Spinors, Twistors and Classical Geometry
The paper studies explicitly the Hitchin system restricted to the Higgs fields on a fixed, very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate to both the geometry of Penrose's twistor spaces and several classical results.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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Інститут математики НАН України
2021
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Spinors, Twistors and Classical Geometry. Nigel J. Hitchin. SIGMA 17 (2021), 090, 9 pages |
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| author | Hitchin, Nigel J. |
| author_facet | Hitchin, Nigel J. |
| citation_txt | Spinors, Twistors and Classical Geometry. Nigel J. Hitchin. SIGMA 17 (2021), 090, 9 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The paper studies explicitly the Hitchin system restricted to the Higgs fields on a fixed, very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate to both the geometry of Penrose's twistor spaces and several classical results.
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 090, 9 pages
Spinors, Twistors and Classical Geometry
Nigel J. HITCHIN
Mathematical Institute, Woodstock Road, Oxford, OX2 6GG, UK
E-mail: hitchin@maths.ox.ac.uk
URL: https://people.maths.ox.ac.uk/hitchin/
Received August 07, 2021, in final form October 07, 2021; Published online October 10, 2021
https://doi.org/10.3842/SIGMA.2021.090
Abstract. The paper studies explicitly the Hitchin system restricted to the Higgs fields on
a fixed very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate
to both the geometry of Penrose’s twistor spaces and several classical results.
Key words: spinor; twistor; quadric; stable bundle
2020 Mathematics Subject Classification: 14H60; 32L25
Dedicated to Roger Penrose
on the occasion of his 90th birthday
1 Introduction
In 1974 I returned to Oxford after three postdoctoral years in the USA and found there a dis-
tinctly new style of mathematics emanating from Roger Penrose’s group. From seminars and
talking to students and postdocs I gradually learned what twistor theory was all about and
it subsequently became a central idea in much of my work, in four and fewer dimensions and
higher, especially in the context of hyperkähler geometry. In fact my papers listed in the bibli-
ography [6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17] are all about the twistor interpretation of various
constructions in differential geometry.
This article is based on a recent encounter with spinors and twistors in trying to understand in
more detail the integrable system associated to moduli spaces of Higgs bundles, part of a project
with Tamás Hausel, and by looking at curves of genus 2 and 3 we come face to face with some
classical results in projective geometry. A common theme seems to be families of quadrics, in
particular the Klein quadric which is Penrose’s compactified, complexified Minkowski space, and
looking at these issues through twistor-coloured glasses reveals some new approaches to these
moduli spaces.
To be more specific, we observe a relationship between the intersection of two quadrics in P5
viewed as both a twistor space and a moduli space of stable bundles on a curve of genus 2, and
use the formalism of self-duality in four dimensions to describe the integrable system for a curve
of genus 3. The classical contributors are Plücker, Salmon, Cayley, Hesse . . . but the modern
reader is referred to [2] for an excellent account of many of these themes.
2 Spinors and twistors
2.1 Four dimensions
The first thing one learns in twistor theory is how to deal with four dimensions using two-
component spinors. If S+, S− are 2-dimensional complex vector spaces with skew forms ϵ+, ϵ−
This paper is a contribution to the Special Issue on Twistors from Geometry to Physics in honor of Roger
Penrose. The full collection is available at https://www.emis.de/journals/SIGMA/Penrose.html
mailto:hitchin@maths.ox.ac.uk
https://people.maths.ox.ac.uk/hitchin/
https://doi.org/10.3842/SIGMA.2021.090
https://www.emis.de/journals/SIGMA/Penrose.html
2 N.J. Hitchin
then the tensor product V = S+ ⊗ S− has an induced non-degenerate symmetric bilinear form.
In particular a null vector v can be written v = ϕ⊗ ψ.
The exterior product now has a decomposition Λ2V = S2
+ ⊕ S2
− as the direct sum of two
3-dimensional spaces, the second symmetric power of S+ or S−. If V is the cotangent space T ∗
of a manifold these are the self-dual and anti-self-dual 2-forms. The spaces S2
+, S
2
− are also
naturally the 3-dimensional Lie algebras of trace zero endomorphisms of S+, S−.
The curvature tensor of a metric, as a section of Λ2T ∗ ⊗ Λ2T ∗, decomposes into the scalar
curvature, the trace-free Ricci tensor which lies in S2
+ ⊗ S2
− and the self-dual and anti-self-dual
components of the Weyl tensor which lie in S4
+, S
4
− respectively.
2.2 Linear twistor theory
The twistor interpretation of complexified Minkowski space begins with P3, complex projective
3-space. The lines in P3 are parametrized by a 4-dimensional quadric Q4 ⊂ P5 and the conformal
geometry on Q4 is defined by asserting that two points are null separated if the corresponding
lines in P3 intersect.
A point in the twistor space P3 defines the family of lines passing through it which is a null
plane in Q4, an α-plane. A plane in P3 (equivalently a point in the dual projective space)
defines the lines within it which consists also of a null plane but belongs to a different family,
the β-planes.
A line in the quadric is a null geodesic and this is given by a pair of a point in P3 lying
in a plane – the line is the intersection of the corresponding α-plane and β-plane. Thus the
basic constituents of the conformal structure are encoded in purely geometrical terms within
a three-dimensional space.
Underlying this correspondence is the special isomorphism Spin(6,C) ∼= SL(4,C). The first
group consists of the conformal transformations of the quadric, and the twistor space and its
dual are the projective spaces of the two spin representations V+, V−.
2.3 Nonlinear twistor theory
The nonlinear graviton construction [23] replaces P3 by a more general complex 3-manifold Z,
which contains a family of rational curves – the twistor lines – copies of P1 with normal bundle
N ∼= O(1)⊕O(1) where O(n) denotes the line bundle of degree n on P1. Kodaira’s deformation
theorem implies that there is a complete four-dimensional family M4 of such lines and null
separation is defined as in the linear case by intersection of the rational curves.
The Weyl tensor is conformally invariant and in this case the component in S4
− vanishes. The
lines through a point in Z define null surfaces like the α-planes but the non-vanishing of the
self-dual component of the Weyl tensor obstructs the existence of β-planes.
3 Integrable systems
These are the so-called Hitchin systems introduced in [11]. The setting is as follows in the
simplest case: we have a compact Riemann surface, or algebraic curve C, of genus g ≥ 2 and
a stable rank 2 vector bundle E over C. The moduli space N of such bundles is a projective
variety of dimension 3g−3 and the cotangent space at a point E is H0(C,End0E⊗K) the space
of holomorphic trace zero sections Φ of the bundle of endomorphisms twisted byK, the canonical
bundle of holomorphic 1-forms on C. There is a natural map trΦ2 from H0(C,End0E ⊗ K)
to H0
(
C,K2
)
. Both spaces have dimension 3g − 3 and so we have a 2n-dimensional mani-
fold T ∗N with n holomorphic functions, quadratic in the fibres. These Poisson-commute and
Spinors, Twistors and Classical Geometry 3
define a completely integrable system. They are quadratic and so on open sets they describe
geodesic flows which are completely integrable.
The simplest example is g = 2 with Λ2E the trivial bundle, in which case N is P3. The
integrable system was calculated in [24], with a beautiful formula in [3] and a less attractive
but more concrete expression in [5]. Nevertheless, if you take a specific E and ask what the
quadratic map from C3 to C3 looks like, it is difficult to realize.
As far as I am aware there is no discussion in the literature of an explicit form for g = 2
and bundles of odd degree (and Λ2E fixed) where, thanks to [19, 21] we know that the moduli
space N is the intersection of two quadrics in P5.
We can describe this situation as three quadratic forms on C3, or in classical geometrical
language a net of conics. We begin next, given our familiarity with the quadrics in twistor
theory, to look at this intersection, its relationship with stable bundles, and the associated net.
4 The intersection of two quadrics
4.1 Projective bundles
Complexified compactified Minkowski space is a quadric Q1 in P5. We are now given another
generic quadric Q2. This means the zero sets of two quadratic forms q1, q2 on C6 and indeed
a pencil of quadrics z1q1+z2q2 for [z1, z2] homogeneous coordinates of a point z ∈ P1. The singu-
lar quadrics in the pencil occur when det(z1q1+z2q2) = 0 which gives six points a1, . . . , a6 ∈ P1.
A point x ∈ Q1 ∩Q2 lies in all quadrics Qz of the pencil.
Each point z ̸= ai in P1 defines a nonsingular quadric Qz and we can take the corresponding
space of α-planes, a projective space Pz. This is not in fact well-defined because there is no
absolute way to distinguish α- and β-planes, but if we take a double covering of P1 branched
over the six points ai there is a well-defined choice. The curve C so constructed is of genus 2
and has an involution σ exchanging the sheets of the covering.
When z = ai the quadric is singular but the planes in it (just one family now) are also
parametrized by P3. One way to think of this is via the group Spin(6,C) with its two 4-
dimensional spin representations V+ and V− – they become isomorphic as representations of
Spin(5,C). The result is a 4-dimensional manifold M4 which is a P3-fibration over the curve C.
A point x ∈ Q1 ∩ Q2 now defines a twistor line in each fibre Pz of M4 and hence a P1
bundle over C – and this is the projectivized bundle P(E) in the description of the moduli space
in [19, 21]! Of course, the main work in those papers is proving that every stable bundle is
obtained this way.
Fixing Λ2E means that E and F define the same projective bundle if F = E ⊗ L and L2 is
trivial. Then the moduli space of projective bundles is N/Γ where Γ = H1(C,Z2) ∼= Z4
2, and
since tensoring with L has no effect on End0E, this moduli space is all we need.
Twistors give us an insight into another construction of the moduli space which will be useful
in the next section. (This is Atiyah’s Smith’s Prize essay [1] when he was a graduate student in
Cambridge at the same time as Penrose.) It is known classically that through a general point
x ∈ Q1 ∩ Q2 there pass four lines. Choose a pair x ∈ ℓ ⊂ Q1 ∩ Q2, then it gives a line in
each quadric of the pencil and hence a point in a plane in each P3 fibre Pz. The point defines
a section over C and since x ∈ ℓ the section lies in the ruled surface P(E). Equivalently we have
a sub-line bundle of E. We can normalize the choice of representative vector bundle by asking
that the subbundle is trivial and then E is an extension O → E → L where L has degree 1.
The extension class lies in the sheaf cohomology space H1(C,L∗) which is dual to H0(C,LK)
and by Riemann–Roch these are two-dimensional. The projective space of a 2-dimensional vector
space V is canonically isomorphic to its dual (to each subspace of V associate its annihilator
in V ∗), so the extension class [α] is determined by a section s of LK, which is of degree 3.
4 N.J. Hitchin
Then s vanishes on a divisor p+q+r, and its annihilator satisfies [α]s = 0 in the duality pairing
H1(C,L∗)⊗H0(C,LK) → H1(C,K) = C.
What Atiyah shows is that the four choices of line define four unordered triples of points
in C: (p, q, r), (p, σ(q), σ(r)), (σ(p), q, σ(r)), (σ(p), σ(q), r). Then, taking the images of p, q, r
under the projection π : C → P1, a model for Q1 ∩Q2/Γ is a double covering of the symmetric
product S3
(
P1
) ∼= P3 branched over six planes (ai, π(q), π(r)), 1 ≤ i ≤ 6.
4.2 The net of conics
We shall use Atiyah’s approach to calculate trΦ2 : H0(C,End0E ⊗ K) → H0
(
C,K2
)
in this
genus 2 case. Via the projection π : C → P1 the canonical bundle is isomorphic to π∗O(1). Each
section of K therefore has a divisor of the form p + σ(p). The 3-dimensional space of sections
of K2 ∼= π∗O(2) is given by pulling back H0
(
P1,O(2)
)
.
To work with the extension we adopt a Dolbeault approach and take a C∞ splitting of E.
Then the holomorphic structure is defined by the new ∂̄-operator (s1, s2) 7→
(
∂̄s1 + αs2, ∂̄s2
)
for α ∈ Ω0,1(C,L∗) representing a class [α] in H1(C,L∗). As we have seen this is determined by
a divisor D = p+ q + r of a section s such that [α] annihilates s, or equivalently αs = ∂̄t.
A section
Φ =
(
a b
c −a
)
of End0E ⊗K is now holomorphic if
∂̄c = 0, ∂̄a+ αc = 0, ∂̄b− 2αa = 0. (4.1)
Multiplication by s, a section of KL, gives an exact sequence of sheaves
0 → OC(L
∗)
s→ OC(K) → OD(K) → 0
and in the exact cohomology sequence to say that [α] annihilates s means it lies in the image of
H0(D,K) → H1(C,L∗). This implies that α = ∂̄u/s where u is a C∞ section of K holomorphic
in a neighbourhood of D and vanishing outside a slightly larger one. Here u is well-defined
modulo the restriction of a section of K to D.
A bundle E is called very stable if there is no nilpotent section of EndE ⊗K.
Proposition 4.1. Let E be a very stable rank 2 bundle of odd degree on a curve C of genus 2.
Then the net of conics defined by tr Φ2 is spanned by quadratic forms of the form x2, y2, z2,
geometrically three double lines.
Proof. We use the description of the bundle as an extension. If the degree one line bundle L∗K
has a section then there is a clear nilpotent Φ, taking a = 0 = c in (4.1), so under the assumption
that E is very stable, H0(C,L∗K) = 0. Now K(−q − r) ∼= L∗(p) and if this has a section then
L ∼= OC(p), but then a holomorphic form vanishing at p gives a section of L∗K ∼= K(−p). It
follows that K(−q − r) has no sections and by adding one-forms that are non-zero at q and r
we can choose u to vanish at q, r.
First consider the case c = 0 then a in (4.1) is a holomorphic 1-form h. By Riemann–Roch
H0(C,L∗K) = 0 implies H1(C,L∗K) =0, therefore [α]a ∈ H1(C,L∗K) vanishes which means
we can solve (4.1) uniquely for b. This gives two sections Φ1, Φ2 of End0E⊗K taking h = h1, h2
spanning H0(C,K).
To determine b explicitly, the equation ∂̄b− 2αa = 0 gives sb− 2uhi as a holomorphic section
of K2 which, by the choice of u, vanishes at q and r. Take h1 to vanish at q and h2 to vanish
Spinors, Twistors and Classical Geometry 5
at r then this is a multiple λ1 of h1h2. So bi = (2uhi + λih1h2)/s and regularity of b at p gives
λ1h2(p) + 2u(p) = 0, λ2h1(p) + 2u(p) = 0.
To find another independent section take c, a holomorphic section of LK, to be s itself. Since
α = ∂̄u/s we can now take a = −u to solve ∂̄a + αc = 0. We require ∂̄b = 2(−u)∂̄u/s =
∂̄
(
−u2
)
/s. Now sb + u2, a holomorphic section of K2, vanishes at q, r so sb = −u2 + λ3h1h2.
Here we have λ3h1h2(p) = u2(p) and hence, since λ1h2(p) + 2u(p) = 0 and λ2h1(p) + 2u(p) = 0,
it follows that λ1λ2 = 4u2/h1h2(p) = 4λ3 and we have Φ3.
Since K = π∗O(1), holomorphic 1-forms are pulled back from linear forms az + b so take
h1 = 1, h2 = z. Then for Φ = x1Φ1 + x2Φ2 + x3Φ3 we get
− tr Φ2 = (x1 + zx2)
2 + x3z(λ1x1 + λ2x2 + λ1λ2x3/4).
Taking coefficients of 1, z, z2, the 3-dimensional space of conics is now spanned by x21, x
2
2 and
2x1x2 + x3(λ1x1 + λ2x2 + λ1λ2x3)/4. Completing the square in the last term gives
λ1λ2
4
(
x3 +
2
λ1λ2
(λ1x1 + λ2x2)
)2
− λ1
λ2
x21 −
λ2
λ1
x22,
so there is a basis in which the net is the three-parameter family of quadratic forms spanned
by x2, y2, z2. ■
Remark 4.2. What is notable here is that the structure is independent of the curve C and also
of the bundle so long as it is very stable. The above method can be used for other types of stable
bundle E, for example a generic situation where there is a nilpotent Φ gives the net spanned
by x2, y2, yz. In Atiyah’s description the bundle P(E) for this case is defined by a generic point
on the diagonal (a, b, b) in S3P1.
Before considering genus 3, there is another aspect to the intersection of quadrics which
deserves to be mentioned.
4.3 A nonlinear twistor space
One outcome of the paper [7] was the observation that the conics on the intersection of two
quadrics have normal bundle O(1)⊕O(1) and hence, as in Section 2.3, form a natural example
of a nonlinear twistor space. I gave the problem to Jacques Hurtubise, my student at the time,
and in [18] he produced concrete expressions for the conformal structure and Weyl tensor. Here
we consider a more geometrical (but less explicit) viewpoint, making contact with the discussion
above.
Firstly, a conic in a projective space is the intersection of a plane with a quadric, and the
conics in Q1 ∩ Q2 are defined by taking an α-plane in one of the quadrics of the pencil and
intersecting it with a generic one. This means taking z ∈ P1 and an α-plane in Qz, but this is
simply a point inM4, our P3-bundle over C. We conclude thatM4 is the corresponding complex
space-time with a conformal structure. Actually we must remove from M4 the α-planes which
intersect Q1 ∩ Q2 in a pair of lines – this is a quartic surface in each P3 fibre, the degree 4
property giving the four lines through a point as noted above.
A point x ∈ Q1 ∩ Q2, according to the nonlinear graviton construction, must define a null
surface in M4. We have already seen how we obtain a ruled surface from x – the projective
bundle of a stable vector bundle E. This is in fact the null surface: a point over z ∈ P1 is an
α-plane in Qz through x, and hence all the conics pass through x and are null separated. Then
the conformal structure on M4 is defined by the stable bundles on C!
6 N.J. Hitchin
5 Genus 3 curves
5.1 The Coble quartic
Recall that if C has genus 3 and is non-hyperelliptic it is a quartic curve in P2 and then the
canonical bundle K is the restriction of O(1) so that the 3-dimensional space of holomorphic 1-
forms can be identified with linear forms on C3, and likewise the 6-dimensional space of sections
of K2 with quadratic forms on C3.
The moduli space N of (semi)stable rank two bundles with Λ2E trivial on a quartic curve C
is again known explicitly [20] as a special singular quartic hypersurface in P7, initially studied
by A. Coble in the context of abelian varieties. The decomposable bundles E = L⊕L∗ describe
a Kummer variety which is the singular locus. This property together with the invariance under
Z6
2
∼= H1(C,Z2) characterize it. Another property, described by Christian Pauly in [22], is that,
like its cousin the Kummer quartic surface, its dual is another copy.
Pauly expresses the self-duality in the following way in terms of bundles: if E is a very stable
rank 2 bundle with Λ2E ∼= O, then there is a unique stable bundle F with Λ2F ∼= K such that
dimH0(C,E ⊗ F ) = 4. He now associates to this a net of quadrics in P3.
5.2 Families of quadrics
Let E be a very stable bundle with corresponding bundle F as above and denote by V the
4-dimensional space H0(C,E ⊗ F ). The bundle E has a holomorphic skew-symmetric form ϵ
and F a skew form ϵ′ with values in K. Hence, as in Section 2.1, we have a quadratic form on V
with values in the 3-dimensional space H0(C,K). This is Pauly’s net of quadrics. But we can
go further, following the spinor description of four dimensional geometry.
There is a natural homomorphism
Λ2V → H0
(
C,Λ2(E ⊗ F )
)
→ H0
(
C, S2E ⊗K
)
by contracting with ϵ′ and symmetrizing E, or taking the self-dual component in the language
of Section 2.1. But S2E ∼= End0E and so we have a map from the 6-dimensional space Λ2V
to the 3g − 3 = 6-dimensional space H0(C,End0E ⊗ K). Leaving aside for the moment the
question of whether this is an isomorphism, it offers a way of evaluating trΦ2 on the image.
On a 4-manifold, a metric on the tangent space T induces one on the self-dual 2-forms, and
identifying S2
+ with End0 S+ this can be taken as − tr a2. The inner product on the self-dual
component α+, β+ of α, β ∈ Λ2T ∗ is given by
(α+, β+)ω =
1
2
[(α, β)ω + α ∧ β] ,
where ω is the volume form. In our situation we have an inner product on V with values in
sections of K, so choose a basis v1, . . . , v4 and take v1 ∧ v2 ∧ v3 ∧ v4 to trivialize Λ4V and we
have ((vi ∧ vj)+, (vi ∧ vk)+) = (vi ∧ vj , vi ∧ vk)/2 but
((v1 ∧ v2)+, (v3 ∧ v4)+) =
1
2
[
(v1 ∧ v2, v3 ∧ v4) +
√
det(vi, vj)
]
.
This expression looks odd – we know that it must take values in H0
(
C,K2
)
, quadratic functions
on C3, but the last term involves the square root of a quartic.
The only resolution is that det(vi, vj) = p2 modulo the quartic equation of C. We therefore
have two quartic curves in P2, the original one with equation det(vi, vj) − p2 = 0 and another
one X, depending on E with equation det(vi, vj) = 0. The fact that the difference of the
quartic expressions is p2, a square, means that X and C meet tangentially at 8 points instead
of 16. Pauly shows in fact that the space of tangential quartics to C essentially parametrizes
the projective bundles arising from the Coble quartic.
Spinors, Twistors and Classical Geometry 7
Remark 5.1. The eight points of intersection determine a pencil of quartics det(vi, vj)−zp2 = 0,
z ∈ P1 and do not pick out X to define the bundle E on C. In fact [22], E has 8 maximal
subbundles L∗
i of degree −1 and 28 sections sij of LiLj , i ̸= j, which vanish at the points when
the pair coincide. If the divisor of sij is x+y then the line in P2 joining x and y in C intersects X
as a bitangent. This fixes X in the pencil.
Remark 5.2. Choosing a square root K1/2 of the line bundle K, the bundle E′ = F ⊗K−1/2
has Λ2E′ trivial since Λ2F ∼= K. The projective bundle P(F ) is independent of the choice of
square root and so the self-duality can be regarded as a transformation on the moduli space of
projective bundles. In the spinor description above E, F play the role of self-dual or anti-self-
dual spin spaces and the inner product changes sign on the
√
det(vi, vj) factor. Hence we have
the involution z 7→ −z on the pencil.
5.3 The discriminant
If qi, 1 ≤ i ≤ m are symmetric n × n matrices and Q = z1q1 + · · · + zmqm then the discrim-
inant is the hypersurface in Pm−1 defined by detQ = 0. In our context of a quadratic map
H0(C,End0E ⊗K) → H0
(
C,K2
)
it is the space of linear functions on H0
(
C,K2
)
which have
a non-zero critical point. By Serre duality, the discriminant lies naturally in P
(
H1(C,K∗)
)
.
In the case of a net of conics, with m = n = 3, the discriminant is a plane cubic curve, but
as we saw in Section 4.2 the case for trΦ2 in genus 2 gave three lines, a non-generic situation.
However, before considering a particular case of the genus 3 discriminant we return to an issue
from the last section:
Proposition 5.3. The map Λ2V → H0
(
C, S2E ⊗K
)
is an isomorphism.
Proof. The sections of K given by (vi, vj) and p define the quadratic form on Λ2V relative to
a basis. Suppose α ∈ Λ2V maps to zero in H0
(
C, S2E ⊗K
)
, then since the form is trΦ2 on the
image, α is orthogonal to all vectors and so the discriminant vanishes identically.
We described the form as Q2+pQ0 where Q0 is the exterior product Q0(α, β) = α∧β and Q2
is the inner product on Λ2V induced by Q on V . Then a calculation gives
det(Q2 + pQ0) =
(
detQ− p2
)3
.
We should be careful in interpreting this formula. The discriminant hypersurface lies in P5 =
P
(
H1(C,K∗)
)
and is defined by a polynomial in H0
(
C,K2
)
, without imposing any relations
like
(
x2
)(
y2
)
= (xy)2. These relations define a Veronese surface, an embedding of P2 in P5 and
detQ − p2 is the equation of C ⊂ P2, so this represents the intersection of the discriminant
with P2. However, since it is not identically zero, the map in the proposition is injective and
since both spaces have dimension 6, it is an isomorphism. ■
5.4 Syzygetic tetrads
Consider finally the special case where the quartic X defining the vector bundle E consists of
four lines. Then C has equation ℓ1ℓ2ℓ3ℓ4 = p2 for linear forms ℓi. Since X meets C tangentially,
these lines are four bitangents, a so-called syzygetic tetrad – the 8 points of intersection lie on
a conic p = 0. It is known classically that any quartic can be so expressed and in 315 different
ways [2]. This only accounts for a finite number of bundles E, but it is a case which we can
work out explicitly and see that, unlike the case of genus 2, there are continuous parameters in
the equivalence class of the family of quadrics.
One should note that bitangents are related to spinors on C – in two dimensions the spin
bundle is a square root K1/2 of the canonical bundle and a holomorphic section is a solution to
8 N.J. Hitchin
the Dirac equation. There are
∣∣H1(C,Z2)
∣∣ = 22g = 26 = 64 of these and 28 have a section s.
Then s2 is a section of K with two double zeros – the bitangent. The notion of bitangent is
thus intrinsic for C, as is the syzygetic tetrad, for this is given by four square roots whose tensor
product is K2.
Proposition 5.4. Let E be the rank 2 bundle on a quartic curve C defined by a syzygetic tetrad.
Then the family of quadrics is spanned by equations of the form
x2i + 2ai(x1y1 + x2y2 + x3y3), y2i + 2bi(x1y1 + x2y2 + x3y3).
for i = 1, 2, 3. The six coefficients ai, bi define the equation of C.
Proof. We normalize the equations of the four lines and then the quadratic form Q on V =
H0(C,E ⊗ F ) has diagonal entries x, y, z, x + y + z. Write a general quadratic polynomial in
(x, y, z) as
v1yz + v2zx+ v3xy + (u1x+ u2y + u3z)(x+ y + z).
Then, using the inner products of self-dual components as in Section 5.2, the 6 × 6 symmetric
matrix of linear forms, with p =
∑3
i=1 aiui + bivi, can be written
u1 0 0 p 0 0
0 u2 0 0 p 0
0 0 u3 0 0 p
p 0 0 v1 0 0
0 p 0 0 v2 0
0 0 p 0 0 v3
so the family of quadrics is spanned, for i = 1, 2, 3, by
x2i + 2ai(x1y1 + x2y2 + x3y3), y2i + 2bi(x1y1 + x2y2 + x3y3). (5.1)
Remark 5.5. One may regard these expressions as the relations in a commutative algebra.
In the more general context of Higgs bundles such algebras include the cohomology of Grass-
mannians and products thereof [4]. So for genus two the relations x2 = y2 = z2 = 0 generate
the cohomology of P1 × P1 × P1 and the above algebra can be viewed as a deformation of
H∗((P1
)6
,C
)
.
It is clear from the matrix that the discriminant is the degree 6 hypersurface which consists of
the union of three singular quadrics uivi = p2, i = 1, 2, 3 and we are concerned with equivalence
up to projective transformations – can the relations in (5.1) be reduced to standard form as in
the genus 2 situation? Although what is at issue now are three separate quadrics qi rather than
the net they generate we can still consider the discriminant
det
(
3∑
i=1
ziqi
)
= z1z2z3(z1z2z3 − 2(a1b1z2z3 + a2b2z3z1 + a3b3z1z2)(z1 + z2 + z3))
and we observe three invariants aibi, which are essential parameters for the quadratic map. The
six coefficients ai, bi correspond to the 3g−3 = 6 moduli of the curve C, so for this case different
quartic curves can define isomorphic algebras with relations of the form (5.1). ■
The decomposition of the discriminant above into three components reflects the fact that in
this case the bundle End0E is a direct sum U1⊕U2⊕U3 where U2
i and U1U2U3 are trivial. This
Spinors, Twistors and Classical Geometry 9
relates to the syzygetic tetrad as follows: each element of the tetrad gives a square root of K:
K1/2, K1/2U1, K
1/2U2, K
1/2U3 and since the product of these is K2 we have U1U2U3 trivial.
Writing the last relation additively (considering the divisor class of the Ui) we obtain U1 +
U2 + U3 = 0 which gives a 2-dimensional vector subspace of H1(C,Z2). It is in fact isotropic
with respect to the intersection form. This is where the number 315 arises – counting isotropic
planes gives
(
26 − 1
)(
25 − 2
)
/|GL(2,Z2)| = 315.
Acknowledgements
The author thanks Tamás Hausel for helpful comments.
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https://doi.org/10.1112/plms/s3-5.4.407
https://doi.org/10.1017/CBO9781139084437
https://doi.org/10.1007/s002200050438
https://doi.org/10.1007/s002200050438
https://arxiv.org/abs/solv-int/9710025
https://arxiv.org/abs/1506.02404
https://doi.org/10.1017/S0305004100055924
https://doi.org/10.1112/plms/s3-43.1.133
https://doi.org/10.1007/BFb0066025
https://doi.org/10.1007/BF01208717
https://doi.org/10.1215/S0012-7094-87-05408-1
https://doi.org/10.4310/jdg/1214457032
https://doi.org/10.1007/s00220-013-1689-y
https://arxiv.org/abs/1210.0424
https://doi.org/10.1007/s13163-014-0150-x
https://arxiv.org/abs/1403.7133
https://doi.org/10.1093/qmath/haaa051
https://arxiv.org/abs/2008.05915
https://doi.org/10.1007/BF01214418
https://doi.org/10.1007/BF01214418
https://doi.org/10.1007/BF01000339
https://doi.org/10.2307/1970807
https://doi.org/10.1016/0040-9383(68)90001-3
https://doi.org/10.1307/mmj/1039029982
https://doi.org/10.1007/bf00763433
https://doi.org/10.1215/S0012-7094-96-08525-7
https://arxiv.org/abs/alg-geom/9410015
https://arxiv.org/abs/alg-geom/9410015
1 Introduction
2 Spinors and twistors
2.1 Four dimensions
2.2 Linear twistor theory
2.3 Nonlinear twistor theory
3 Integrable systems
4 The intersection of two quadrics
4.1 Projective bundles
4.2 The net of conics
4.3 A nonlinear twistor space
5 Genus 3 curves
5.1 The Coble quartic
5.2 Families of quadrics
5.3 The discriminant
5.4 Syzygetic tetrads
References
|
| id | nasplib_isofts_kiev_ua-123456789-211437 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T17:00:27Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hitchin, Nigel J. 2026-01-02T08:33:26Z 2021 Spinors, Twistors and Classical Geometry. Nigel J. Hitchin. SIGMA 17 (2021), 090, 9 pages 1815-0659 2020 Mathematics Subject Classification: 14H60; 32L25 arXiv:2108.02603 https://nasplib.isofts.kiev.ua/handle/123456789/211437 https://doi.org/10.3842/SIGMA.2021.090 The paper studies explicitly the Hitchin system restricted to the Higgs fields on a fixed, very stable rank 2 bundle in genus 2 and 3. The associated families of quadrics relate to both the geometry of Penrose's twistor spaces and several classical results. The author thanks Tamás Hausel for helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spinors, Twistors and Classical Geometry Article published earlier |
| spellingShingle | Spinors, Twistors and Classical Geometry Hitchin, Nigel J. |
| title | Spinors, Twistors and Classical Geometry |
| title_full | Spinors, Twistors and Classical Geometry |
| title_fullStr | Spinors, Twistors and Classical Geometry |
| title_full_unstemmed | Spinors, Twistors and Classical Geometry |
| title_short | Spinors, Twistors and Classical Geometry |
| title_sort | spinors, twistors and classical geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211437 |
| work_keys_str_mv | AT hitchinnigelj spinorstwistorsandclassicalgeometry |