ker x : Я°(Х, Kxf2 -» Я°(Х, Kf)
in the following way: the canonical map фкх :X->F8~X is an embedding
since X is not hyperelliptic, and the symmetric elements of Ker x are exactly
the homogeneous polynomials of degree 2 over?8~x which vanish on X; when X
is neither trigonal nor isomorphic to a planar quintic, X С ?8~х is isomorphic
to the algebraic subscheme or complex submanifold defined by these equations.
These algebraic statements now give us the following results on the period map
Corollary 10.25 (Infinitesimal Torelli theorem for curves) Let X be a non-
hyperelliptic curve. Then the local period map
V : В
is an embedding at the point 0 € В corresponding to X.
Proof By theorem 10.24 and lemma 10.22, dVl is injective at 0 when X is not
Corollary 10.26 Ifg>5,a generic curve X is determined by its infinitesimal
variation of Hodge structure and by the isomorphism
Я1H(Х) (H X, C) #1i0(X))*.
In particular, assume that we have two curves X and X , an isomorphism
compatible with the intersection forms i : Hl(X, C) HX(X , C), and a germ
of isomorphisms j : (B,0) (Bf,0f) between the bases of the local universal
deformations ofX and X respectively, giving an identification of the variations
of Hodge structure in the neighbourhood of X and ofX :
Vх : В
Vх : B
Then X and X are isomorphic.
Grass(g, Hx(Xf, Q).
Proof The first statement follows from the fact that the differential dVx at the
point 0 6 В corresponding to X gives a symmetric map (relative to the Serre
duality Hh0(X) (HX(X, C) HX °(X))*):
The isomorphism Я1 0^) (Hl(X, С) Я1>0(Х))* induced by the intersection
form on HX(X, C) makes it possible to dualise this map to a symmetric map
x : HX °(X) °(X) -» W*. Lemma 10.22 and theorem 10.24 then sho>v
that if X is not hyperelliptic, trigonal or a planar quintic, X can be identified with
the subscheme of Р(Я10(Х)*) defined by the symmetric elements of Ker x.
The second result follows immediately from this, since by differentiation,
the commutative diagram above gives an identification of the infinitesimal vari-
variations of Hodge structures at the points b and j(b), Vb e B, compatible with
the duality isomorphisms A0.10) since i preserves the intersection form, i.e. a
commutative diagram where the vertical arrows are isomorphisms:
Uom(Hx0(Xb), Hx(Xb, C) Hx<°(Xb))
But then for generic b, we must have Xb Xjqj), since Xb and Xj^) are
determined by Ker [ib and Ker x^^) respectively. It then follows that Xb X^)
for every b e B. (The moduli space of the curves is separated (Deligne &
Mumford 1967).) D
The result shown above is the generic Torelli theorem for curves of genus > 5.
It says essentially that the global period map, which to a curve X associates the
polarised Hodge structure on Я1 (X, Z), is of degree 1 on its image. In the second
volume, we will prove a similar statement due to Donagi for most of the families