Abstract. We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.By a theorem of Hormander [6], if Xx, . . . , X¡ are vector fields on a manifold N with the property that their commutators up to a certain order span the tangent space at every point, then the differential operator D = 2 X2 j is hypoelliptic; that is, the solutions of the equation Df = g with g G C°°, are also C°°. Especially interesting is the case where N is a nilpotent Lie group and the Xj's are generators of its Lie algebra; in particular, these "sublaplacians" play a central role in the Rothschild-Stein regularity theory of second order hypoelliptic equations [9].In this paper we introduce a class of step-2 nilpotent groups (type H) whose standard sublaplacians are shown to admit explicit fundamental solutions of an elementary form. This phenomenon had been originally observed in the case of the Heisenberg group by Folland [4] (cf. also [7]). For the class introduced here, heuristic evidence suggests that it should be "the largest" with that property; in any case, it yields the previously known examples together with infinitely many new ones. The analytic hypoellipticity of the differential operators involved follows as an immediate consequence of the formula for their fundamental solution.Groups of type H arise in a natural manner from the so-called compositions of quadratic forms (or "orthogonal multiplications"), a notion that has found various applications in algebra and topology [2], [3], [8]; we use a classical result from this theory to give a measure of the size of the class of sublaplacians so obtained. Besides, the relationship between that notion and nilpotent groups may have some interest on its own.Independently, B. Helffer found similar solutions for a class of groups that turned out to be equivalent to ours (personal communication). Finally, we wish to thank L. Rothschild and the reviewer for their useful suggestions, in particular those concerning the question of analytic hypoellipticity discussed at the end of §2.
Let
S
S
be a nonsingular complex algebraic variety and
V
\mathcal {V}
a polarized variation of Hodge structure of weight
2
p
2p
with polarization form
Q
Q
. Given an integer
K
K
, let
S
(
K
)
{S^{(K)}}
be the space of pairs
(
s
,
u
)
(s,u)
with
s
∈
S
s \in S
,
u
∈
V
s
u \in {\mathcal {V}_s}
integral of type
(
p
,
p
)
(p,p)
, and
Q
(
u
,
u
)
≤
K
Q(u,u) \leq K
. We show in Theorem 1.1 that
S
(
K
)
{S^{(K)}}
is an algebraic variety, finite over
S
S
. When
V
\mathcal {V}
is the local system
H
2
p
(
X
s
,
Z
)
{H^{2p}}({X_s},\mathbb {Z})
/torsion associated with a family of nonsingular projective varieties parametrized by
S
S
, the result implies that the locus where a given integral class of type
(
p
,
p
)
(p,p)
remains of type
(
p
,
p
)
(p,p)
is algebraic.
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