Noether-Lefschetz locus for Beilinson-Hodge cycles onopen complete intersections
S. Saito
The infinitesimal method in Hodge theory is fruitful in various aspects of algebraic
geometry. The idea originates from Griffiths work [Gri] where the Poincaré residue representation of cohomology of a hypersurface played a crucial role in proving the infinitesimal
Torelli theorem for hypersurfaces. Since then many important applications of the idea
have been made in different geometric problems such as the generic Torelli problem andthe Noether-Lefschetz theorem for Hodge cycles and the study of algebraic cycles (see[G1], Lectures 7 and 8). In this lecture we explain that the method can be applied tostudy an analog of the Noether-Lefschetz theorem in the context of Beilinson’s Hodgeconjecture.
Let X ⊂ Pn be a smooth projective variety over C. Recall that the Hodge conjecture
for X predicts that the space of Hodge cycles in codimension q on X is generated bycohomology classes of algebraic subvarieties on X, namely the cycle class map from thethe Chow group to the singular cohomology of X:
CH q (X) ⊗ Q → H 2q (X, Q(q)) ∩ F q H 2q (X, C)√is surjective, where Q(q) = (2π −1)q Q ⊂ C and F ∗ denotes the Hodge filtration. LetH 2q (X, Q(q))triv ⊂ H 2q (X, Q(q)) ∩ F q H 2q (X, C),
be the (one-dimensional) Q-subspace generated by the class of the section on X of a linear
subspace of codimension q in Pn . It is called the space of trivial cycles.
Now let S be a non-singular quasi-projective variety over C and assume that we aregiven X ,→ PnS , an algebraic family over S of smooth projective varieties. Let Xt be thefiber of X over t ∈ S. Then the Noether-Lefschetz locus for Hodge cycles in codimension
q on X /S is defined to beqq 2q2q2q
SNL = {t ∈ S| F H (Xt , C) ∩ H (Xt , Q(q)) 6= H (Xt , Q(q))triv }.

It is the locus of such t ∈ S that there exist non-trivial Hodge cycles in codimension q onq
Xt and hence that the Hodge conjecture is non-trivial for Xt . One can prove SNL is theunion of countable number of (not necessarily proper) closed algebraic subsets of S.
Now we take S to be the moduli space of smooth hypersurfaces of degree d in P3 andq
X /S to be the universal family of hypersurfaces. Let SN L denote SN
L for q = 1. It is
the locus of those surfaces that possess curves which are not complete intersections of thegiven surface with another surface. The celebrated theorem of Noether-Lefschetz affirmsthat every component of SN L has positive codimension in S when d ≥ 4. M. Green andC. Voisin have refined it to show the following striking theorem:1

Theorem 0.1 ([G2], [G3], [V]) For every irreducible component T of SN L , codim(T ) ≥d − 3. If d ≥ 5, the only irreducible component of SN L having codimension d − 3 is thefamily of surfaces of degree d containing a line.
Let U be a (non-complete) smooth variety over C. Beilinson’s Hodge conjecture for
U predicts the surjectivity of the regulator maps:
regUq : CH q (U, q) ⊗ Q → H q (U, Q(q)) ∩ F q H q (U, C),
where CH q (U, q) denotes Bloch’s higher Chow group and H q (U, C) is endowed with themixed Hodge structure defined by Deligne (see [J] for the definitions). Taking a smoothcompactification U ⊂ X with Z = X \ U , a simple normal crossing divisor on X, we havethe following formula for the values of regUq on decomposable elements in CH q (U, q);
regUq ({g1 , . . . , gq }) =dgq
dg1
∧ ··· ∧∈ H 0 (X, ΩqX (log Z)) = F q H q (U, C),
g1gq∗
where {g1 , . . . , gq } ∈ CH q (U, q) is the products of gj ∈ CH 1 (U, 1) = Γ(U, OZar). Suchqelements of CH (U, q) is called decomposable and we letCH q (U, q)dec ⊂ CH q (U, q) ⊗ Q
denote the subspace generated by decomposable elements. The description of regUq on
non-decomposable elements are rather complicate. We put
H q (U, Q(q))triv := regUq (CH q (U, q)dec ) ⊂ H q (U, Q(q)) ∩ F q H q (U, C).
Now let S be a non-singular quasi-projective variety over C and assume given U → S, analgebraic family over S of non-complete smooth varieties. Let Ut be the fiber of U overt ∈ S. Then the Noether-Lefschetz locus for Beilinson-Hodge cycles on U/S is defined asqqq qq
SN
L = {t ∈ S| H (Ut , Q(q)) ∩ F H (Ut , C) 6= H (Ut , Q(q))triv }.

In this lecture we explain some results on Noether-Lefschetz locus for Beilinson-Hodge
cycles that are analogous to 0.1 proved in [AS1] and [AS2].

References[AS1]

M. Asakura and S. Saito, Beilinson’s Hodge conjecture for open complete intersections, Math. Zeit. 252 (2006), 251–273[AS2]

M. Asakura and S. Saito, Beilinson’s Hodge conjecture for open complete intersections, II, preprint[G1]

M. Green Infinitesimal methods in Hodge theory, Lecture Notes in Math., 1594
(1993), 1-92 Springer-Verlag[G2]

M. Green, A new proof of the explicit Noether-Lefschetz theorem, J. DifferentialGeometry, 27 (1988), 155-159
2

[G3]

M. Green, Components of maximal dimension in the Noether-Lefschetz locus,J. Differential Geometry, 29 (1989), 295-302[Gri]

P. Griffiths Periods of certain rational integrals:I and II, Ann. of Math., 90(1969), 460-541[J]

U.Jannsen, Mixed motives and Algebraic K-Theory, Lecture Notes in Math.,
1400 (1980), Springer-Verlag[V]

C. Voisin, Une précision concernant le théorem de Noether, Math. Ann., 280(1988), 605-611

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo,
153-8914 JAPANsshuji@msb.biglobe.ne.jp3