Introduction
0.1 Sums of squares
Let
$\mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
be the space of real polynomials of degree
${\leqslant}d$
in two variables. Consider the cone
$P_{d}\subset \mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
of polynomials that are positive semidefinite, i.e. that only take nonnegative values. Since an odd degree polynomial changes sign, we will assume that
$d$
is even.
It has been known since Hilbert that every polynomial
$f\in P_{d}$
is a sum of four squares in the field
$\mathbb{R}(x_{1},x_{2})$
of rational functions ([Reference HilbertHil93], see [Reference LandauLan03, p. 282]). If
$d\leqslant 4$
, Hilbert [Reference HilbertHil88] has shown the stronger statement that
$f$
is a sum of three squares in
$\mathbb{R}[x_{1},x_{2}]$
, but this does not extend in any way to degrees
$d\geqslant 6$
. Indeed, there exist polynomials
$f\in P_{6}$
that are not sums of squares of polynomials (Hilbert [Reference HilbertHil88]), and that are not sums of three squares of rational functions (Cassels, Ellison and Pfister [Reference Cassels, Ellison and PfisterCEP71]). Motzkin’s polynomial
$1+x_{1}^{2}x_{2}^{4}+x_{2}^{2}x_{1}^{4}-3x_{1}^{2}x_{2}^{2}$
is an example of both phenomena.
Sums of squares of polynomials form a closed cone
$\unicode[STIX]{x1D6F4}_{d}\subset P_{d}$
[Reference Berg, Christensen and JensenBCJ79, Theorem 3] and hence are not dense in
$P_{d}$
as soon as
$d\geqslant 6$
. At least when
$d=6$
, the structure of the cone
$\unicode[STIX]{x1D6F4}_{d}$
is very well understood [Reference BlekhermanBle13, Reference Blekherman, Hauenstein, Ottem, Ranestad and SturmfelsBHORS12].
Our goal is to complete the picture by studying the set
$Q_{d}\subset P_{d}$
of polynomials that are sums of three squares in
$\mathbb{R}(x_{1},x_{2})$
. It is easily seen to be a countable union of closed subsets of
$P_{d}$
, indexed by the degrees of the denominators of the rational functions that appear in a representation of
$f\in Q_{d}$
as a sum of three squares. Our main contribution is a proof of the density of this subset.
Theorem 0.1. The subset
$Q_{d}\subset P_{d}$
is a countable union of closed semialgebraic subsets of
$P_{d}$
that has empty interior if
$d\geqslant 6$
. It is dense in
$P_{d}$
.
When
$d\geqslant 6$
, the set
$P_{d}\setminus \unicode[STIX]{x1D6F4}_{d}$
of positive semidefinite polynomials that are not sums of squares of polynomials is a nonempty open subset of
$P_{d}$
. As a consequence of Theorem 0.1,
$Q_{d}$
is dense in this open subset, showing the existence of many polynomials
$f\in \mathbb{R}[x_{1},x_{2}]_{d}$
that are sums of three squares in
$\mathbb{R}(x_{1},x_{2})$
but not sums of squares of polynomials. The first examples of such polynomials had been constructed by Leep and Starr [Reference Leep and StarrLS01, Theorem 2].
0.2 Strategy of the proof
Our starting point is Colliot-Thélène’s Hodge-theoretic proof of the Cassels–Ellison–Pfister theorem [Reference Colliot-ThélèneCol93]: he associates to a polynomial
$f$
its homogenization
$F\in \mathbb{R}[X_{0},X_{1},X_{2}]$
, and the real algebraic surface defined by
$S:=\{Z^{2}+F(X_{0},X_{1},X_{2})=0\}$
. He then interprets the polynomials
$f$
that are sums of three squares in
$\mathbb{R}(x_{1},x_{2})$
as those for which the complex surface
$S_{\mathbb{C}}$
carries an extra line bundle of a particular kind, and concludes by applying the Noether–Lefschetz theorem.
As a consequence,
$Q_{d}$
may be viewed as a union of Noether–Lefschetz loci in
$P_{d}$
. Over
$\mathbb{C}$
, density results for Noether–Lefschetz loci have been first obtained by Ciliberto, Harris and Miranda, and by Green [Reference Ciliberto, Harris and MirandaCHM88], and we adapt these arguments over
$\mathbb{R}$
.
We rely on a real analogue of Green’s infinitesimal criterion [Reference Ciliberto, Harris and MirandaCHM88, § 5], [Reference VoisinVoi02, § 17.3.4], which was developed for other purposes in a joint work with Wittenberg [Reference Benoist and WittenbergBW18, § 7.2]. Section 1 is devoted to establishing this criterion in a form suitable for our needs: Proposition 1.3. One way to verify the hypothesis of Green’s criterion is to construct Noether–Lefschetz loci of the expected dimension. Following Ciliberto and Lopez [Reference Ciliberto and LopezCL91], we do so in § 2 by considering Noether–Lefschetz loci associated to determinantal curves, a strategy independently adopted by Bruzzo, Grassi and Lopez in [Reference Bruzzo, Grassi and LopezBGL17]. Finally, § 3 contains the proof of Theorem 0.1.
0.3 Level of function fields
The argument described above may be adapted to other families of real surfaces: here is another application of it. Recall that if
$K$
is a field, Pfister [Reference PfisterPfi65, Satz 4] has shown that the smallest integer
$s$
such that
$-1$
is a sum of
$s$
squares in
$K$
is a power of
$2$
(or
$+\infty$
): it is the level
$s(K)$
of
$K$
. Moreover, if
$X$
is an integral variety over
$\mathbb{R}$
of dimension
$n$
without real points,
$s(\mathbb{R}(X))\leqslant 2^{n}$
[Reference PfisterPfi67, Theorem 2].
Let us restrict to varieties that are smooth degree
$d$
surfaces
$S\subset \mathbb{P}_{\mathbb{R}}^{3}$
, defined by a degree
$d$
homogeneous equation
$F\in \mathbb{R}[X_{0},X_{1},X_{2},X_{3}]_{d}$
. Let
$\unicode[STIX]{x1D6E9}_{d}\subset \mathbb{P}(\mathbb{R}[X_{0},X_{1},X_{2},X_{3}]_{d})$
be the set of those surfaces that have no real points. As before, we assume that
$d$
is even, since otherwise
$\unicode[STIX]{x1D6E9}_{d}=\varnothing$
. If
$d=2$
, any such surface
$S$
is isomorphic to the anisotropic quadric
$\{X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}=0\}$
, and
$s(\mathbb{R}(S))=2$
. On the other hand, it follows from the Noether–Lefschetz theorem applied as in [Reference Colliot-ThélèneCol93] that if
$d\geqslant 4$
, a very general
$S\in \unicode[STIX]{x1D6E9}_{d}$
satisfies
$s(\mathbb{R}(S))=4$
. In § 4, we will show the following.
Theorem 0.2. The set of surfaces
$S\in \unicode[STIX]{x1D6E9}_{d}$
such that
$s(\mathbb{R}(S))=2$
is dense in
$\unicode[STIX]{x1D6E9}_{d}$
.
0.4 Conventions about algebraic varieties over
$\mathbb{R}$
An algebraic variety
$X$
over
$\mathbb{R}$
is a separated scheme of finite type over
$\mathbb{R}$
. We denote its complexification by
$X_{\mathbb{C}}$
. Its set of complex points
$X(\mathbb{C})$
is endowed with an action of
$G:=\operatorname{Gal}(\mathbb{C}/\mathbb{R})\simeq \mathbb{Z}/2\mathbb{Z}$
such that the complex conjugation
$\unicode[STIX]{x1D70E}\in G$
acts antiholomorphically. The real points of
$X$
are then the fixed points
$X(\mathbb{R})=X(\mathbb{C})^{G}$
.
Conversely, suppose that the set of complex points of a reduced quasi-projective algebraic variety
$X_{\mathbb{C}}$
over
$\mathbb{C}$
is endowed with an action of
$G$
given by an antiholomorphic involution. If the isomorphism
$X(\mathbb{C})\simeq \overline{X(\mathbb{C})}$
that the involution induces between
$X(\mathbb{C})$
and its conjugate variety is algebraic, Galois descent shows that
$X_{\mathbb{C}}$
is naturally the complexification of an algebraic variety
$X$
over
$\mathbb{R}$
. We refer to [Reference SilholSil89, I § 1] or [Reference MangolteMan17, § 2.2] for more details.
1 Green’s infinitesimal criterion over
$\mathbb{R}$
This section is devoted to an adaptation over
$\mathbb{R}$
of [Reference Ciliberto, Harris and MirandaCHM88, § 5], where Green studies the relation between infinitesimal variations of Hodge structure and density of Noether–Lefschetz loci. We follow the exposition of [Reference VoisinVoi02, § 17.3.4].
1.1 Variations of Hodge structure over real varieties
Let
$B_{\mathbb{C}}$
be a smooth algebraic variety over
$\mathbb{C}$
, and let
$\mathbb{H}_{\mathbb{Q}}^{2}$
be a
$\mathbb{Q}$
-local system on
$B(\mathbb{C})$
carrying a weight
$2$
variation of Hodge structure: in particular, the holomorphic vector bundle
${\mathcal{H}}^{2}:=\mathbb{H}_{\mathbb{Q}}^{2}\otimes _{\mathbb{Q}}{\mathcal{O}}_{B(\mathbb{C})}$
is endowed with a Hodge filtration by holomorphic subbundles
$F^{2}{\mathcal{H}}^{2}\subset F^{1}{\mathcal{H}}^{2}\subset {\mathcal{H}}^{2}$
, whose graded pieces
${\mathcal{H}}^{2,0}$
,
${\mathcal{H}}^{1,1}$
and
${\mathcal{H}}^{0,2}$
may be viewed as
$C^{\infty }$
complex subbundles of
${\mathcal{H}}^{2}$
. Let
${\mathcal{H}}_{\mathbb{R}}^{2}\subset {\mathcal{H}}^{2}$
and
${\mathcal{H}}_{\mathbb{R}}^{1,1}\subset {\mathcal{H}}^{1,1}$
be the
$C^{\infty }$
real subbundles consisting of sections with values in the real local system
$\mathbb{H}_{\mathbb{R}}^{2}:=\mathbb{H}_{\mathbb{Q}}^{2}\otimes _{\mathbb{Q}}\mathbb{R}$
, and
$\unicode[STIX]{x1D6FB}:{\mathcal{H}}^{2}\rightarrow {\mathcal{H}}^{2}\otimes \unicode[STIX]{x1D6FA}_{B(\mathbb{C})}^{1}$
be the connection induced by
$\mathbb{H}_{\mathbb{Q}}^{2}$
. Finally, we still denote by
${\mathcal{H}}^{2}$
the total space of the geometric vector bundle associated to
${\mathcal{H}}^{2}$
, whose fiber over
$b\in B(\mathbb{C})$
is
${\mathcal{H}}_{b}^{2}=\mathbb{H}_{\mathbb{Q},b}^{2}\otimes _{\mathbb{Q}}\mathbb{C}$
, and we use the same convention for its subbundles.
Assume now that
$B_{\mathbb{C}}$
is the complexification of an algebraic variety
$B$
over
$\mathbb{R}$
, and that we are given an action of
$G$
on the
$\mathbb{Q}$
-local system
$\mathbb{H}_{\mathbb{Q}}^{2}$
that is compatible with that on
$B(\mathbb{C})$
. We consider the induced action of
$G$
on (the total space of)
${\mathcal{H}}^{2}$
induced by the maps:
$$\begin{eqnarray}{\mathcal{H}}_{b}^{2}=\mathbb{H}_{\mathbb{ Q},b}^{2}\otimes _{\mathbb{ Q}}\mathbb{C}\xrightarrow[{}]{\unicode[STIX]{x1D70E}\otimes \unicode[STIX]{x1D70E}}\mathbb{H}_{\mathbb{Q},\unicode[STIX]{x1D70E}(b)}^{2}\otimes _{\mathbb{ Q}}\mathbb{C}={\mathcal{H}}_{\unicode[STIX]{x1D70E}(b)}^{2},\end{eqnarray}$$
where
$\unicode[STIX]{x1D70E}$
acts naturally on the first factor and via complex conjugation on the second: in particular,
$\unicode[STIX]{x1D70E}$
acts
$\mathbb{C}$
-antilinearly in the fibers of
${\mathcal{H}}^{2}$
. We make the assumption that this action of
$\unicode[STIX]{x1D70E}$
preserves the factors
${\mathcal{H}}^{p,q}$
of the Hodge decomposition. Consequently, there are induced actions of
$G$
on
${\mathcal{H}}^{1,1}$
and
${\mathcal{H}}_{\mathbb{R}}^{1,1}$
.
We will consider the
$G$
-module
$\mathbb{Z}(1):=2\unicode[STIX]{x1D70B}\sqrt{-1}\cdot \mathbb{Z}\subset \mathbb{C}$
. It is isomorphic to
$\mathbb{Z}$
with an action of
$G$
by multiplication by
$-1$
. We also denote by
$\mathbb{Z}(1)$
the
$G$
-equivariant constant local system on
$B(\mathbb{C})$
with fibers
$\mathbb{Z}(1)$
, and if
$M$
is any
$G$
-module or
$G$
-equivariant sheaf on
$B(\mathbb{C})$
, we define
$M(1)$
to be the tensor product
$M\otimes _{\mathbb{Z}}\mathbb{Z}(1)$
. If
$b\in B(\mathbb{R})$
, we view
${\mathcal{H}}_{\mathbb{R},b}^{1,1}(1)$
as a
$G$
-stable subspace of
${\mathcal{H}}_{b}^{2}$
via the embeddings
${\mathcal{H}}_{\mathbb{R},b}^{1,1}\subset {\mathcal{H}}_{b}^{2}$
and
$\mathbb{Z}(1)\subset \mathbb{C}$
.
1.2 The infinitesimal criterion
We fix
$b\in B(\mathbb{R})$
, and we choose a
$G$
-stable connected analytic neighbourhood
$\unicode[STIX]{x1D6E5}$
of
$b\in B(\mathbb{C})$
, on which
$\mathbb{H}_{\mathbb{Q}}^{2}$
is trivialized, and such that
$\unicode[STIX]{x1D6E5}(\mathbb{R}):=\unicode[STIX]{x1D6E5}\cap B(\mathbb{R})$
is connected and contractible. Such a neighbourhood exists: choose any connected contractible neighbourhood of
$b$
in
$B(\mathbb{C})$
, intersect it with its image by
$\unicode[STIX]{x1D70E}$
, remove an appropriate closed subset of
$\unicode[STIX]{x1D6E5}(\mathbb{R})$
, and retain the connected component containing
$b$
.
The trivialization of
$\mathbb{H}_{\mathbb{Q}}^{2}$
over
$\unicode[STIX]{x1D6E5}$
gives rise to an isomorphism
${\mathcal{H}}^{2}|_{\unicode[STIX]{x1D6E5}}\simeq \unicode[STIX]{x1D6E5}\times {\mathcal{H}}_{b}^{2}$
, that is
$G$
-equivariant by unicity of the trivialization.
Proposition 1.1. Suppose that there exists
$\unicode[STIX]{x1D706}\in {\mathcal{H}}_{\mathbb{R},b}^{1,1}(1)^{G}$
such that the map
$$\begin{eqnarray}\overline{\unicode[STIX]{x1D6FB}}(\unicode[STIX]{x1D706}):T_{B(\mathbb{C}),b}\longrightarrow {\mathcal{H}}_{b}^{0,2}\end{eqnarray}$$
induced by evaluating the connection
$\unicode[STIX]{x1D6FB}$
on
$\unicode[STIX]{x1D706}$
is surjective.
Then there exists an open cone
$\unicode[STIX]{x1D6FA}\subset {\mathcal{H}}_{\mathbb{R},b}^{2}(1)^{G}$
such that for every
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$
, there exists
$b^{\prime }\in \unicode[STIX]{x1D6E5}(\mathbb{R})$
and
$\unicode[STIX]{x1D714}^{\prime }\in {\mathcal{H}}_{\mathbb{R},b^{\prime }}^{1,1}(1)^{G}$
such that
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}^{\prime }$
under the identification
${\mathcal{H}}_{b}^{2}\simeq {\mathcal{H}}_{b^{\prime }}^{2}$
given by the trivialization.
Proof. The trivialization of
$\mathbb{H}_{\mathbb{Q}}^{2}$
over
$\unicode[STIX]{x1D6E5}$
yields an isomorphism
${\mathcal{H}}_{\mathbb{R}}^{2}|_{\unicode[STIX]{x1D6E5}}\simeq \unicode[STIX]{x1D6E5}\times {\mathcal{H}}_{\mathbb{R},b}^{2}$
. Under our surjectivity hypothesis, the composition of the inclusion
${\mathcal{H}}_{\mathbb{R}}^{1,1}|_{\unicode[STIX]{x1D6E5}}\subset {\mathcal{H}}_{\mathbb{R}}^{2}|_{\unicode[STIX]{x1D6E5}}$
and of the projection to
${\mathcal{H}}_{\mathbb{R},b}^{2}$
gives a map
$$\begin{eqnarray}\unicode[STIX]{x1D719}:{\mathcal{H}}_{\mathbb{R}}^{1,1}|_{\unicode[STIX]{x1D6E5}}\rightarrow {\mathcal{H}}_{\mathbb{ R},b}^{2}\end{eqnarray}$$
that is submersive at
$(b,\unicode[STIX]{x1D706})$
by [Reference VoisinVoi02, Lemme 17.21]. The map
$\unicode[STIX]{x1D719}$
is equivariant with respect to the natural action of
$G$
on both sides. Consequently, tensoring by
$\mathbb{Z}(1)$
and taking
$G$
-invariants gives rise to a map
$$\begin{eqnarray}\unicode[STIX]{x1D719}^{\prime }:{\mathcal{H}}_{\mathbb{ R}}^{1,1}(1)^{G}|_{\unicode[STIX]{x1D6E5}(\mathbb{R})}\rightarrow {\mathcal{H}}_{\mathbb{R},b}^{2}(1)^{G},\end{eqnarray}$$
where
${\mathcal{H}}_{\mathbb{R}}^{1,1}(1)^{G}|_{\unicode[STIX]{x1D6E5}(\mathbb{R})}$
is the
$C^{\infty }$
real vector bundle on
$\unicode[STIX]{x1D6E5}(\mathbb{R})$
with fiber
${\mathcal{H}}_{\mathbb{R},b^{\prime }}^{1,1}(1)^{G}$
at
$b^{\prime }\in \unicode[STIX]{x1D6E5}(\mathbb{R})$
.
It is well known that the fixed locus
$M^{G}$
of a
$G$
-action on a
$C^{\infty }$
manifold
$M$
is again a manifold, whose tangent space
$T_{x}(M^{G})$
at a fixed point
$x\in M^{G}$
is
$(T_{x}M)^{G}$
(see the much more general [Reference AudinAud91, I § 2.1]). This implies that
$\unicode[STIX]{x1D719}^{\prime }$
is still submersive at
$(b,\unicode[STIX]{x1D706})$
. Consequently, the image of
$\unicode[STIX]{x1D719}^{\prime }$
contains an open set of
${\mathcal{H}}_{\mathbb{R},b}^{2}(1)^{G}$
. Since this image is obviously a cone, it contains an open cone
$\unicode[STIX]{x1D6FA}\subset {\mathcal{H}}_{\mathbb{R},b}^{2}(1)^{G}$
. This cone has the required property.◻
1.3 Density
In the classical complex case, the set of points in the base where Green’s criterion may be verified is the complement of a complex-analytic subset. Consequently, assuming the base connected, it is dense if it is nonempty. The same holds in our setting if one restricts to a connected component of
$B(\mathbb{R})$
, as one sees by adapting the argument to the real-analytic category.
Proposition 1.2. Let
$K\subset B(\mathbb{R})$
be a connected component. The set of
$b\in K$
for which there exists
$\unicode[STIX]{x1D706}\in {\mathcal{H}}_{\mathbb{R},b}^{1,1}(1)^{G}$
such that the map (1.2) is surjective is either empty, or dense in
$K$
.
Proof. We prove that the complement
$W\subset K$
of this set is a real-analytic subset of
$K$
. This concludes because the set of
$x\in K$
such that a neighbourhood of
$x$
is included in
$W$
is easily seen to be open and closed, hence equal to
$K$
or empty, and it follows that the complement of
$W$
is empty or dense in
$K$
, as wanted.
Since
$F^{1}{\mathcal{H}}^{2}\subset {\mathcal{H}}^{2}$
is holomorphic and
${\mathcal{H}}_{\mathbb{R}}^{2}\subset {\mathcal{H}}^{2}$
is a real-analytic subbundle, we deduce that
${\mathcal{H}}_{\mathbb{R}}^{1,1}=F^{1}{\mathcal{H}}^{2}\cap {\mathcal{H}}_{\mathbb{R}}^{2}$
is a real-analytic subbundle of
${\mathcal{H}}^{2}$
. Since the action of
$G$
on
$B(\mathbb{C})$
is real-analytic, and since the compatible action on
${\mathcal{H}}^{2}$
is induced by an action on
$\mathbb{H}_{\mathbb{Q}}^{2}$
, the
$G$
-action on
${\mathcal{H}}^{2}$
is real-analytic, and hence so is the subbundle
${\mathcal{H}}_{\mathbb{R}}^{1,1}(1)^{G}\subset {\mathcal{H}}^{2}|_{B(\mathbb{R})}$
. Since the connection
$\unicode[STIX]{x1D6FB}$
induces a morphism
$\overline{\unicode[STIX]{x1D6FB}}:{\mathcal{H}}^{1,1}\otimes T_{B(\mathbb{C})}\rightarrow {\mathcal{H}}^{0,2}$
of holomorphic vector bundles [Reference VoisinVoi02, (10.2.1)], the set
$Z\subset {\mathcal{H}}_{\mathbb{R}}^{1,1}(1)^{G}$
consisting of the
$(b,\unicode[STIX]{x1D706})$
for which the map (1.2) is not surjective is a real-analytic subset of
${\mathcal{H}}_{\mathbb{R}}^{1,1}(1)^{G}$
.
We may deduce that
$W:=\{b\in B(\mathbb{R})\mid {\mathcal{H}}_{\mathbb{R},b}^{1,1}(1)^{G}\subset Z\}$
is a real-analytic subset of
$B(\mathbb{R})$
. To check it locally at
$b\in B(\mathbb{R})$
, choose a trivialization of
${\mathcal{H}}_{\mathbb{R}}^{1,1}(1)^{G}$
in a neighbourhood
$U\subset B(\mathbb{R})$
of
$b$
with fiber
$\mathbb{R}^{k}$
and notice that
$W\cap U$
is real-analytic as the intersection of the real-analytic subsets
$(Z\cap (U\times \{v\}))_{v\in \mathbb{R}^{k}}$
of
$U$
. ◻
1.4 Families of real varieties
Let us specialize Propositions 1.1 and 1.2 to variations of Hodge structure of geometric origin.
Let
$\unicode[STIX]{x1D70B}:{\mathcal{S}}\rightarrow B$
be a smooth projective morphism of smooth algebraic varieties over
$\mathbb{R}$
: both
${\mathcal{S}}(\mathbb{C})$
and
$B(\mathbb{C})$
are endowed with an action of
$G$
such that
$\unicode[STIX]{x1D70E}$
acts antiholomorphically, and the map
$\unicode[STIX]{x1D70B}:{\mathcal{S}}(\mathbb{C})\rightarrow B(\mathbb{C})$
is
$G$
-equivariant.
The local system
$\mathbb{H}_{\mathbb{Q}}^{2}:=R^{2}\unicode[STIX]{x1D70B}_{\ast }\mathbb{Q}$
on
$B(\mathbb{C})$
underlies a weight
$2$
variation of Hodge structure, whose connection is the Gauss–Manin connection. The action of
$G$
on
${\mathcal{S}}(\mathbb{C})$
induces an action of
$G$
on
$\mathbb{H}_{\mathbb{Q}}^{2}$
that is compatible with the
$G$
-action on
$B(\mathbb{C})$
. Let us verify the assumption made in § 1.1 that the induced
$\mathbb{C}$
-antilinear action on
${\mathcal{H}}^{2}$
preserves the Hodge decomposition. By (1.1), this action may be written as a composition:
$$\begin{eqnarray}{\mathcal{H}}_{b}^{2}=\mathbb{H}_{\mathbb{ Q},b}^{2}\otimes _{\mathbb{ Q}}\mathbb{C}\xrightarrow[{}]{\operatorname{Id}\otimes \unicode[STIX]{x1D70E}}\mathbb{H}_{\mathbb{Q},b}^{2}\otimes _{\mathbb{ Q}}\mathbb{C}\xrightarrow[{}]{\unicode[STIX]{x1D70E}\otimes \operatorname{Id}}\mathbb{H}_{\mathbb{Q},\unicode[STIX]{x1D70E}(b)}^{2}\otimes _{\mathbb{ Q}}\mathbb{C}={\mathcal{H}}_{\unicode[STIX]{x1D70E}(b)}^{2}.\end{eqnarray}$$
The first arrow is the conjugation with respect to the real structure
${\mathcal{H}}_{\mathbb{R},b}^{2}\subset {\mathcal{H}}_{b}^{2}$
, and hence exchanges the factors
${\mathcal{H}}_{b}^{p,q}$
and
${\mathcal{H}}_{b}^{q,p}$
of the Hodge decomposition. The second arrow is obtained by functoriality from the antiholomorphic map
$\unicode[STIX]{x1D70E}:{\mathcal{S}}_{\unicode[STIX]{x1D70E}(b)}(\mathbb{C})\rightarrow {\mathcal{S}}_{b}(\mathbb{C})$
, and hence also exchanges the factors of the Hodge decomposition by the argument of [Reference SilholSil89, I Lemma 2.4]. Consequently, we are indeed in the setting of § 1.1.
Let
$b\in B(\mathbb{R})$
and
$b\in \unicode[STIX]{x1D6E5}\subset B(\mathbb{C})$
be as in § 1.2. Griffiths [Reference VoisinVoi02, Théorème 10.21] has computed the map
$\overline{\unicode[STIX]{x1D6FB}}(\unicode[STIX]{x1D706})$
of (1.2) as the composition of the Kodaira–Spencer map and of the contracted cup-product with
$\unicode[STIX]{x1D706}$
induced by the pairing
$\unicode[STIX]{x1D6FA}_{{\mathcal{S}}_{b,\mathbb{C}}}^{1}\otimes T_{{\mathcal{S}}_{b,\mathbb{C}}}\rightarrow {\mathcal{O}}_{{\mathcal{S}}_{b,\mathbb{C}}}$
:
$$\begin{eqnarray}T_{B_{\mathbb{C}},b}\longrightarrow H^{1}({\mathcal{S}}_{b,\mathbb{C}},T_{{\mathcal{S}}_{b,\mathbb{C}}})\stackrel{\unicode[STIX]{x1D706}}{\longrightarrow }H^{2}({\mathcal{S}}_{b,\mathbb{C}},{\mathcal{O}}_{{\mathcal{S}}_{b,\mathbb{C}}}).\end{eqnarray}$$
Propositions 1.1 and 1.2 then become as follows.
Proposition 1.3. Suppose that there exists
$\unicode[STIX]{x1D706}\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b}(\mathbb{C}))(1)^{G}$
such that the composition (1.4) is surjective. Then there exists an open cone
$\unicode[STIX]{x1D6FA}\subset H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))^{G}$
such that for every
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$
, there exists
$b^{\prime }\in \unicode[STIX]{x1D6E5}(\mathbb{R})$
and
$\unicode[STIX]{x1D714}^{\prime }\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}))(1)^{G}$
such that
$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}^{\prime }$
under the identification
$H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{C})\simeq H^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{C})$
given by the trivialization.
Moreover, the set of
$b\in B(\mathbb{R})$
for which there exists such a
$\unicode[STIX]{x1D706}$
is dense in every connected component
$K$
of
$B(\mathbb{R})$
that it meets.
Remark 1.4. In order to verify the hypothesis of Proposition 1.3, one has to construct a class
$\unicode[STIX]{x1D706}\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b}(\mathbb{C}))(1)^{G}$
. A natural source of such cohomology classes are cycle classes
$[{\mathcal{L}}]\in H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))$
of line bundles
${\mathcal{L}}$
on
${\mathcal{S}}_{b}$
that are defined over
$\mathbb{R}$
. Indeed,
$[{\mathcal{L}}]$
is of type
$(1,1)$
by Hodge theory, and analyzing the
$G$
-action on the exponential exact sequence [Reference SilholSil89, I (4.11), Lemma 4.12] shows that it belongs to
$H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))^{G}$
.
Remark 1.5. It is possible, in the setting of Proposition 1.3, that the set
$\unicode[STIX]{x1D6F4}$
of
$b\in B(\mathbb{R})$
such that there exists a
$\unicode[STIX]{x1D706}\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b}(\mathbb{C}))(1)^{G}$
for which (1.4) is surjective is dense in some connected component of
$B(\mathbb{R})$
, but does not meet another one.
This happens when
${\mathcal{S}}\rightarrow B$
is the universal family of smooth quartic surfaces in
$\mathbb{P}_{\mathbb{R}}^{3}$
. In this case, we will show in Lemma 4.3 that
$\unicode[STIX]{x1D6F4}$
is dense in the connected components of
$B(\mathbb{R})$
parameterizing surfaces without real points. However,
$\unicode[STIX]{x1D6F4}$
cannot intersect the connected components corresponding to surfaces
$S$
whose real locus is a union of
$10$
spheres (that exist by [Reference SilholSil89, (3.3) p. 189]). Indeed, one computes using [Reference SilholSil89, VIII § 3] that for such surfaces,
$H_{\mathbb{R}}^{1,1}(S(\mathbb{C}))(1)^{G}$
has rank
$1$
, and hence is generated by
$\unicode[STIX]{x1D706}:=[{\mathcal{O}}_{\mathbb{P}_{\mathbb{R}}^{3}}(1)]$
. Since the line bundle
${\mathcal{O}}_{\mathbb{P}_{\mathbb{R}}^{3}}(1)$
is defined on the whole family, its cohomology class
$\unicode[STIX]{x1D706}$
remains Hodge under small deformations showing that (1.4) vanishes.
Remark 1.6. If
$b\in B(\mathbb{R})$
, let
$F_{\infty }:=\unicode[STIX]{x1D70E}\otimes \operatorname{Id}$
be the
$\mathbb{C}$
-linear involution of
${\mathcal{H}}_{b}^{2}=\mathbb{H}_{\mathbb{Q},b}^{2}\otimes _{\mathbb{Q}}\mathbb{C}$
induced by
$\unicode[STIX]{x1D70E}$
. It exchanges the factors of the Hodge decomposition, and hence preserves
${\mathcal{H}}_{b}^{1,1}$
. Since
${\mathcal{H}}_{\mathbb{R},b}^{1,1}(1)^{G}$
is Zariski-dense in its complexification
$({\mathcal{H}}_{b}^{1,1})^{F_{\infty }=-1}$
, and since the surjectivity of (1.4) depends algebraically on
$\unicode[STIX]{x1D706}$
, it would be sufficient, in the hypotheses of Proposition 1.3, to require that
$\unicode[STIX]{x1D706}\in ({\mathcal{H}}_{b}^{1,1})^{F_{\infty }=-1}$
.
Remark 1.7. If the family
${\mathcal{S}}\rightarrow B$
is induced by a linear system in a fixed variety
$X$
over
$\mathbb{R}$
, it may be better to apply Proposition 1.1 to the variation of Hodge structure given by the vanishing cohomology.
2 An explicit Noether–Lefschetz locus
To verify the hypothesis of Proposition 1.3, we need to construct an appropriate cohomology class
$\unicode[STIX]{x1D706}$
. In the complex setting, several strategies are available to do so: the original degeneration method of Ciliberto, Harris and Miranda [Reference Ciliberto, Harris and MirandaCHM88], computations with jacobian rings [Reference KimKim91, Theorem 2], use of explicit Noether–Lefschetz loci [Reference Ciliberto and LopezCL91], or the much more general arguments of Voisin [Reference VoisinVoi06].
Here, we adapt the strategy of Ciliberto and Lopez [Reference Ciliberto and LopezCL91, Lemma 1.2, Theorem 1.3 and their proofs]: we take for
$\unicode[STIX]{x1D706}$
the class of a determinantal curve. This section is devoted to working out this idea in the generality we need. We first analyze Green’s criterion when
$\unicode[STIX]{x1D706}$
is the class of a curve in § 2.1, and specialize to the case of a determinantal curve in § 2.2. The main difference with [Reference Ciliberto and LopezCL91] is that we argue purely cohomologically rather than geometrically on the Noether–Lefschetz loci.
2.1 Applying Green’s criterion to the class of a curve
In this paragraph, we fix smooth projective complex varieties
$C\subset S\subset X$
, where
$C$
is a curve,
$S$
a surface and
$X$
a threefold. The image in
$H^{2}(S(\mathbb{C}),\mathbb{C})$
of the Betti cohomology class
$\unicode[STIX]{x1D706}\in H^{2}(S(\mathbb{C}),\mathbb{Z}(1))$
of
$C$
in
$S$
is of type
$(1,1)$
by Hodge theory. We may thus view it as an element
$\unicode[STIX]{x1D706}\in H^{1}(S,\unicode[STIX]{x1D6FA}_{S}^{1})=H^{1,1}(S)$
. We study the composition
$$\begin{eqnarray}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D706}}:H^{0}(S,N_{S/X})\rightarrow H^{1}(S,T_{S})\stackrel{\unicode[STIX]{x1D706}}{\longrightarrow }H^{2}(S,{\mathcal{O}}_{S})\end{eqnarray}$$
of the boundary map of the normal exact sequence
$0\rightarrow T_{S}\rightarrow T_{X}|_{S}\rightarrow N_{S/X}\rightarrow 0$
and of the contracted cup-product with
$\unicode[STIX]{x1D706}$
. To do so, we consider the two exact sequences
$$\begin{eqnarray}\displaystyle & \displaystyle 0\rightarrow N_{C/S}\rightarrow N_{C/X}\rightarrow N_{S/X}|_{C}\rightarrow 0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle 0\rightarrow {\mathcal{O}}_{S}\rightarrow {\mathcal{O}}_{S}(C)\rightarrow {\mathcal{O}}_{S}(C)|_{C}\rightarrow 0, & \displaystyle\end{eqnarray}$$
and we recall that
$N_{C/S}\simeq {\mathcal{O}}_{S}(C)|_{C}$
.
Proposition 2.1. The map
$\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D706}}$
of (2.1) coincides with the composition
$$\begin{eqnarray}H^{0}(S,N_{S/X})\rightarrow H^{0}(C,N_{S/X}|_{C})\rightarrow H^{1}(C,N_{C/S})\rightarrow H^{2}(S,{\mathcal{O}}_{S})\end{eqnarray}$$
of the restriction map and of the boundary maps of (2.2) and (2.3).
We first recall some properties of extension classes used in the proof. Let
$$\begin{eqnarray}0\rightarrow {\mathcal{A}}\xrightarrow[{}]{f}{\mathcal{B}}\xrightarrow[{}]{g}{\mathcal{C}}\rightarrow 0\end{eqnarray}$$
be an exact sequence of locally free sheaves on
$S$
. Choose an open cover
$(U_{i})$
of
$S$
such that
$g|_{U_{i}}$
admits a section
$\unicode[STIX]{x1D70E}_{i}:{\mathcal{C}}|_{U_{i}}\rightarrow {\mathcal{B}}|_{U_{i}}$
. Setting
$\unicode[STIX]{x1D70F}_{i,j}:=\unicode[STIX]{x1D70E}_{j}-\unicode[STIX]{x1D70E}_{i}\in H^{0}(U_{i}\cap U_{j},{\mathcal{A}}\otimes {\mathcal{C}}^{\vee })$
gives rise to a cocycle
$(\unicode[STIX]{x1D70F}_{i,j})$
whose cohomology class
$\unicode[STIX]{x1D709}\in H^{1}(S,{\mathcal{A}}\otimes {\mathcal{C}}^{\vee })$
is independent of the choices. It is the extension class of (2.4). Direct computations with cocycles show that the extension class of the tensor product
$0\rightarrow {\mathcal{A}}\otimes {\mathcal{L}}\xrightarrow[{}]{f}{\mathcal{B}}\otimes {\mathcal{L}}\xrightarrow[{}]{g}{\mathcal{C}}\otimes {\mathcal{L}}\rightarrow 0$
by a line bundle
${\mathcal{L}}$
on
$S$
is equal to
$\unicode[STIX]{x1D709}$
, that the extension class of the dual
$0\rightarrow {\mathcal{C}}^{\vee }\xrightarrow[{}]{g^{\vee }}{\mathcal{B}}^{\vee }\xrightarrow[{}]{f^{\vee }}{\mathcal{A}}^{\vee }\rightarrow 0$
is equal to
$-\unicode[STIX]{x1D709}$
, and that the boundary maps
$H^{q}(S,{\mathcal{C}})\rightarrow H^{q+1}(S,{\mathcal{A}})$
in the long exact sequence of cohomology associated to (2.4) are induced by the cup-product by
$\unicode[STIX]{x1D709}$
.
Proof of Proposition 2.1.
The proposition follows from the compatibility of the two boundary maps appearing in the commutative diagram of coherent sheaves on
$S$
and in the pull-back diagram
in addition to the fact that the boundary map
$H^{1}(S,T_{S})\rightarrow H^{2}(S,{\mathcal{O}}_{S})$
associated to the first line of (2.6) is induced by the cup-product with its extension class, that turns out to be equal to
$\unicode[STIX]{x1D706}$
. To verify this fact, we rather consider the twist of (2.6) by
${\mathcal{O}}_{S}(-C)$
and we prove that the extension on the first line is canonically dual to the extension
$$\begin{eqnarray}0\rightarrow \unicode[STIX]{x1D6FA}_{S}^{1}(C)\rightarrow D({\mathcal{O}}_{S}(C))\rightarrow {\mathcal{O}}_{S}(C)\rightarrow 0\end{eqnarray}$$
defined by Atiyah in [Reference AtiyahAti57, § 4], and whose extension class (the Atiyah class) is equal to
$-\unicode[STIX]{x1D706}$
by [Reference AtiyahAti57, Proposition 12] (the factor
$2\unicode[STIX]{x1D70B}i$
in [Reference AtiyahAti57, Proposition 12] corresponds to the comparison between Chern classes in Betti and de Rham cohomology, and is accounted for here by our definition of
$\mathbb{Z}(1)$
).
To check this duality statement, choose
$x\in S$
and local sections
$f\in {\mathcal{O}}_{S,x}$
and
$v\in T_{S}(-C)_{x}$
that coincide in
${\mathcal{O}}_{C,x}$
and hence induce
$(f,v)\in {\mathcal{E}}(-C)_{x}$
. Let
$s\in {\mathcal{O}}_{S}(C)_{x}$
and
$\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{S}^{1}(C)_{x}$
giving rise to a local section
$(s,\unicode[STIX]{x1D6FD})\in D({\mathcal{O}}_{S}(C))_{x}$
in the notations of [Reference AtiyahAti57, § 4]. A direct computation shows that
$(f,v)\cdot (s,\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D6FD}(v)+sf-ds(v)$
is well defined,
${\mathcal{O}}_{S,x}$
-linear, and induces the required duality.◻
Corollary 2.2. If the groups
$H^{1}(S,N_{S/X}(-C))$
,
$H^{1}(C,N_{C/X})$
and
$H^{2}(S,{\mathcal{O}}_{S}(C))$
vanish, the map
$\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D706}}$
of (2.1) is surjective.
2.2 The case of a determinantal curve
We now restrict the situation to the case where
$C$
is a determinantal curve. Let
$X$
be a smooth projective connected complex threefold, and
$n\geqslant 1$
be an integer. Let
$L,H_{1},\ldots ,H_{n}$
be base-point free line bundles on
$X$
, and define
$H:=H_{1}+\cdots +H_{n}$
. In this paragraph, we make the following assumptions.
Hypotheses 2.3. For every
$1\leqslant j,k\leqslant n$
:
(i)
$H^{2}(X,{\mathcal{O}}_{X})=0$
;(ii)
$H^{1}(X,H_{j})=H^{2}(X,H_{j}-H_{k})=H^{3}(X,-H_{j})=0$
;(iii)
$H^{0}(X,K_{X}+L)=H^{1}(X,L)=H^{1}(X,K_{X}+L-H_{j})=H^{2}(X,L-H_{j})=0$
.
The particular case considered in [Reference Ciliberto and LopezCL91] is
$X=\mathbb{P}_{\mathbb{C}}^{3}$
,
$H_{j}={\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{3}}(1)$
and
$L={\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{3}}(2)$
.
Choose
$M=(M_{i,j})_{1\leqslant i\leqslant n+1,1\leqslant j\leqslant n}$
a
$(n+1)\times n$
matrix with
$M_{i,j}\in H^{0}(X,H_{j})$
. Let
$C\subset X$
be defined by the vanishing of the maximal minors of
$M$
, and
$S\subset X$
be the zero-locus of a section
$\unicode[STIX]{x1D70F}\in H^{0}(X,H+L)$
vanishing on
$C$
.
Lemma 2.4. If
$M$
is general,
$C$
is a smooth curve, possibly empty. Once such a matrix
$M$
is fixed, if
$\unicode[STIX]{x1D70F}$
is general,
$S$
is a smooth surface, possibly empty.
Proof. Let
${\mathcal{M}}:=\bigoplus _{1\leqslant i\leqslant n+1,1\leqslant j\leqslant n}H^{0}(X,H_{j})$
be the parameter space for such matrices, and let
${\mathcal{C}}\subset {\mathcal{M}}\times X$
be the universal variety defined by the vanishing of maximal minors. We consider the fiber
${\mathcal{C}}_{x}$
of the second projection
${\mathcal{C}}\rightarrow X$
at
$x\in X(\mathbb{C})$
. Since the
$H_{j}$
are base-point free, evaluation at
$x$
with respect to local trivializations yields a linear surjection
$\operatorname{ev}_{x}:{\mathcal{M}}\rightarrow M_{n+1,n}(\mathbb{C})$
, and
${\mathcal{C}}_{x}=\{M\in {\mathcal{M}}\mid \operatorname{rank}(\operatorname{ev}_{x}(M))<n\}$
. It follows that
${\mathcal{C}}_{x}\subset {\mathcal{M}}$
is irreducible of codimension
$2$
and that its singular locus
$\operatorname{Sing}({\mathcal{C}}_{x})\subset {\mathcal{M}}$
, being the inverse image by
$\operatorname{ev}_{x}$
of the set of matrices of rank
${<}n-1$
, has codimension
$6$
[Reference Bruns and VetterBV88, Proposition 1.1]. We deduce that
${\mathcal{C}}$
is irreductible of dimension
$\dim ({\mathcal{M}})+1$
and that
$\dim (\operatorname{Sing}({\mathcal{C}}))<\dim ({\mathcal{M}})$
. It follows that the generic fiber of the first projection
${\mathcal{C}}\rightarrow {\mathcal{M}}$
has dimension
$1$
(but may be empty if this projection is not dominant) and does not meet
$\operatorname{Sing}({\mathcal{C}})$
, and hence is smooth by generic smoothness as we are in characteristic
$0$
. Consequently, we may choose
$M\in {\mathcal{M}}$
so that
$C$
is a smooth curve.
That
$S$
may be chosen smooth follows from an easy variant of the results of [Reference Kleiman and AltmanKA79]. Since
$C$
is defined by the vanishing of sections of
$H$
,
${\mathcal{I}}_{C}(H)$
is generated by its global sections, and since
$L$
is base-point free, so is
${\mathcal{I}}_{C}(H+L)$
. Fix
$x\in C$
, and let
$\mathfrak{m}_{x}\subset {\mathcal{O}}_{X,x}$
be the maximal ideal. As
$C$
is a smooth curve, the evaluation map
$H^{0}(X,{\mathcal{I}}_{C}(H+L))\rightarrow {\mathcal{O}}_{X}(H+L)\otimes (\mathfrak{m}_{x}/\mathfrak{m}_{x}^{2})$
has rank
$2$
, so that the set of
$\unicode[STIX]{x1D70F}\in H^{0}(X,{\mathcal{I}}_{C}(H+L))$
whose zero-locus is singular at
$x$
has codimension
$2$
. A dimension count shows that the zero-locus of a general
$\unicode[STIX]{x1D70F}$
is smooth along
$C$
. The base locus of the linear system
$H^{0}(X,{\mathcal{I}}_{C}(H+L))$
being
$C$
, it follows from Bertini’s theorem [Reference JouanolouJou83, I Théorème 6.10 2)] that the zero-locus of a general
$\unicode[STIX]{x1D70F}$
is smooth off
$C$
. We deduce that if
$\unicode[STIX]{x1D70F}$
is general,
$S$
is smooth.◻
From now on, we suppose that
$M$
and
$\unicode[STIX]{x1D70F}$
have been chosen general in the sense of Lemma 2.4. Our goal is to show in Proposition 2.5 below that, under Hypotheses 2.3, the morphism
$\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D706}}$
defined in (2.1) is surjective. We first explain the tools that will allow to carry out the relevant coherent cohomology computations.
By Lemma 2.4, the curve
$C$
is a determinantal curve of the expected codimension in
$X$
. It follows that its ideal sheaf
${\mathcal{I}}_{C}\subset {\mathcal{O}}_{X}$
is resolved by the Eagon–Northcott complex (also called the Hilbert–Burch complex in this particular case, see [Reference EisenbudEis05, Theorem A2.60, Example A2.67]):
$$\begin{eqnarray}0\rightarrow \bigoplus _{j=1}^{n}{\mathcal{O}}_{X}(-H-H_{j})\stackrel{M}{\longrightarrow }{\mathcal{O}}_{X}(-H)^{\oplus n+1}\rightarrow {\mathcal{I}}_{C}\rightarrow 0,\end{eqnarray}$$
in which the first map is given by the matrix
$M$
and the second one by the
$n+1$
maximal minors of
$M$
. Restricting (2.7) to
$C$
using right exactness of the tensor product, and noticing that the kernel of
$\bigoplus _{j=1}^{n}{\mathcal{O}}_{C}(-H-H_{j})\stackrel{M}{\longrightarrow }{\mathcal{O}}_{C}(-H)^{\oplus n+1}$
is a line bundle on
$C$
that may be computed by calculating its determinant, one gets
$$\begin{eqnarray}0\rightarrow K_{C}^{-1}\otimes K_{X}|_{C}\rightarrow \bigoplus _{j=1}^{n}{\mathcal{O}}_{C}(-H-H_{j})\stackrel{M}{\longrightarrow }{\mathcal{O}}_{C}(-H)^{\oplus n+1}\rightarrow N_{C/X}^{\vee }\rightarrow 0.\end{eqnarray}$$
The dual of the Eagon–Northcott complex is still a resolution by [Reference EisenbudEis05, Theorem A2.60], of a sheaf on
$X$
that we denote by
${\mathcal{Q}}$
:
$$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{X}\rightarrow {\mathcal{O}}_{X}(H)^{\oplus n+1}\stackrel{M^{\mathsf{T}}}{\longrightarrow }\bigoplus _{j=1}^{n}{\mathcal{O}}_{X}(H+H_{j})\rightarrow {\mathcal{Q}}\rightarrow 0.\end{eqnarray}$$
It follows from Cramer’s rule that the maximal minors of
$M$
vanish on the support of
${\mathcal{Q}}$
. Consequently,
${\mathcal{Q}}$
may be computed after restriction to
$C$
, and the dual of (2.8) shows that
${\mathcal{Q}}=K_{C}\otimes K_{X}^{-1}|_{C}$
. Finally, there is an obvious morphism between (2.9) and the dual of (2.8), where the left vertical arrow is the zero map.
Proposition 2.5. The map
$\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D706}}$
of (2.1) is surjective.
Proof. By Corollary 2.2, it suffices to show the vanishing of the three groups
$H^{1}(S,N_{S/X}(-C))$
,
$H^{1}(C,N_{C/X})$
and
$H^{2}(S,{\mathcal{O}}_{S}(C))$
. Twisting the natural exact sequence
$$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{X}(-S)\rightarrow {\mathcal{I}}_{C}\rightarrow {\mathcal{O}}_{S}(-C)\rightarrow 0\end{eqnarray}$$
by
${\mathcal{O}}_{X}(S)$
and taking cohomology, we see that the vanishing of
$H^{1}(S,N_{S/X}(-C))$
follows from that of
$H^{2}(X,{\mathcal{O}}_{X})$
and
$H^{1}(X,{\mathcal{I}}_{C}(S))$
which are deduced from Hypotheses 2.3 using (2.7).
A diagram chase using (2.10) (or, more conceptually, an analysis of the second hypercohomology spectral sequence of this exact sequence of complexes) shows that in order to prove the vanishing of
$H^{1}(C,N_{C/X})$
, it suffices to check that
$H^{2}(X,{\mathcal{I}}_{C}(H))=H^{1}(X,{\mathcal{I}}_{C}(H+H_{j}))=0$
for every
$1\leqslant j\leqslant n$
. In turn, these vanishings follow from the Hypotheses 2.3 and from (2.7).
By Serre duality and adjunction, the vanishing of
$H^{2}(S,{\mathcal{O}}_{S}(C))$
is equivalent to that of
$H^{0}(S,K_{S}(-C))=H^{0}(S,(K_{X}+S)|_{S}(-C))$
. To prove it, twist (2.11) by
$K_{X}+S$
, take cohomology, and notice that
$H^{1}(X,K_{X})=H^{0}(X,{\mathcal{I}}_{C}(K_{X}+S))=0$
by (2.7) and Hypotheses 2.3.◻
3 Sums of three squares
In this section, we prove Theorem 0.1. We fix an even integer
$d=2\unicode[STIX]{x1D6FF}$
with
$\unicode[STIX]{x1D6FF}\geqslant 1$
.
We explain in § 3.1 the connection between sums of squares in
$\mathbb{R}(x_{1},x_{2})$
and line bundles on double covers of
$\mathbb{P}_{\mathbb{R}}^{2}$
, relating our problem to the study of Noether–Lefschetz loci. In §§ 3.2–3.3, we apply the results of § 2 to verify Green’s infinitesimal criterion for the family of real double covers of
$\mathbb{P}_{\mathbb{R}}^{2}$
, and the proof of Theorem 0.1 is completed in §§ 3.4–3.5.
3.1 Sums of squares and line bundles
We first recall two lemmas already used by Colliot-Thélène [Reference Colliot-ThélèneCol93].
Lemma 3.1. Let
$K$
be a field of characteristic
$\neq 2$
, and
$f\in K$
, and
$L:=K[\sqrt{-f}]$
. Then the following are equivalent:
(i)
$f$
is a sum of three squares in
$K$
;(ii)
$-1$
is a sum of two squares in
$L$
.
Proof. This is [Reference Colliot-ThélèneCol93, Lemma 1.2] (see also [Reference LamLam80, ch. 11 Theorem 2.7]). ◻
Lemma 3.2. Let
$S$
be a smooth projective geometrically connected variety over
$\mathbb{R}$
. Then the following are equivalent:
(i)
$-1$
is a sum of two squares in
$\mathbb{R}(S)$
;(ii) the pull-back map
$\operatorname{Pic}(S)\rightarrow \operatorname{Pic}(S_{\mathbb{C}})^{G}$
is not surjective.
Proof. This is [Reference van HamelvHam00, ch. I, Corollary 2.5]. For the convenience of the reader, we recall the argument. Condition (i) is equivalent to the nontrivial quaternion algebra over
$\mathbb{R}$
splitting over
$\mathbb{R}(S)$
[Reference Gille and SzamuelyGS06, Proposition 1.1.7], hence to the pull-back map
$\mathbb{Z}/2\mathbb{Z}\simeq \operatorname{Br}(\mathbb{R})\rightarrow \operatorname{Br}(\mathbb{R}(S))$
being zero. The exact sequence [Reference Colliot-ThélèneCol93, Lemma 1.1],
$$\begin{eqnarray}0\rightarrow \operatorname{Pic}(S)\rightarrow \operatorname{Pic}(S_{\mathbb{C}})^{G}\rightarrow \operatorname{Br}(\mathbb{R})\rightarrow \operatorname{Br}(\mathbb{R}(S)),\end{eqnarray}$$
shows that this is equivalent to (ii). ◻
In § 3.4, Lemmas 3.1 and 3.2 will be applied to a positive semidefinite polynomial
$f\in K=\mathbb{R}(x_{1},x_{2})$
, and to the quadratic extension
$L=K[\sqrt{-f}]=\mathbb{R}(S)$
associated to the double cover
$S\rightarrow \mathbb{P}_{\mathbb{R}}^{2}$
determined by
$f$
.
3.2 A real double cover containing a determinantal curve
In this paragraph, we construct varieties
$C$
,
$S$
, and
$X$
to which the results of § 2.2 apply.
Let
$\unicode[STIX]{x1D6E4}:=\{X_{0}^{2}+X_{1}^{2}+X_{2}^{2}=0\}\subset \mathbb{P}_{\mathbb{R}}^{2}$
be the anisotropic conic over
$\mathbb{R}$
. There is an isomorphism between its complexification
$\unicode[STIX]{x1D6E4}_{\mathbb{C}}$
and the projective line
$\mathbb{P}_{\mathbb{C}}^{1}$
. The line bundle
${\mathcal{L}}:={\mathcal{O}}_{\unicode[STIX]{x1D6E4}_{\mathbb{C}}}(1)$
is, however, not defined over
$\mathbb{R}$
for the real structure we consider on
$\unicode[STIX]{x1D6E4}_{\mathbb{C}}$
(as the zero locus of a real section of
${\mathcal{L}}$
would be a real point of
$\unicode[STIX]{x1D6E4}$
). Instead, it has a so-called quaternionic structure: it may be equipped with an isomorphism
$\unicode[STIX]{x1D719}:{\mathcal{L}}\stackrel{{\sim}}{\longrightarrow }\unicode[STIX]{x1D70E}^{\ast }\overline{{\mathcal{L}}}$
such that
$\unicode[STIX]{x1D719}\circ (\unicode[STIX]{x1D70E}^{\ast }\overline{\unicode[STIX]{x1D719}})=-\text{Id}$
, where
$\overline{{\mathcal{L}}}$
denotes the conjugate line bundle on the conjugate variety
$\overline{\unicode[STIX]{x1D6E4}_{\mathbb{C}}}$
, and the real structure
$\unicode[STIX]{x1D6E4}$
of
$\unicode[STIX]{x1D6E4}_{\mathbb{C}}$
is viewed as an isomorphism
$\unicode[STIX]{x1D70E}:\unicode[STIX]{x1D6E4}_{\mathbb{C}}\stackrel{{\sim}}{\longrightarrow }\overline{\unicode[STIX]{x1D6E4}_{\mathbb{C}}}$
(see for instance [Reference Biswas, Huisman and HurtubiseBHH10, § 3]). The isomorphism
$\unicode[STIX]{x1D719}$
induces a
$\mathbb{C}$
-antilinear automorphism
$\unicode[STIX]{x1D70E}^{\ast }$
of
$H^{0}(\unicode[STIX]{x1D6E4}_{\mathbb{C}},{\mathcal{L}})$
such that
$\unicode[STIX]{x1D70E}^{\ast }\circ \unicode[STIX]{x1D70E}^{\ast }=-1$
. One may then choose a basis
$(A,B)$
of
$H^{0}(\unicode[STIX]{x1D6E4}_{\mathbb{C}},{\mathcal{L}})$
on which the action of
$\unicode[STIX]{x1D70E}^{\ast }$
is given by
$$\begin{eqnarray}\unicode[STIX]{x1D70E}^{\ast }A=B\quad \text{and}\quad \unicode[STIX]{x1D70E}^{\ast }B=-A.\end{eqnarray}$$
In view of (3.1), the isomorphism
$\unicode[STIX]{x1D6E4}_{\mathbb{C}}\xrightarrow[{}]{{\sim}}\mathbb{P}_{\mathbb{C}}^{1}$
defined by
$x\mapsto [A(x):B(x)]$
induces an action of
$G$
on
$\mathbb{P}^{1}(\mathbb{C})$
given by
$\unicode[STIX]{x1D70E}([a:b])=[\overline{b}:-\overline{a}]$
.
We define
$Y:=\unicode[STIX]{x1D6E4}\times \mathbb{P}_{\mathbb{R}}^{2}$
and
$X:=Y_{\mathbb{C}}\simeq \unicode[STIX]{x1D6E4}_{\mathbb{C}}\times \mathbb{P}_{\mathbb{C}}^{2}$
to be the complexification of
$Y$
with the induced real structure. We set
$n:=2+\unicode[STIX]{x1D6FF}$
, and introduce the following line bundles on
$X$
:
$H_{j}:=p_{1}^{\ast }{\mathcal{O}}_{\unicode[STIX]{x1D6E4}_{\mathbb{C}}}(1)$
if
$j=1,2$
,
$H_{j}=p_{2}^{\ast }{\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{2}}(1)$
if
$j\in \{3,\ldots ,2+\unicode[STIX]{x1D6FF}\}$
,
$H=\sum _{j}H_{j}$
and
$L={\mathcal{O}}_{X}$
. Note that Hypotheses 2.3 are satisfied.
Let
$\mathbb{P}$
be the parameter space for
$(n+1)\times n$
matrices
$M=(M_{i,j})_{1\leqslant i\leqslant n+1,1\leqslant j\leqslant n}$
with
$M_{i,j}\in H^{0}(X,H_{j})$
such that no column is identically zero, and whose columns are well defined up to multiplication by a scalar:
$\mathbb{P}$
is a product of
$n$
projective spaces. To such a matrix
$M$
, we associate the variety
$C\subset X$
defined by the vanishing of all the maximal minors of
$M$
. Let
$\unicode[STIX]{x1D70F}\in H^{0}(X,H)$
be a section vanishing on
$C$
, and
$S\subset X$
be the zero-locus of
$\unicode[STIX]{x1D70F}$
.
Lemma 3.3. There exist
$M$
and
$\unicode[STIX]{x1D70F}$
such that:
(i)
$C$
is a smooth curve and
$S$
is a smooth surface;(ii) the projection
$p_{2}|_{S}:S\rightarrow \mathbb{P}_{\mathbb{C}}^{2}$
is a finite double cover ramified along a smooth degree
$d$
curve
$D\subset \mathbb{P}_{\mathbb{C}}^{2}$
;(iii) the subvarieties
$C$
and
$S$
of
$X$
are defined over
$\mathbb{R}$
.
Proof. Lemma 2.4 shows that there is a nonempty Zariski-open subset of
$\mathbb{P}$
over which
$C$
is smooth, and that for a generic choice of
$\unicode[STIX]{x1D70F}$
,
$S$
is also smooth.
Let us show that if
$M$
is general, the projection
$S\rightarrow \mathbb{P}_{\mathbb{C}}^{2}$
is finite when
$\unicode[STIX]{x1D70F}$
is the first maximal minor
$\det (M_{i,j})_{1\leqslant i,j\leqslant n}$
of
$M$
(and consequently when
$\unicode[STIX]{x1D70F}$
is general). It suffices to exhibit one such
$M$
, for which we use homogeneous coordinates
$A,B$
on
$\unicode[STIX]{x1D6E4}_{\mathbb{C}}$
as above and
$X_{0},X_{1},X_{2}$
on
$\mathbb{P}_{\mathbb{C}}^{2}$
. One can take
$$\begin{eqnarray}M=\left(\begin{array}{@{}ccc@{}}A & 0 & X_{0}\\ B & A & X_{1}\\ 0 & B & X_{2}\end{array}\right)\end{eqnarray}$$
when
$\unicode[STIX]{x1D6FF}=1$
. One verifies that a general matrix whose first and second columns are
$(A,B,0,\ldots ,0)$
and
$(0,0,A,B,0,\ldots ,0)$
works when
$\unicode[STIX]{x1D6FF}\geqslant 2$
.
We have shown the existence of a nonempty Zariski-open subset
$U\subset \mathbb{P}$
for which
$C$
is a smooth curve, and such that
$S$
is smooth with finite projection
$S\rightarrow \mathbb{P}_{\mathbb{C}}^{2}$
for a general choice of
$\unicode[STIX]{x1D70F}$
. If
$P(X_{0},X_{1},X_{2})A^{2}+Q(X_{0},X_{1},X_{2})AB+R(X_{0},X_{1},X_{2})B^{2}=0$
is the equation in
$X$
of such a surface
$S$
, where
$P,Q$
and
$R$
have degree
$\unicode[STIX]{x1D6FF}$
, a direct computation shows that the projection
$S\rightarrow \mathbb{P}_{\mathbb{C}}^{2}$
is finite of degree
$2$
with ramification locus
$D\subset \mathbb{P}_{\mathbb{C}}^{2}$
defined by the equation
$Q^{2}=4PR$
of degree
$2\unicode[STIX]{x1D6FF}=d$
, and that the smoothness of
$S$
implies that of
$D$
.
Let
$U^{\prime }\subset \mathbb{P}$
be the open set consisting of matrices whose first two columns are not proportional, and notice that
$U\subset U^{\prime }$
. There are fixed point free actions of
$\mathbb{Z}/2\mathbb{Z}$
on
$U^{\prime }$
and
$U$
obtained by exchanging the first two columns. The quotients are smooth complex algebraic varieties
$U/(\mathbb{Z}/2\mathbb{Z})\subset U^{\prime }/(\mathbb{Z}/2\mathbb{Z})$
.
Letting
$\unicode[STIX]{x1D70E}$
act on
$H^{0}(\mathbb{P}_{\mathbb{C}}^{2},{\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{2}}(1))$
using the natural real structure
${\mathcal{O}}_{\mathbb{P}_{\mathbb{R}}^{2}}(1)$
of
${\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{2}}(1)$
and on
$H^{0}(\mathbb{P}_{\mathbb{C}}^{1},{\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{1}}(1))$
using (3.1), we obtain a
$G$
-action on
$\mathbb{P}$
, which descends to a
$G$
-action on
$U/(\mathbb{Z}/2\mathbb{Z})$
and
$U^{\prime }/(\mathbb{Z}/2\mathbb{Z})$
, endowing these algebraic varieties over
$\mathbb{C}$
with a real structure by § 0.4. It is obvious that
$U^{\prime }/(\mathbb{Z}/2\mathbb{Z})$
has a real point for this real structure (for instance, choose
$M_{1,1}=A$
,
$M_{1,2}=B$
, and
$M_{i,j}=0$
otherwise). Since
$U^{\prime }/(\mathbb{Z}/2\mathbb{Z})$
is smooth and irreducible, the implicit function theorem shows that its real points are Zariski-dense [Reference BenoistBen17, Proposition 1.1], and we deduce that
$U/(\mathbb{Z}/2\mathbb{Z})$
also has a real point. Choose
$M$
to be a matrix lifting this real point: then
$C\subset X$
is defined over
$\mathbb{R}$
. Finally, choose
$\unicode[STIX]{x1D70F}$
general and defined over
$\mathbb{R}$
to ensure that
$S\subset X$
is also defined over
$\mathbb{R}$
.◻
Remark 3.4. A variant of our strategy would have been to choose
$n=1+\unicode[STIX]{x1D6FF}$
,
$H_{1}:=p_{1}^{\ast }{\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{1}}(2)$
(which has the advantage of being defined over
$\mathbb{R}$
),
$H_{j}=p_{2}^{\ast }{\mathcal{O}}_{\mathbb{P}_{\mathbb{C}}^{2}}(1)$
if
$j\in \{2,\ldots ,1+\unicode[STIX]{x1D6FF}\}$
and
$L={\mathcal{O}}_{X}$
. Unfortunately, with these choices, assertion (ii) of Lemma 3.3 would not hold.
3.3 Verifying Green’s criterion for a family of double covers
To be able to apply Proposition 1.3 to the family of double covers of
$\mathbb{P}_{\mathbb{R}}^{2}$
, we need a connectedness result going back to Hilbert [Reference HilbertHil88, p. 344], which we state first.
Let
$V:=\mathbb{C}[X_{0},\ldots X_{N}]_{d}$
be the space of degree
$d$
homogeneous polynomials
$F$
in
$N+1$
variables, endowed with its natural real structure, and let
$B\subset V$
be the Zariski-open subset parametrizing polynomials
$F$
whose zero locus
$\{F=0\}\subset \mathbb{P}_{\mathbb{C}}^{N}$
is a smooth hypersurface. Define
$P_{d}^{\operatorname{sm}}\subset B(\mathbb{R})$
to be the locus where
$F$
is positive semidefinite, and let
$P_{d}^{+}\subset V(\mathbb{R})$
be the set of polynomials
$F$
such that
$F(x)>0$
if
$x\in \mathbb{R}^{N+1}\setminus \{0\}$
.
Proposition 3.5. The set
$P_{d}^{\operatorname{sm}}$
is open, connected and equal to
$B(\mathbb{R})\cap P_{d}^{+}$
.
Proof. Let
$F\in P_{d}^{\operatorname{sm}}$
and
$x\in \mathbb{R}^{N+1}\setminus \{0\}$
. If
$F(x)>0$
did not hold, then
$F(x)=0$
, and
$x$
would be a smooth point of
$\{F=0\}$
. Consequently, the differential
$dF_{x}$
would be surjective and
$F$
would take negative values near
$x$
, which is a contradiction. This shows that
$P_{d}^{\operatorname{sm}}=B(\mathbb{R})\cap P_{d}^{+}$
. Since
$P_{d}^{+}$
is open and convex [Reference BenoistBen17, Lemma 4.2], it follows that
$P_{d}^{\operatorname{sm}}$
is open and
$P_{d}^{+}$
is connected.
The complement of
$P_{d}^{\operatorname{sm}}$
in
$P_{d}^{+}$
consist of polynomials
$F$
such that
$\{F=0\}$
is singular and has no real points. It follows that
$\{F=0\}$
has at least two singular points (any of them and its distinct complex conjugate). But the set polynomials
$F\in V$
such that
$\{F=0\}$
has at least two singular points has codimension
${\geqslant}2$
, so that
$P_{d}^{\operatorname{sm}}$
is the complement in
$P_{d}^{+}$
of a semialgebraic set of codimension
${\geqslant}2$
. Since the open set
$P_{d}^{+}\subset V(\mathbb{R})$
is connected, so is
$P_{d}^{\operatorname{sm}}$
.◻
For the remainder of this section, we restrict to the case
$N=2$
. We consider the universal family
${\mathcal{S}}\subset \mathbb{P}(1,1,1,\unicode[STIX]{x1D6FF})\times B$
with projection
$\unicode[STIX]{x1D70B}:{\mathcal{S}}\rightarrow B$
, parametrizing double covers of
$\mathbb{P}^{2}$
ramified over a smooth curve of degree
$d$
: if
$b\in B(\mathbb{C})$
corresponds to the polynomial
$F$
, one has
${\mathcal{S}}_{b}=\{Z^{2}+F(X_{0},X_{1},X_{2})=0\}\subset \mathbb{P}(1,1,1,\unicode[STIX]{x1D6FF})$
. The map
$\unicode[STIX]{x1D70B}$
is a smooth projective morphism of algebraic varieties over
$\mathbb{R}$
. We are interested in the fibers of
$\unicode[STIX]{x1D70B}$
over
$P_{d}^{\operatorname{sm}}\subset B(\mathbb{R})$
.
Lemma 3.6. For a dense set of
$b\in P_{d}^{\operatorname{sm}}$
, there exists
$\unicode[STIX]{x1D706}\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b}(\mathbb{C}))(1)^{G}$
such that the composition
$$\begin{eqnarray}T_{B_{\mathbb{C}},b}\longrightarrow H^{1}({\mathcal{S}}_{b,\mathbb{C}},T_{{\mathcal{S}}_{b,\mathbb{C}}})\stackrel{\unicode[STIX]{x1D706}}{\longrightarrow }H^{2}({\mathcal{S}}_{b,\mathbb{C}},{\mathcal{O}}_{{\mathcal{S}}_{b,\mathbb{C}}})\end{eqnarray}$$
of the Kodaira–Spencer map and of the contracted cup-product with
$\unicode[STIX]{x1D706}$
is surjective.
Proof. It follows from the conditions listed in Lemma 3.3 that the surface
$S$
constructed there is isomorphic to a real member of the family
$\unicode[STIX]{x1D70B}:{\mathcal{S}}\rightarrow B$
: there exists
$b_{0}\in B(\mathbb{R})$
such that
$S\simeq {\mathcal{S}}_{b_{0}}$
. Since
$S$
projects to
$\unicode[STIX]{x1D6E4}$
,
$S(\mathbb{R})=\varnothing$
, and we deduce that
$b_{0}\in P_{d}^{\operatorname{sm}}$
. By Remark 1.4, the cohomology class
$\unicode[STIX]{x1D706}_{0}$
of the line bundle
${\mathcal{L}}={\mathcal{O}}_{S}(C)$
associated to the curve constructed in Lemma 3.3 belongs to
$H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b_{0}}(\mathbb{C}))(1)^{G}$
. Moreover, Proposition 2.5, shows that the contracted cup-product
$H^{1}({\mathcal{S}}_{b_{0},\mathbb{C}},T_{{\mathcal{S}}_{b_{0},\mathbb{C}}})\stackrel{\unicode[STIX]{x1D706}_{0}}{\longrightarrow }H^{2}({\mathcal{S}}_{b_{0},\mathbb{C}},{\mathcal{O}}_{{\mathcal{S}}_{b_{0},\mathbb{C}}})$
is surjective.
The deformation theory of the family of smooth double covers of
$\mathbb{P}^{2}$
, carried out in [Reference ManettiMan99, p. 260], shows that the Kodaira–Spencer map
$T_{B_{\mathbb{C}},b_{0}}\rightarrow H^{1}({\mathcal{S}}_{b_{0},\mathbb{C}},T_{{\mathcal{S}}_{b_{0},\mathbb{C}}})$
is surjective unless
$d=6$
. Indeed, [Reference ManettiMan99, (1.3’)] applied with
$Y=\mathbb{P}^{2}$
shows that the cokernel of this map embeds into
$H^{1}(\mathbb{P}^{2},T_{\mathbb{P}^{2}})\oplus H^{1}(\mathbb{P}^{2},T_{\mathbb{P}^{2}}(-\unicode[STIX]{x1D6FF}))$
, that vanishes when
$\unicode[STIX]{x1D6FF}\neq 3$
and is one-dimensional when
$\unicode[STIX]{x1D6FF}=3$
(as one computes using the Euler exact sequence and Serre duality). When
$d\neq 6$
, this shows at once that the map (3.2) is surjective for
$(b,\unicode[STIX]{x1D706})=(b_{0},\unicode[STIX]{x1D706}_{0})$
. In the exceptional case
$d=6$
, the double covers are
$K3$
surfaces, and the image of the Kodaira–Spencer map
$T_{B_{\mathbb{C}},b_{0}}\rightarrow H^{1}({\mathcal{S}}_{b_{0},\mathbb{C}},T_{{\mathcal{S}}_{b_{0},\mathbb{C}}})$
is included in the subspace of polarized infinitesimal deformations, that preserve the line bundle
${\mathcal{O}}_{\mathbb{P}^{2}}(1)$
. This subspace has codimension
$1$
(see [Reference HuybrechtsHuy16, ch. 6, 2.4]). Since the image of
$H^{0}(S,N_{S/X})\rightarrow H^{1}(S,T_{S})$
in (2.1) lands in this subspace, we still deduce from Proposition 2.5 the surjectivity of (3.2) for
$(b,\unicode[STIX]{x1D706})=(b_{0},\unicode[STIX]{x1D706}_{0})$
.
The lemma then follows from the last statement of Proposition 1.3, which applies because
$P_{d}^{\operatorname{sm}}\subset B(\mathbb{R})$
is open and connected by Proposition 3.5.◻
3.4 Density
We are now ready to give the proof of the density statement of Theorem 0.1. Recall from § 0.1 that
$P_{d}$
(respectively
$Q_{d}$
) is the subset of
$\mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
consisting of polynomials that are positive semidefinite (respectively sums of three squares in
$\mathbb{R}(x_{1},x_{2})$
).
Proposition 3.7. The set
$Q_{d}$
is dense in
$P_{d}$
.
Proof. Fix an open subset
$U\subset P_{d}$
, and
$b\in U\subset \mathbb{R}[X_{0},X_{1},X_{2}]_{d}$
corresponding to the inhomogeneous polynomial
$f\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
. Our goal is to construct
$b^{\prime }\in U$
such that the associated polynomial
$f^{\prime }\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
is a sum of three squares in
$\mathbb{R}(x_{1},x_{2})$
.
Replacing
$f$
with
$f+t(1+x_{1}^{d}+x_{2}^{d})$
for
$t\in \mathbb{R}_{{>}0}$
small enough, we may assume that
$b\in P_{d}^{\operatorname{sm}}$
(see § 3.3). Up to changing
$b\in U$
again, Lemma 3.6 allows us to suppose that there exists
$\unicode[STIX]{x1D706}\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b}(\mathbb{C}))(1)^{G}$
such that (3.2) is surjective. Consequently, we can choose an open cone
$\unicode[STIX]{x1D6FA}\subset H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))^{G}$
as in Proposition 1.3.
Shrinking
$U$
, we may assume it is of the form
$\unicode[STIX]{x1D6E5}(\mathbb{R})$
for some
$\unicode[STIX]{x1D6E5}\subset B(\mathbb{C})$
as in § 1.2. Denote by
${\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})}$
the inverse image of
$\unicode[STIX]{x1D6E5}(\mathbb{R})$
by
$\unicode[STIX]{x1D70B}:{\mathcal{S}}(\mathbb{C})\rightarrow B(\mathbb{C})$
. By a
$G$
-equivariant version of Ehresmann’s theorem [Reference DimcaDim85, Lemma 4], it is possible, after shrinking
$\unicode[STIX]{x1D6E5}$
, to ensure that there is a
$G$
-equivariant diffeomorphism commuting with the projection to
$\unicode[STIX]{x1D6E5}(\mathbb{R})$
:
$$\begin{eqnarray}{\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})}\stackrel{{\sim}}{\longrightarrow }{\mathcal{S}}_{b}(\mathbb{C})\times \unicode[STIX]{x1D6E5}(\mathbb{R}).\end{eqnarray}$$
Let us now look at the Hochschild–Serre spectral sequence computing the
$G$
-equivariant cohomology of
${\mathcal{S}}_{b}(\mathbb{C})$
:
$$\begin{eqnarray}E_{2}^{p,q}=H^{p}(G,H^{q}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z}(1)))\;\Longrightarrow \;H_{G}^{p+q}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z}(1)).\end{eqnarray}$$
Since
$H_{1}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z})=0$
(see [Reference DimcaDim85, Proposition 6(i), (ii)]), [Reference van HamelvHam99, Lemma 2.3] shows that the cokernel of the edge morphism
$\unicode[STIX]{x1D700}:H_{G}^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z}(1))\rightarrow H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z}(1))^{G}$
is isomorphic to
$\mathbb{Z}/2\mathbb{Z}$
. Let
$\unicode[STIX]{x1D6EC}$
be the image in
$H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))^{G}$
of the complement of the image of
$\unicode[STIX]{x1D700}$
. Since
$\unicode[STIX]{x1D6EC}$
is a translate of a lattice in the real vector space
$H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))^{G}$
, it meets the open cone
$\unicode[STIX]{x1D6FA}$
. Consequently, we can find a class
$\unicode[STIX]{x1D714}\in H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z}(1))^{G}$
not in the image of
$\unicode[STIX]{x1D700}$
, whose image in
$H^{2}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{R}(1))^{G}$
, still denoted by
$\unicode[STIX]{x1D714}$
, belongs to
$\unicode[STIX]{x1D6FA}$
. By construction of
$\unicode[STIX]{x1D6FA}$
, there exists
$b^{\prime }\in \unicode[STIX]{x1D6E5}(\mathbb{R})$
such that the parallel transport
$\unicode[STIX]{x1D714}^{\prime }\in H^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{R}(1))^{G}$
belongs to
$H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}))(1)^{G}$
.
For every
$k\geqslant 0$
, the two restriction maps
$H^{k}({\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})},\mathbb{Z}(1))\rightarrow H^{k}({\mathcal{S}}_{b}(\mathbb{C}),\mathbb{Z}(1))$
and
$H^{k}({\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})},\mathbb{Z}(1))\rightarrow H^{k}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))$
are
$G$
-equivariant isomorphisms by (3.3) and contractibility of
$\unicode[STIX]{x1D6E5}(\mathbb{R})$
. We deduce that the restriction map from the Hochschild–Serre spectral sequence for the
$G$
-equivariant cohomology of
${\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})}$
$$\begin{eqnarray}E_{2}^{p,q}=H^{p}(G,H^{q}({\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})},\mathbb{Z}(1)))\;\Longrightarrow \;H_{G}^{p+q}({\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})},\mathbb{Z}(1))\end{eqnarray}$$
to that (3.4) for
${\mathcal{S}}_{b}(\mathbb{C})$
is an isomorphism in page
$2$
, and hence an isomorphism. The same goes for the restriction map between the Hochschild–Serre spectral sequences for
${\mathcal{S}}(\mathbb{C})|_{\unicode[STIX]{x1D6E5}(\mathbb{R})}$
and
${\mathcal{S}}_{b^{\prime }}(\mathbb{C})$
. We can thus deduce from the corresponding property of
$\unicode[STIX]{x1D714}$
that
$\unicode[STIX]{x1D714}^{\prime }\in H^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{R}(1))^{G}$
lifts to a class
$\unicode[STIX]{x1D714}^{\prime }\in H^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))^{G}$
not in the image of the edge map
$\unicode[STIX]{x1D700}^{\prime }:H_{G}^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))\rightarrow H^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))^{G}$
.
Since
$\operatorname{Pic}^{0}({\mathcal{S}}_{b^{\prime },\mathbb{C}})=0$
by [Reference DimcaDim85, Proposition 6(i)], the Lefschetz
$(1,1)$
theorem shows that
$$\begin{eqnarray}\operatorname{Pic}({\mathcal{S}}_{b^{\prime },\mathbb{C}})\stackrel{{\sim}}{\longrightarrow }\operatorname{Hdg}^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1)):=H^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))\cap H^{1,1}({\mathcal{S}}_{b^{\prime }}(\mathbb{C})),\end{eqnarray}$$
and hence that
$\unicode[STIX]{x1D714}^{\prime }\in \operatorname{Hdg}^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))^{G}$
is the class of a line bundle
${\mathcal{L}}\in \operatorname{Pic}({\mathcal{S}}_{b^{\prime },\mathbb{C}})^{G}$
. If
${\mathcal{L}}$
were induced by a real line bundle on
${\mathcal{S}}_{b^{\prime }}$
, the existence of a cycle class map with value in
$G$
-equivariant Betti cohomology [Reference KrasnovKra91, § 1.3] would show that
$\unicode[STIX]{x1D714}^{\prime }$
lifts to
$H_{G}^{2}({\mathcal{S}}_{b^{\prime }}(\mathbb{C}),\mathbb{Z}(1))$
, and thus we would have a contradiction.
By implication (ii)
$\;\Longrightarrow \;$
(i) of Lemma 3.2, we deduce that
$-1$
is a sum of two squares in
$\mathbb{R}({\mathcal{S}}_{b^{\prime }})$
. Lemma 3.1 then shows that the polynomial
$f^{\prime }\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
associated to
$b^{\prime }$
is a sum of three squares in
$\mathbb{R}(x_{1},x_{2})$
, which is what we wanted.◻
3.5 The set of sums of three squares
We finally complete the proof of Theorem 0.1.
Proof of Theorem 0.1.
Let
$Q_{d}^{N}\subset \mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
be the set of polynomials
$f$
such that there exist polynomials
$g,h_{1},h_{2},h_{3}\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}N}$
of degree
${\leqslant}N$
satisfying
$$\begin{eqnarray}fg^{2}=h_{1}^{2}+h_{2}^{2}+h_{3}^{2}\quad \text{and}\quad g\neq 0.\end{eqnarray}$$
It is an immediate consequence of the Tarski–Seidenberg theorem [Reference Bochnak, Coste and RoyBCR98, Theorem 2.2.1] that
$Q_{d}^{N}$
is a semialgebraic subset of
$P_{d}$
. Let us prove that
$Q_{d}^{N}$
is a closed subset of
$\mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
by adapting [Reference Berg, Christensen and JensenBCJ79, Theorem 3]. Consider the norm
$\Vert h\Vert :=\sup _{p\in [0,1]^{2}}h(p)$
on
$\mathbb{R}[x_{1},x_{2}]$
. Let
$(f_{j})_{j\in \mathbb{N}}$
be a sequence of elements of
$Q_{d}^{N}$
converging to
$f\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
. Since the case
$f=0$
is trivial, we may assume that the
$f_{j}$
are nonzero. Choose
$g_{j},h_{j,1},h_{j,2},h_{j,3}\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}N}$
with
$g_{j}\neq 0$
such that
$$\begin{eqnarray}f_{j}g_{j}^{2}=h_{j,1}^{2}+h_{j,2}^{2}+h_{j,3}^{2}.\end{eqnarray}$$
Up to scaling
$g_{j}$
and the
$h_{j,i}$
, we may assume that
$\Vert f_{j}g_{j}^{2}\Vert =1$
and as a consequence that
$\Vert h_{j,i}\Vert \leqslant 1$
for
$1\leqslant i\leqslant 3$
. Extracting subsequences, we may ensure that the sequences
$(g_{j})$
and
$(h_{j,i})$
converge to polynomials
$g,h_{i}\in \mathbb{R}[x_{1},x_{2}]_{{\leqslant}N}$
. Taking the limit, we see that
$\Vert fg^{2}\Vert =1$
so that
$g\neq 0$
, and that (3.5) holds. Since
$Q_{d}=\bigcup _{N\in \mathbb{N}}Q_{d}^{N}$
, this proves the first assertion of Theorem 0.1.
If
$d\geqslant 6$
, it is a consequence of the Noether–Lefschetz theorem applied as in [Reference Colliot-ThélèneCol93] that
$Q_{d}$
has empty interior. More precisely, every open subset of
$\mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
contains a polynomial whose coefficients are algebraically independent over
$\mathbb{Q}$
, and such a polynomial cannot be a sum of three squares of rational functions by [Reference Colliot-ThélèneCol93, Theorem 3.1]. One could also argue as in [Reference Colliot-ThélèneCol93, Remark 4.3]:
$Q_{d}$
is included in a countable union of proper closed algebraic subvarieties of
$\mathbb{R}[x_{1},x_{2}]_{{\leqslant}d}$
, and hence has empty interior by a Baire category argument.
Finally, we have proven the last assertion of Theorem 0.1 in Proposition 3.7. ◻
4 Surfaces whose function field has level
$2$
The proof of Theorem 0.2 is analogous to that of Proposition 3.7. We fix an even integer
$d\geqslant 2$
.
Consider
$X:=\mathbb{P}_{\mathbb{C}}^{3}$
, define
$B\subset \mathbb{P}(H^{0}(X,{\mathcal{O}}_{X}(d)))$
to be the subset parametrizing equations
$F$
defining smooth surfaces
$S\subset X$
, and
$\unicode[STIX]{x1D70B}:{\mathcal{S}}\rightarrow B$
to be the universal surface. Endow
$X$
,
$B$
and
${\mathcal{S}}$
with their natural real structures. Recall from § 0.3 that
$\unicode[STIX]{x1D6E9}_{d}\subset B(\mathbb{R})$
is the set of equations
$F$
whose associated surfaces
$S=\{F=0\}$
have no real points. Since
$\mathbb{R}^{4}\setminus \{0\}$
is connected,
$\unicode[STIX]{x1D6E9}_{d}$
consists of equations, well defined up to a scalar, that are either positive or negative on
$\mathbb{R}^{4}\setminus \{0\}$
and Proposition 3.5 implies that
$\unicode[STIX]{x1D6E9}_{d}\subset B(\mathbb{R})$
is open and connected.
Set
$n:=d$
if
$d\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
and
$n:=d-2$
if
$d\equiv 2\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
. Define
$H_{j}:={\mathcal{O}}_{X}(1)$
for
$1\leqslant j\leqslant n$
,
$L:={\mathcal{O}}_{X}$
if
$d\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
and
$L:={\mathcal{O}}_{X}(2)$
if
$d\equiv 2\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
. Note that Hypotheses 2.3 are satisfied. When
$d\equiv 2\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
, those are exactly the original choices of Ciliberto and Lopez [Reference Ciliberto and LopezCL91].
Consider
$(n+1)\times n$
matrices
$M=(M_{i,j})_{1\leqslant i\leqslant n+1,1\leqslant j\leqslant n}$
with
$M_{i,j}\in H^{0}(X,H_{j})$
. To such a matrix
$M$
, we associate the variety
$C\subset X$
defined by the vanishing of all the maximal minors of
$M$
. We also consider a section
$\unicode[STIX]{x1D70F}\in H^{0}(X,{\mathcal{O}}(d))$
vanishing on
$C$
with zero locus
$S\subset X$
. The following is an analogue of Lemma 3.3.
Lemma 4.1. It is possible to find
$M$
and
$\unicode[STIX]{x1D70F}$
such that:
(i)
$C$
is a smooth curve and
$S$
is a smooth surface;(ii) the subvarieties
$C$
and
$S$
of
$X$
are defined over
$\mathbb{R}$
;(iii)
$S(\mathbb{R})=\varnothing$
.
Proof. Since
$n\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
, one may consider a particular choice
$M^{0}$
of
$M$
, that is defined over
$\mathbb{R}$
, and whose
$n\times n$
submatrix
$(M_{i,j}^{0})_{1\leqslant i,j\leqslant n}$
is diagonal by blocks with blocks:
$$\begin{eqnarray}\left(\begin{array}{@{}cccc@{}}X_{0} & -X_{1} & X_{2} & X_{3}\\ X_{1} & X_{0} & X_{3} & -X_{2}\\ X_{2} & X_{3} & -X_{0} & X_{1}\\ X_{3} & -X_{2} & -X_{1} & -X_{0}\end{array}\right).\end{eqnarray}$$
Computing that the determinant of every such block is
$\sum _{0\leqslant i,j\leqslant 3}X_{i}^{2}X_{j}^{2}$
, one sees that
$\unicode[STIX]{x1D702}^{0}:=\det ((M_{i,j}^{0})_{1\leqslant i,j\leqslant n})\in H^{0}(X,{\mathcal{O}}_{X}(n))$
does not vanish on
$X(\mathbb{R})$
. Let
$M$
be a general small real perturbation of
$M^{0}$
: the property that
$\unicode[STIX]{x1D702}:=\det ((M_{i,j})_{1\leqslant i,j\leqslant n})$
does not vanish on
$X(\mathbb{R})$
persists. Choose
$\unicode[STIX]{x1D70F}$
to be a general small real perturbation of
$\unicode[STIX]{x1D702}$
if
$d\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
(respectively of
$(X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2})\cdot \unicode[STIX]{x1D702}$
if
$d\equiv 2\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
). By Lemma 2.4, the curve
$C$
and the surface
$S$
are smooth, and the pair
$(M,\unicode[STIX]{x1D70F})$
satisfies the required conditions.◻
Remark 4.2. When
$d\equiv 2\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
, it is not possible to run our strategy with
$n=d$
and
$L={\mathcal{O}}_{X}$
as when
$d\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
. Indeed, the degree of the determinantal curve
$C$
, that is equal to
$n(n+1)/2$
by [Reference Harris and TuHT84, Proposition 12(a)], would be odd. Consequently,
$C$
would have a real point and assertion (iii) of Lemma 4.1 could not hold.
We deduce an analogue of Lemma 3.6.
Lemma 4.3. For a dense set of
$b\in \unicode[STIX]{x1D6E9}_{d}$
, there exists
$\unicode[STIX]{x1D706}\in H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b}(\mathbb{C}))(1)^{G}$
such that the composition of the Kodaira–Spencer map and of the contracted cup-product with
$\unicode[STIX]{x1D706}$
$$\begin{eqnarray}T_{B_{\mathbb{C}},b}\longrightarrow H^{1}({\mathcal{S}}_{\mathbb{ C},b},T_{{\mathcal{S}}_{\mathbb{C},b}})\stackrel{\unicode[STIX]{x1D706}}{\longrightarrow }H^{2}({\mathcal{S}}_{\mathbb{ C},b},{\mathcal{O}}_{{\mathcal{S}}_{\mathbb{C},b}})\end{eqnarray}$$
is surjective.
Proof. The surface
$S$
constructed in Lemma 4.1 is isomorphic to a particular member
${\mathcal{S}}_{b_{0}}$
of the family
$\unicode[STIX]{x1D70B}:{\mathcal{S}}\rightarrow B$
, corresponding to a point
$b_{0}\in \unicode[STIX]{x1D6E9}_{d}$
. By Remark 1.4, the cohomology class
$\unicode[STIX]{x1D706}_{0}$
of the real curve
$C\subset S$
constructed in Lemma 4.1 belongs to
$H_{\mathbb{R}}^{1,1}({\mathcal{S}}_{b_{0}}(\mathbb{C}))(1)^{G}$
. By Proposition 2.5, the map (4.1) for
$b=b_{0}$
and
$\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{0}$
is surjective. The lemma then follows from the last part of Proposition 1.3, which applies because
$\unicode[STIX]{x1D6E9}_{d}\subset B(\mathbb{R})$
is open and connected.◻
We can now give the following proof.
Proof of Theorem 0.2.
Fix an open subset
$U\subset \unicode[STIX]{x1D6E9}_{d}$
, and
$b\in U$
according to Lemma 4.3. Running the proof of Proposition 3.7 (replacing
$P_{d}^{\operatorname{sm}}$
by
$\unicode[STIX]{x1D6E9}_{d}$
) shows that there exists
$b^{\prime }\in U$
such that
$\operatorname{Pic}({\mathcal{S}}_{b^{\prime }})\rightarrow \operatorname{Pic}({\mathcal{S}}_{b^{\prime },\mathbb{C}})^{G}$
is not surjective. Implication (ii)
$\;\Longrightarrow \;$
(i) of Lemma 3.2 completes the proof.◻
Acknowledgements
I would like to thank Olivier Wittenberg for numerous discussions on related topics, as well as suggestions to improve this paper.
