# Higher complex structures

V.V. Fock, A. Thomas  Université de Strasbourg, CNRS, IRMA UMR 7501, 67084
Strasbourg, France<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce and analyze a new geometric structure on topological surfaces
generalizing the complex structure. To define this so called higher complex
structure we use the punctual Hilbert scheme of the plane. The moduli space of
higher complex structures is defined and is shown to be a generalization of
the classical Teichmüller space. We give arguments for the conjectural
isomorphism between the moduli space of higher complex structures and
Hitchin’s component.

## Introduction

Poincaré’s uniformization theorem links complex structures on a surface
$\Sigma$ to homomorphisms from the fundamental group of the surface to
$PSL(2,\mathbb{R})$, the automorphism group of the hyperbolic plane. With
this, Teichmüller space $\mathcal{T}_{\Sigma}$ can be identified with the
connected component of the character variety of faithful and discrete
representations of the fundamental group in $PSL(2,\mathbb{R})$:

$\mathcal{T}_{\Sigma}\cong\operatorname{Hom}^{\text{discrete}}(\pi_{1}(\Sigma),PSL(2,\mathbb{R}))/PSL(2,\mathbb{R}).$

In his celebrated paper [Hi92], Nigel Hitchin proves the existence of a
connected component of the character variety for an adjoint group of a split
real form of any complex simple Lie group (for instance $PSL(n,\mathbb{R})$)
consisting of faithful and discrete representations and parametrized by
holomorphic differentials. These components generalize Teichmüller space. His
methods are analytic, using the theory of Higgs bundles. Teichmüller space has
also geometric descriptions: it is the moduli space of hyperbolic structures
(metrics with constant curvature -1) and also the moduli space of complex
structures. The natural question is then if there is a geometric description
of Hitchin’s component.

In this paper, we describe and analyze a new geometric structure on surfaces
generalizing the complex structure. Conjecturally, the moduli space of the so
called higher complex structure is isomorphic to Hitchin’s component. This
would give a purely geometric approach to higher Teichmüller theory. We give
arguments in favor of the isomorphism.

The search for a geometric origin of Hitchin’s component is of course not new.
Goldman, Guichard-Wienhard, Labourie and others describe Hitchin’s component
via geometric structures on bundles over the surface. For $PSL(3,\mathbb{R})$,
this geometric structure is the convex projective structure described by
Goldman in [Go68]. For $n=4$, Guichard and Wienhard describe convex foliated
structure on the unit tangent bundle in [GW08]. Labourie introduces the
concept of an Anosov representation in [La06]. The drawback of these
constructions is that the bundle on which the geometric structure is defined
is not canonically associated to the surface.

All these generalizations are rigid geometric structures (meaning that the
local automorphism group is finite dimensional). Our generalization is not
rigid in this sense but behaves as a generalization of complex structures
(with local automorphism group holomorphic functions which are infinite
dimensional).

We also signal a link to W-algebras since this paper gives partial answers to
questions raised in an article of the first author together with Bilal and
Kogan [BFK91]. This connection will not be treated here.

The paper is structured as follows: first, we review in 1 the approach to
complex structures via the Beltrami differential, the most appropriate
approach for our generalization. Then in Section 2, we introduce the tool we
need for generalizing the complex structure: the punctual Hilbert scheme of
the plane and its zero-fiber. In the main part 3, we define and analyze the
higher complex structure. To define the moduli space of higher complex
structures, we enlarge the group of diffeomorphisms and look at higher complex
structures modulo Hamiltonian diffeomorphisms of the cotangent space
$T^{*}\Sigma$ preserving the zero section setwise. We show that higher complex
structure is locally trivializable. The moduli space of higher complex
structures is analyzed and compared to Hitchin’s component. In the last
section 4, we give arguments in favor for the conjectural isomorphism to
Hitchin’s component by constructing a spectral curve associated to a higher
complex structure. In the appendix 5 we put details of proofs and
computations.

Throughout the paper, $\Sigma$ denotes a connected oriented closed real
surface of genus $g\geq 2$. A complex local coordinate on $\Sigma$ is denoted
by $z=x+iy$ and its conjugate coordinates on $T^{*\mathbb{C}}\Sigma$ by $p$
and $\bar{p}$. The space of sections of a bundle $B$ is denoted by
$\Gamma(B)$.

##  1\. Complex Structures on Surfaces

In this section we review the approach to complex and almost complex
structures on a surface via the Beltrami differential. This will prepare the
reader to the generalization which will follow in Section 3.

Recall that a complex structure is a complex atlas with holomorphic transition
functions. Retaining only the information that every tangent space
$T_{z}\Sigma$ has the structure of a complex vector space, i.e. is equipped
with an endomorphism $J(z)$ whose square is $-\operatorname{id}$, we get an
almost complex structure. A theorem due to Gauss and Korn-Lichtenstein states
that every almost complex structure on a surface comes from a complex
structure (i.e. can be integrated to a complex structure). Thus, both notions
are equivalent on a surface.

So the complex structure is encoded in the operators $J(z)$. These can be
better understood by diagonalization. The characteristic polynomial being
$X^{2}+1$, the eigenvalues are $\pm i$. So we need to complexify the tangent
space to see the eigendirections:

$T^{\mathbb{C}}\Sigma=T^{1,0}\Sigma\oplus T^{0,1}\Sigma$

where $T^{1,0}\Sigma$ is the eigenspace associated to eigenvalue $i$.
Furthermore, the eigendirections are conjugated to each other:
$T^{1,0}\Sigma=\overline{T^{0,1}\Sigma}$.

Therefore the complex structure is entirely encoded by $T^{0,1}\Sigma$, i.e. a
direction in the complexified tangent space. Thus, we can see a complex
structure as a section of the projectivized cotangent bundle
$\mathbb{P}(T^{\mathbb{C}}\Sigma)$.

Let’s describe this viewpoint in coordinates. For this, fix a reference
complex coordinate $z=x+iy$ on $\Sigma$. This gives a basis
$(\partial,\bar{\partial})$ in $T^{\mathbb{C}}\Sigma$ where
$\partial=\frac{1}{2}(\partial_{x}-i\partial_{y})\text{ and
}\bar{\partial}=\frac{1}{2}(\partial_{x}+i\partial_{y})$. The generator of the
linear subspace $T^{1,0}_{z}\Sigma$ can be normalize to be
$\bar{\partial}-\mu(z,\bar{z})\partial$, (where $\mu$ is a coordinate on
$\mathbb{C}P^{1}$, and thus can take infinite value). The coefficient $\mu$ is
called Beltrami differential. There is one condition on $\mu$, coming from the
fact that the vector $\bar{\partial}-\mu\partial$ and its conjugate
$\partial-\bar{\mu}\bar{\partial}$ have to be linearly independent as
eigenvectors corresponding to different eigenvalues. A simple computation
shows that this is equivalent to $\mu\bar{\mu}\neq 1$. If we restrict
ourselves to complex structures compatible with orientation of the surface
(i.e. homotopy equivalent to the reference complex structure) this condition
gives $\mu\bar{\mu}<1$.

We only have seen the Beltrami differential in a local chart. Changing
coordinates $z\mapsto w(z)$ gives
$\mu(z,\bar{z})\mapsto\frac{d\bar{z}/d\bar{w}}{dz/dw}\mu(z,\bar{z})$, so $\mu$
is of type $(-1,1)$, i.e. a section of $K^{-1}\otimes\bar{K}$ where
$K=T^{*(1,0)}\Sigma$ the canonical line bundle.

To sum up, we look at a complex structure on a surface as a given linear
direction in every complexified tangent space, which is the same as a 1-jet of
a curve at the origin. For the generalization, we need to consider rather the
cotangent space $T^{*}\Sigma$ (the operators $J(z)$ also act in
$T^{*}_{z}\Sigma$). Higher complex structures will be given by a $n$-jet of a
curve in the complexified cotangent space. To describe this idea precisely in
geometric terms, we use the punctual Hilbert scheme of the plane.

##  2\. Punctual Hilbert scheme of the plane

We present here the tool necessary for the higher complex structure: the
punctual Hilbert scheme of the plane. The Hilbert scheme is the parameter
space of all subschemes of an algebraic variety. In general this scheme can be
quite complicated but here we are in a very specific case of 0-dimensional
subschemes of $\mathbb{C}^{2}$. Nothing new is presented here, a classical
reference is Nakajima’s book [Na99].

Consider $n$ points in the plane $\mathbb{C}^{2}$ as an algebraic variety,
i.e. defined by some ideal $I$ in $\mathbb{C}[x,y]$. The function space
$\mathbb{C}[x,y]/I$ is $n$-dimensional, since a function on $n$ points is
defined by its $n$ values. So the ideal $I$ is of codimension $n$. This gives
a simple example of a subscheme of dimension zero. We define the length of a
zero-dimensional subscheme to be the dimension of its function space. So the
variety of $n$ distinct points is of length $n$. We will see that we get more
interesting examples when two or several points collapses into one single
point. The moduli space of zero-dimensional subschemes of length $n$ is called
the punctual Hilbert scheme:

######  Definition 1.

The punctual Hilbert scheme $\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ of
length $n$ of the plane is the set of ideals of $\mathbb{C}\left[x,y\right]$
of codimension $n$:

$\operatorname{Hilb}^{n}(\mathbb{C}^{2})=\\{I\text{ ideal of
}\mathbb{C}\left[x,y\right]\mid\dim(\mathbb{C}\left[x,y\right]/I)=n\\}.$

The subspace of $\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ consisting of all
ideals supported on 0, i.e. whose associated algebraic variety is $(0,0)$, is
called the zero-fiber of the punctual Hilbert scheme and is denoted by
$\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$.

######  Remark.

In the literature the punctual Hilbert scheme of a space $X$ is often denoted
by $X^{[n]}$. Here we will deal with several notions of Hilbert schemes, so we
prefer the notation $\operatorname{Hilb}^{n}(X)$.

A theorem of Grothendieck and Fogarty asserts that
$\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ is a smooth and irreducible variety
of dimension $2n$ (see [Fo68]). The zero-fiber
$\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$ is a irreducible variety of
dimension $n-1$, but it is in general not smooth.

Let’s get a feeling for the form of a generic ideal in the Hilbert scheme.
Given $n$ generic distinct points in $\mathbb{C}^{2}$, there is a unique
polynomial $Q$ of degree $n-1$ such that these points belong to the graph of
$Q$, the Lagrange interpolation polynomial. So we can choose $y=Q(x)$ to be in
$I$. If we denote by $x_{i}$ the $x$-coordinate of the $i$-th point, we get
$\prod_{i}(x-x_{i})$ in $I$. These two relations determine already the $n$
points. The ideal $I$ then has two generators and can be put into the form

$I=\left\langle
x^{n}+t_{1}x^{n-1}+\cdots+t_{n},-y+\mu_{1}+\mu_{2}x+...+\mu_{n}x^{n-1}\right\rangle.$

A point in the zero-fiber of the Hilbert scheme is obtained by collapsing all
$n$ points to the origin. A generic point (see [Ia72]) is obtained when the
Lagrange interpolation polynomial admits a limit (for exemple if all points
glide along a given curve to the origin). The only condition is that the
constant term of $Q$, which is $\mu_{1}$, has to be zero. Since all $x_{i}$
become 0, all $t_{i}$ as well. So we get an ideal of
$\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$ of the form

$I=\left\langle x^{n},-y+\mu_{2}x+...+\mu_{n}x^{n-1}\right\rangle.$

There are other points in the zero-fiber if $n>2$. For instance in
$\operatorname{Hilb}^{3}_{0}$ we find the ideal $\left\langle
x^{2},xy,y^{2}\right\rangle$ which is not of the above form (it has three
generators).

Notice that for $n=2$, we just get a linear nonvertical direction at the
origin, i.e. a point in an affine space.

There are several equivalent ways to look on the punctual Hilbert scheme. We
discuss two of them: as blow-up of a configuration space and as the set of
commuting matrices.

When we consider the set of $n$-tuples points as an algebraic variety, we
don’t see their order. So we see them as a point in the configuration space of
(not necessarily distinct) $n$ points, which is the quotient of
$(\mathbb{C}^{2})^{n}$ by the symmetric group $\mathcal{S}_{n}$. This quotient
space is singular, since the action of the symmetric group is not free. The
theorem of Grothendieck and Fogarty asserts also that
$\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ is a minimal resolution of
singularities by blow-ups of the configuration space
$\operatorname{Sym}^{n}(\mathbb{C}^{2})$. The map from the Hilbert scheme to
the configuration space associates to an ideal its support (points in the
plane with multiplicity). This is called the Chow map.

To give a link to commuting matrices, we associate to an ideal $I$ of
codimension $n$ the linear maps $M_{x}$ and $M_{y}$, which are multiplication
by $x$ and $y$ respectively in the $n$-dimensional vector space
$\mathbb{C}[x,y]/I$. These are two commuting operators admitting a cyclic
vector (the image of $1\in\mathbb{C}[x,y]$ in the quotient). So we get a map
from the punctual Hilbert scheme to the space of commuting matrices with
cyclic vector modulo conjugation. This is in fact a bijection:

$\operatorname{Hilb}^{n}(\mathbb{C}^{2})\cong\\{A,B\in\operatorname{End}{\mathbb{C}}^{n},v\in\mathbb{C}^{n}\mid[A,B]=0,\text{$v$
is a cyclic vector}\\}/\text{ conj.}$

The inverse map associates to an equivalence class $[(A,B)]$ the ideal
$I=\\{P\in\mathbb{C}[x,y]\mid P(A,B)=0\\}$. The existence of the cyclic vector
garantees that $I$ is of codimension $n$.

To end the section, we also introduce the notion of the reduced Hilbert
scheme, denoted by $\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$.
Geometrically, it corresponds to configurations of $n$ points with barycenter
0. The punctual Hilbert scheme is topologically a direct product of
$\mathbb{C}^{2}$ and the reduced Hilbert scheme. So the dimension of
$\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$ is $2n-2$. Its precise
definition as a symplectic quotient will be given in Section 4. The main
property is that the zero-fiber is a Lagrangian subspace of the reduced
Hilbert scheme (see Proposition 4).

##  3\. Higher complex structures

In this section, we define the higher complex structure and explore its main
properties. In order to define a moduli space for higher complex structures,
we need to enlarge the group of diffeomorphisms of $\Sigma$ to the space of
Hamiltonian diffeomorphisms of the cotangent space $T^{*}\Sigma$ preserving
the zero section. We then explore the local and global theory of that new
structure.

###  3.1. Definition and basic properties

In Section 1, we saw that a complex structure on a surface $\Sigma$ is
uniquely given by a section $\sigma$ of $\mathbb{P}(T^{*\mathbb{C}}\Sigma)$,
the (pointwise) projectivized complexified cotangent space, such that at any
point $z\in\Sigma$, $\sigma(z)$ and $\bar{\sigma}(z)$ are distinct. In the
previous section, we saw that the projectivization is a special case of the
zero-fiber of the punctual Hilbert scheme for $n=2$:
$\mathbb{P}(T^{*\mathbb{C}}\Sigma)=\operatorname{Hilb}^{2}_{0}(T^{*\mathbb{C}}\Sigma)$.
It is now easy to guess the generalization.

######  Definition 2.

A higher complex structure of order $n$ on a surface $\Sigma$, for short
$n$-complex structure, is a section $I$ of
$\operatorname{Hilb}^{n}_{0}(T^{*\mathbb{C}}\Sigma)$ such that at each point
$z$ we have that the sum $I(z)+\bar{I}(z)$ is the maximal ideal supported at
zero of $T_{z}^{*\mathbb{C}}\Sigma$.

######  Remark.

The space $\operatorname{Hilb}^{n}(T^{*\mathbb{C}}\Sigma)$ is the pointwise
application of $\operatorname{Hilb}^{n}$ to $T^{*\mathbb{C}}_{z}\Sigma$ at
every point $z$ of $\Sigma$. So it is a bundle of Hilbert schemes. We call it
Hilbert scheme bundle. The same holds for the zero-fiber.

For $n=2$, the condition that $I+\bar{I}$ is maximal simply reads
$\mu_{2}\bar{\mu}_{2}\neq 1$ which is exactly what we had for the complex
structure. So we recover the complex structure.

In the previous section, we saw that not all points in the zero-fiber can be
written in the form of a Lagrange interpolation polynomial passing through the
origin. Another important consequence of the extra condition is that it rules
out non-generic ideals:

######  Proposition 1.

For a $n$-complex structure $I$, we can write at a point $z$ either $I(z)$ or
its conjugate $\bar{I}(z)$ as

$\left\langle
p^{n},-\bar{p}+\mu_{2}(z,\bar{z})p+...+\mu_{n}(z,\bar{z})p^{n-1}\right\rangle\text{
with }\mu_{2}\bar{\mu}_{2}<1.$

The proof is a simple computation and can be found in the appendix 5.1

We call the coefficients $\mu_{k}$ higher Beltrami differentials. The
proposition allows to think of a $n$-complex structure as a polynomial curve
in the cotangent fiber attached to each point of the surface. You can either
think off a "hairy" surface (with polynomial curved hairs), or a complex
surface germ along the zero-section $\Sigma$ in $T^{*\mathbb{C}}\Sigma$.

In the case where $I$ is of the above form (and not its conjugate $\bar{I}$),
we call the $n$-complex structure compatible. This is analogous to the case if
Beltrami differential is of norm smaller than 1 or bigger than 1. One can pass
from one to the other by conjugation.

Let’s compute the global nature of the higher Beltrami differentials. We will
see that $\mu_{2}$ is just the usual Beltrami differential, so of type
$(-1,1)$. Under a holomorphic coordinate transform $z\to z(w)$, we have
$p=\frac{\partial}{\partial z}\mapsto\frac{dw}{dz}\frac{\partial}{\partial w}$
and similarly for $\bar{p}$. Hence,

$\displaystyle\left\langle
p^{n},-\bar{p}+\mu_{2}(z,\bar{z})p+...+\mu_{n}(z,\bar{z})p^{n-1}\right\rangle$
$\displaystyle\mapsto\left\langle\left(\frac{dw}{dz}\right)^{n}\left(\frac{\partial}{\partial
w}\right)^{n},-\frac{d\bar{w}}{d\bar{z}}\frac{\partial}{\partial\bar{w}}+\frac{dw}{dz}\mu_{2}(z,\bar{z})\frac{\partial}{\partial
w}+...+\left(\frac{dw}{dz}\right)^{n-1}\mu_{n}(z,\bar{z})\left(\frac{\partial}{\partial
w}\right)^{n-1}\right\rangle$
$\displaystyle=\left\langle\left(\frac{\partial}{\partial
w}\right)^{n},-\frac{\partial}{\partial\bar{w}}+\frac{d\bar{z}/d\bar{w}}{dz/dw}\mu_{2}(z,\bar{z})\frac{\partial}{\partial
w}+...+\frac{d\bar{z}/d\bar{w}}{(dz/dw)^{n-1}}\mu_{n}(z,\bar{z})\left(\frac{\partial}{\partial
w}\right)^{n-1}\right\rangle$

Thus, we see that for $m=2,...,n$ we get

$\mu_{m}(w,\bar{w})=\frac{d\bar{z}/d\bar{w}}{(dz/dw)^{m-1}}\mu_{m}(z,\bar{z}).$

So $\mu_{m}$ is of type $(-m+1,1)$, i.e. a section of $K^{-m+1}\otimes\bar{K}$
(where $K$ is the canonical line bundle).

###  3.2. Higher diffeomorphisms

We wish to define a moduli space of higher complex structures which is finite-
dimensional. Higher complex structures "live" in a neighborhood of the zero-
section of $T^{*\mathbb{C}}\Sigma$. A diffeomorphism of $\Sigma$ extends
linearly to $T^{*}\Sigma$ and its complexification. Roughly speaking it can
only act linearly on the polynomial curve (given by the higher complex
structure), so the quotient of all $n$-complex structures by diffeomorphisms
of $\Sigma$ is infinite-dimensional. Therefore, we have to enlarge the
equivalence relation and quotient by a larger group. What we need are
polynomial changes in the cotangent bundle. These can be obtained by
symplectomorhpisms of $T^{*}\Sigma$ generated by a Hamiltonian (a function on
$T^{*}\Sigma$):

######  Definition 3.

A higher diffeomorphism of a surface $\Sigma$ is a Hamiltonian diffeomorphism
of $T^{*}\Sigma$ preserving the zero-section $\Sigma\subset T^{*}\Sigma$
setwise. The set of higher diffeomorphisms is denoted by
$\operatorname{Symp}_{0}(T^{*}\Sigma)$.

A higher vector field is a Hamiltonian vector field on the cotangent bundle
tangent to the zero-section. The space of such vector fields is denoted by
$\operatorname{Ham}_{0}(T^{*}\Sigma)$.

We say that a higher vector field is of order $n$ if its Hamiltonian $H$ is a
homogenous polynomial of degree $n$.

A higher vector field of order 1 is just a vector field of the surface
$\Sigma$ linearly extended to the cotangent space.

Preserving the zero-section means that the Hamiltonian $H$ of a higher vector
field can be chosen to vanish on the zero-section. In coordinate that means
that one can write the Hamiltonian as
$H(z,\bar{z},p,\bar{p})=\sum_{k,l}w_{k,l}(z,\bar{z})p^{k}\bar{p}^{l}$ with
$w_{0,0}=0$. Furthermore, since the Hamiltonian is a real function written in
complex coordinates, we have the condition $w_{l,k}=\overline{w_{k,l}}$.

Let’s see how a higher diffeomorphism acts on the $n$-complex structure.
Roughly speaking a diffeomorphism of the cotangent bundle acts on the space of
sections, so also on the space of $n$-tuples of sections (corresponds to $n$
points in each fiber), so also on the zero-fiber of the Hilbert scheme (which
can be seen as the limit when all $n$ points collapse to the origin). In this
way, a higher diffeomorphism acts on a $n$-complex structure.

To be more precise, we need first to understand the variation of an ideal in
the space of all ideals:

######  Proposition 2.

The space of infinitesimal variations of an ideal $I$ in a ring $A$ is the set
of all $A$-module homomorhpisms from $I$ to $A/I$.

So to compute the variation of an ideal, all we need is to compute the
variation of its generators modulo $I$. These generators are polynomial
functions. A general fact of symplectic geometry asserts that the variation of
a function $f$ under a Hamiltonian flow generated by a Hamiltonian $H$ is
given by the Poisson bracket $\\{H,f\\}$. Therefore to compute the action of a
higher diffeomorphism generated by a Hamiltonian $H$, we only have to compute
the Poisson bracket of $H$ with the generators of $I$, modulo $I$.

In coordinates, we get that the variation of the generators of the ideal $I$
is given by $\\{H,p^{n}\\}\mod I$ and
$\\{H,-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\\}\mod I$.

Since we mod out by $I$, a Hamiltonian of order $n$ or higher will have no
effect on a $n$-complex structure. So only polynomial Hamiltonians of degree
at most $n-1$ act nontrivially. The Hamiltonians of degree $\geq n$ generate a
normal subgroup in $\operatorname{Symp}_{0}(T^{*}\Sigma)$ acting trivially on
$n$-complex structure. Therefore the higher diffeomorphisms act through its
quotient. In the following subsection we compute precisely the variation of
the higher Beltrami differentials under a Hamiltonian and we deduce the local
theory of higher complex structures.

###  3.3. Local theory

In this subsection, we are in an open neighborhood of 0 in $\mathbb{C}$. We
will prove that a higher complex structure can locally be trivialized by a
higher diffeomorphism, which means that we can make
$\mu_{2}(z,\bar{z})=...=\mu_{n}(z,\bar{z})=0$ in a neighborhood of 0. Before
doing so, we have to compute the variation of the higher complex structure by
a higher diffeomorphism.

As seen in the precedent subsection, we have to compute Poisson brackets
modulo the ideal $I$. A small argument will simplify a lot the computations:

######  Lemma 4.

Let $I=\left\langle f_{1},...,f_{r}\right\rangle$ be an ideal of
$\mathbb{C}[z,\bar{z},p,\bar{p}]$ such that $\\{f_{i},f_{j}\\}=0\mod I$ for
all $i$ and $j$. Then for all polynomials $H$ and all $k\in\\{1,...,r\\}$ we
have $\\{H,f_{k}\\}\mod I=\\{H\mod I,f_{k}\\}\mod I$.

###### Proof.

The only thing to show is that if we replace $H$ by $H+gf_{l}$ for some
polynomial $g$ and some $l\in\\{1,...,r\\}$, the expression does not change.
Indeed,
$\\{H+gf_{l},f_{k}\\}=\\{H,f_{k}\\}+g\\{f_{l},f_{k}\\}+\\{g,f_{k}\\}f_{l}=\\{H,f_{k}\\}\mod
I$ using the assumption. ∎

For our ideal $I=\left\langle
p^{n},-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\right\rangle$, we have
$\\{p^{n},-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\\}=np^{n-1}(\partial\mu_{2}p+...+\partial\mu_{n}p^{n-1})=0\mod
I$, so we can use the proposition. A Hamiltonian $H$ modulo $I$ can thus
always be written as $H=\sum_{k}w_{k}p^{k}$.

Another small argument is that the first generator, $p^{n}$, does not change :
$\\{H,p^{n}\\}\mod I=np^{n-1}\partial H\mod I=0$ since there is no constant
term in $H$. So we only have to settle the second generator.

Let’s compute the variation of the second generator under the Hamiltonian
$H=w_{k}p^{k}$.

$\displaystyle\\{w_{k}p^{k},-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\\}=$
$\displaystyle kw_{k}p^{k-1}(\partial\mu_{2}p+...+\partial\mu_{n}p^{n-1})$
$\displaystyle-\partial
w_{k}p^{k}(\mu_{2}+2\mu_{3}p+...+(n-1)\mu_{n}p^{n-2})+\bar{\partial}w_{k}p^{k}$
$\displaystyle=$ $\displaystyle p^{k}(\bar{\partial}w_{k}-\mu_{2}\partial
w_{k}+k\partial\mu_{2}w_{k})$
$\displaystyle+\sum_{l=1}^{n-1-k}p^{k+l}(k\partial\mu_{l+2}w_{k}-(l+1)\mu_{l+2}\partial
w_{k})\mod I$

Thus we obtain

######  Proposition 3.

The variations $\delta\mu_{l}$ are given by

$\delta\mu_{l}=\left\\{\begin{array}[]{cl}(\bar{\partial}-\mu_{2}\partial+k\partial\mu_{2})w_{k}&\text{
if }l=k+1\\\ (k\partial\mu_{l-k+1}-(l-k)\mu_{l-k+1}\partial)w_{k}&\text{ if
}l>k+1\\\ 0&\text{ if }l<k+1.\end{array}\right.$

Now, we are ready to state the local triviality of higher complex structure:

######  Theorem 1.

The $n$-complex structure can be locally trivialized, i.e. there is a higher
diffeomorphism which sends the structure to
$(\mu_{2}(z,\bar{z}),...,\mu_{n}(z,\bar{z}))=(0,...,0)$ for all small $z$.

The proof is in the spirit of the classical proof of Darboux theorem on local
theory of symplectic structures. You find it in the appendix 5.2.

###  3.4. Moduli space

We are finally ready to define and study the moduli space of higher complex
structures. We then show that it is a contractible ball of dimension
$(n^{2}-1)(g-1)$ and describe its tangent and cotangent space.

######  Definition 5.

The space of all compatible $n$-complex structures modulo higher
diffeomorphisms is called the geometric Hitchin space and denoted by
$\hat{\mathcal{T}}^{n}_{\Sigma}$. In formula:

$\hat{\mathcal{T}}^{n}_{\Sigma}=\Gamma(\operatorname{Hilb}^{n}_{0}(T^{*\mathbb{C}}\Sigma))/\operatorname{Symp}_{0}(T^{*}\Sigma)$

Recall that an $n$-complex structure is compatible with orientation if the
ideals $I$ are of the form $\left\langle
p^{n},-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\right\rangle$ with
$\mu_{2}\bar{\mu}_{2}<1$. Using complex conjugation we get another moduli
space for which $\bar{I}$ is of this form.

Since a higher diffeomorphism of order 1 is a usual diffeomorphism and only
Hamiltonians of order at most $n-1$ act non-trivially on $n$-complex
structure, we recover for $n=2$ the usual Teichmüller space:

$\hat{\mathcal{T}}^{2}_{\Sigma}=\mathcal{T}_{\Sigma}.$

Our main result is

######  Theorem 2.

For a surface $\Sigma$ of genus $g\geq 2$ the geometric Hitchin space
$\hat{\mathcal{T}}^{n}_{\Sigma}$ is a contractible manifold of complex
dimension $(n^{2}-1)(g-1)$. In addition, its cotangent space at any point
$\mu=(\mu_{2},...,\mu_{n})$ is given by

$T^{*}_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\bigoplus_{m=2}^{n}H^{0}(\Sigma,K^{m})$

In addition, there is a forgetful map
$\hat{\mathcal{T}}^{n}_{\Sigma}\rightarrow\hat{\mathcal{T}}^{n-1}_{\Sigma}$.

We essentially show the existence of a cotangent space at every point, so we
have a manifold. The dimension of the cotangent space is computed by Riemann-
Roch. We don’t enter into details on issues about infinite-dimensional
manifolds and quotients.

###### Proof.

Contractibility is quite easy: given an equivalence class of a $n$-complex
structure $[(\mu_{2},...,\mu_{n})]$ with $\left|\mu_{2}\right|<1$, we can
retract it via $[(1-t)(\mu_{2},...,\mu_{n})]$ to $[(0,...,0)]$.

To see that it is a manifold, we will examine the infinitesimal variation
around any point. This will also give a description of the tangent and
cotangent space. By definition, we have

$\hat{\mathcal{T}}^{n}_{\Sigma}=\\{(\mu_{2},...,\mu_{n})\mid\mu_{m}\in
K^{-m+1}\otimes\bar{K}\;\forall\,m\text{ and
}\left|\mu_{2}\right|<1\\}/\operatorname{Symp}_{0}(T^{*}\Sigma)$

The infinitesimal variation around $\mu=(\mu_{2},...,\mu_{n})$ is then given
by

$T_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\\{(\delta\mu_{2},...,\delta\mu_{n})\mid\delta\mu_{m}\in
K^{-m+1}\otimes\bar{K}\;\forall\,m\\}/\operatorname{Ham}_{0}(T^{*}\Sigma).$

In the previous subsection, we have seen that every $n$-complex structure is
locally trivializable. So there is an atlas in which
$\mu:=(\mu_{2},...,\mu_{n})=0$. In addition we have computed the action of a
Hamiltonian vector field on the $n$-complex structure in proposition 3.
Locally, we can decompose the Hamiltonian into homogeneous parts of degrees 1
to $n-1$. All higher terms do not affect the $n$-complex structure. By
proposition 3 (with $\mu_{k}=0$ for all $k$), we get

$T_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\\{(\delta\mu_{2},...,\delta\mu_{n})\\}/(\bar{\partial}w_{1},...,\bar{\partial}w_{n-1})$

where $w_{m}$ is a section of $K^{m}$. Thus, the tangent space splits into
parts

$T_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\\{\delta\mu_{2}\in\bar{K}\otimes
K^{-1}\\}/\bar{\partial}w_{1}\oplus...\oplus\\{\delta\mu_{n}\in
K^{-n+1}\otimes\bar{K}\\}/\bar{\partial}w_{n-1}$

To compute the cotangent space, we use the pairing between differential of
type $(1-k,1)$ and of type $(k,0)$ given by integration over the surface. We
get

$\displaystyle(\\{\delta\mu_{m}\\}/\bar{\partial}w_{m-1})^{*}=$
$\displaystyle\;\\{t_{m}\in K^{m}\mid\smallint
t_{m}\bar{\partial}w_{m-1}=0\;\forall\,w_{m-1}\in K^{m-1}\\}$ $\displaystyle=$
$\displaystyle\;\\{t_{m}\in
K^{m}\mid\smallint\bar{\partial}t_{m}w_{m-1}=0\;\forall\,w_{m-1}\in
K^{m-1}\\}$ $\displaystyle=$ $\displaystyle\;\\{t_{m}\in
K^{m}\mid\bar{\partial}t_{m}=0\\}$ $\displaystyle=$
$\displaystyle\;H^{0}(\Sigma,K^{m})$

Therefore

$T^{*}_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\bigoplus_{m=2}^{n}H^{0}(\Sigma,K^{m})$

Now, a standard computation using Riemann-Roch formula shows that (using genus
$g\geq 2$)

$\dim H^{0}(\Sigma,K^{m})=(2m-1)(g-1).$

Therefore

$\dim{\hat{\mathcal{T}}^{n}_{\Sigma}}=\dim{T^{*}_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}}=\sum_{m=2}^{n}\dim{H^{0}(K^{m})}=\sum_{m=2}^{n}(2m-1)(g-1)=(n^{2}-1)(g-1)$

The forgetful map is simply given by the following: to the equivalence class
of a $n$-complex structure given by an ideal $I$, we associate the equivalence
class of $I+\left\langle p,\bar{p}\right\rangle^{n-1}$. This is independent of
coordinates since $\left\langle p,\bar{p}\right\rangle$ is the maximal ideal
supported on the origin. In coordinates, the map just forgets $\mu_{n}$. ∎

Composing the forgetful maps, we get a map from
$\hat{\mathcal{T}}^{n}_{\Sigma}$ to Teichmüller space. To any $n$-complex
structure is therefore associated a complex structure. The cotangent space of
geometric Hitchin space at $I$ is the Hitchin base of holomorphic
differentials where holomorphicity is with respect to the associated complex
structure of $I$.

Since the cotangent space is a complex vector space, we have automatically an
almost complex structure on the geometric Hitchin space. Furthermore, since
it’s the moduli space of a geometric structure on the surface, the mapping
class group acts properly discontinuously on it.

Let’s compare geometric Hitchin space to Hitchin’s component: both are
contractible real manifolds of the same dimension. Hitchin’s component has a
symplectic structure (Goldman’s symplectic structure on character varieties),
but no obvious complex structure, nor forgetful maps. Geometric Hitchin space
has (for the moment) no obvious symplectic structure, but enjoys all the
properties we explained before. An isomorphism between both would enrich both
sides.

######  Conjecture 6.

Geometric Hitchin space $\hat{\mathcal{T}}^{n}_{\Sigma}$ is canonically
isomorphic to Hitchin’s component $\mathcal{T}^{n}_{\Sigma}$.

One way to attack the conjecture is the following idea: To any $n$-complex
structure, we can associate canonically a vector bundle of rank $n$ whose
fiber over a point $z$ is $\mathbb{C}[p,\bar{p}]/I(z)$. The task is then to
find a flat connection in this bundle with monodromy in $PSL_{n}(\mathbb{R})$.
This amounts to a map from $\pi_{1}(\Sigma)$ to $PSL_{n}(\mathbb{R})$, so a
point of the character variety.

In the last section we give an extra argument in favor of this conjecture: as
for Hitchin’s component, we get a spectral curve.

##  4\. A spectral curve

In this final section, we explore and exploit the symplectic structure of the
Hilbert scheme in order to construct a spectral curve associated to a
cotangent vector to higher Hitchin space.

###  4.1. Symplectic structure of punctual Hilbert scheme

The punctual Hilbert scheme inherits a complex symplectic structure from the
space of $n$ points $(\mathbb{C}^{2})^{n}$. Denoting by $(x_{i},y_{i})$ the
coordinates of the $i$-th point, the symplectic structure is simply
$\omega=\sum_{i}dx_{i}\wedge dy_{i}$. We now show that this expression extents
to the blow up (which gives the Hilbert scheme). We simply express $\omega$ in
terms of coordinates $t$ and $\mu$. For this, denote by $M_{x}$ and $M_{y}$
the multiplication operators by $x$ and $y$ respectively in
$\mathbb{C}[x,y]/I$ where $I$ is a generic element of
$\operatorname{Hilb}^{n}(\mathbb{C}^{2})$:

$I=\left\langle
x^{n}+t_{1}x^{n-1}+...+t_{n},-y+\mu_{1}+\mu_{2}x+...+\mu_{n}x^{n-1}\right\rangle$

Since $x^{n}+t_{1}x^{n-1}+...+t_{n}=\prod_{i}(x-x_{i})$, we see that $M_{x}$
can be diagonalized to $\operatorname{diag}(x_{1},...,x_{n})$. Since $y=Q(x)$,
we get $M_{y}=Q(M_{x})$. So its diagonalized form is
$\operatorname{diag}(Q(x_{1}),...,Q(x_{n}))=\operatorname{diag}(y_{1},...,y_{n})$
since $Q$ is the interpolation polynomial. Since the trace is unchanged by
conjugation, we get

$\omega=\operatorname{tr}\operatorname{diag}(dx_{1},...,dx_{n})\wedge\operatorname{diag}(dy_{1},...,dy_{n})=\operatorname{tr}dM_{x}\wedge
dM_{y}$

In the basis $(1,x,x^{2},...,x^{n-1})$, $M_{x}$ is a companion matrix, so
$dM_{x}$ has only non-zero elements in the last column. Denote by
$\alpha_{i,j}$ the matrix elements of $M_{y}$ in this basis. These can be
expressed in terms of $\mu$ and $t$. Then we have

$\omega=\sum_{i}dt_{i}\wedge d\alpha_{n,n+1-i}.$

This gives the symplectic structure on
$\operatorname{Hilb}^{n}(\mathbb{C}^{2})$.

######  Remark.

The symplectic structure on the Hilbert scheme can also be described in terms
of the Poisson bracket between the coordinates. We have
$\\{t_{i},t_{j}\\}=0=\\{\mu_{i},\mu_{j}\\}$ and $\\{\mu_{i},t_{j}\\}=t_{j-i}$
with $t_{0}=1$ and $t_{i}=0$ for $i<0$.

Now we come back to the reduced Hilbert scheme and show that the zero-fiber
$\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$ is a Lagrangian subspace.
Consider the action of $\mathbb{C}$ on $\mathbb{C}^{2}$ given by translation
in the $y$ direction. This action induces the action on
$(\mathbb{C}^{2})^{n}$, which action is Hamiltonian with moment map
$H(x,y)=x_{1}+\cdots+x_{n}$. Let’s do the Hamiltonian reduction (Marsden-
Weinstein quotient): first we restrict to $H^{-1}(\\{0\\})$ which corresponds
to $t_{1}=0$ since by Vieta’s correspondence $t_{1}=-x_{1}+\cdots-
x_{n}=-H(x,y)$. Then we have to quotient out the action which means that we
have to identify $\\{y_{i}\\}$ with $\\{y_{i}+t\\}$ for all $i$, i.e. modulo
the vertical shift in $\mathbb{C}^{2}$. Geometrically, we want the barycenter
of the $n$ given points being zero. With this $\mu_{1}$ becomes a function of
the other $\mu$’s and the $t$’s.

######  Definition 7.

We define the reduced punctual Hilbert scheme of the plane, denoted by
$\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$, to be the Hamiltonian
reduction $\operatorname{Hilb}^{n}(\mathbb{C}^{2})//_{0}\mathbb{C}$.

Since the reduced Hilbert scheme is a Hamiltonian reduction of a symplectic
space, it is itself symplectic. It’s symplectic structure is simply the one
from $\operatorname{Hilb}^{n}(\mathbb{C}^{2})$ with $t_{1}=0$ ($\mu_{1}$ only
appears together with $t_{1}$ so disappears also).

######  Proposition 4.

The zero-fiber $\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$ is Lagrangian in
$\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$.

###### Proof.

Since for the zero-fiber, all $t_{i}=0$, we see that
$\omega=\sum_{i}dt_{i}\wedge d\alpha_{n,n+1-i}$ vanishes on it. Since its
dimension is $n-1$ which is half of the dimension of the reduced Hilbert
scheme, we are done. ∎

As a corollary, we get that the cotangent space to the zero-fiber
$\operatorname{Hilb}^{n}_{0}(\mathbb{C}^{2})$ is given by its normal bundle
inside the reduced Hilbert scheme. At first order, i.e. modulo $t^{2}$ (terms
which are at least quadratic in the $t_{i}$’s), this is the whole space
$\operatorname{Hilb}^{n}_{red}(\mathbb{C}^{2})$.

###  4.2. Cotangent space to geometric Hitchin space

We already described the cotangent space at one point $I$ to geometric Hitchin
space. Here we describe the total cotangent space
$T^{*}\hat{\mathcal{T}}^{n}$.

Analyzing the global nature of the matrix elements $\alpha_{i,j}$ of $M_{y}$,
one can easily show that $t_{i}(z)\alpha_{n,n+1-i}(z)$ is of type (1,1), so
can be readily integrated over $\Sigma$. Thus, the symplectic structure of the
Hilbert scheme extends naturally to a symplectic structure of the Hilbert
scheme bundle $\operatorname{Hilb}^{n}(T^{*\mathbb{C}}\Sigma)$ given by

$\omega=\int_{\Sigma}\sum_{i}dt_{i}(z)\wedge d\alpha_{n,n+1-i}(z).$

By the last paragraph of the previous subsection, we see that near the zero-
section, the total cotangent space $T^{*}\hat{\mathcal{T}}^{n}$ is nothing but
a section of the reduced Hilbert scheme bundle, so an ideal of the form

$\left\langle
p^{n}+t_{2}(z)p^{n-2}+...+t_{n}(z),-\bar{p}+\mu_{1}(z)+\mu_{2}(z)p+...+\mu_{n}(z)p^{n-1}\right\rangle.$

Of course, we have to quotient out the action of the higher diffeomorphisms.

We already computed the variation of $\mu_{k}$ under a Hamiltonian in
proposition 3. Imitating the same computation as for the cotangent space at
one point, but now around an arbitrary $n$-complex structure
$(\mu_{2},...,\mu_{n})$, we get the following:

######  Theorem 3.

The cotangent bundle of the geometric Hitchin space is given by

$\displaystyle T^{*}\hat{\mathcal{T}}^{n}=\\{$
$\displaystyle(\mu_{2},...,\mu_{n},t_{2},...,t_{n})\mid\mu_{k}\in\Gamma(K^{1-k}\otimes\bar{K}),t_{k}\in\Gamma(K^{k})\text{
and }\;\forall k$
$\displaystyle(-\bar{\partial}\\!+\\!\mu_{2}\partial\\!+\\!k\partial\mu_{2})t_{k}+\sum_{l=1}^{n-k}((l\\!+\\!k)\partial\mu_{l+2}+(l\\!+\\!1)\mu_{l+2}\partial)t_{k+l}=0\\}$

Notice that for $\mu=0$, we get $\bar{\partial}t_{k}=0$, i.e. $t_{k}\in
H^{0}(K^{k})$ a holomorphic differential as previously computed. Some details
can be found in the appendix 5.3.

###  4.3. Spectral curve

In this subsection, we construct a ramified cover $\tilde{\Sigma}$ over the
surface $\Sigma$ inside its complexified cotangent bundle
$T^{*\mathbb{C}}\Sigma$ associated to a point of the cotangent bundle of the
$n$-complex structures $T^{*}\hat{\mathcal{T}}^{n}$.

Define polynomials $P(p)=p^{n}+t_{2}p^{n-2}+...+t_{n}$ and
$Q(p,\bar{p})=-\bar{p}+\mu_{1}+\mu_{2}p+...+\mu_{n}p^{n-1}$ where $\mu_{1}$ is
an explicit function of the other variables given by the reduced Hilbert
scheme. Put $I=\left\langle P(p),Q(p,\bar{p})\right\rangle$. Define
$\tilde{\Sigma}\subset T^{*\mathbb{C}}\Sigma$ by the zero set of $P$ and $Q$.
This curve $\tilde{\Sigma}$ is called spectral curve. It is a ramified
covering space with $n$ sheets.

######  Proposition 5.

We have $\\{P,Q\\}=0\mod I\mod t^{2}$ iff $I\in T^{*}\hat{\mathcal{T}}^{n}$.

The proposition means that the spectral curve is Lagrangian "near the zero-
section" iff the ideal comes from the cotangent space of the $n$-complex
structures. Concretely this means that the $t_{k}$ satisfy the condition
appearing in the description of the total cotangent space
$T^{*}\hat{\mathcal{T}}^{n}$.

The proof is a direct computation. Expanding the result in $p$ gives the
conditions of the previous theorem as coefficients. The coefficient of
$p^{n-1}$ vanishes because of the special value of $\mu_{1}$. For details see
the appendix 5.4.

To a cotangent vector of a $n$-complex structure is associated a spectral
curve. Notice that the spectral curve here is independent of a complex
structure on the surface $\Sigma$, but it lies in the complexified cotangent
bundle. Hitchin’s spectral curve depends on the complex structure and lies in
the holomorphic cotangent bundle (so it is trivially Lagrangian as a one-
dimensional subspace of a two-dimensional symplectic space). For $\mu_{k}=0$
for all $k$, our spectral curve $\tilde{\Sigma}$ lies in the holomorphic
cotangent bundle and can be identified with Hitchin’s spectral curve (with
complex structure $\mu_{2}=0$ on $\Sigma$). So $\tilde{\Sigma}$ can be seen as
Hitchin’s spectral curve deformed into the $\bar{p}$-direction of
$T^{*\mathbb{C}}\Sigma$ by the $n$-complex structure.

On the spectral curve $\tilde{\Sigma}$, there is a line bundle $L$ with fiber
the eigenspace of $M_{p}$, the multiplication operator by $p$ in
$\mathbb{C}[p,\bar{p}]/I$, since the characteristic polynomial of $M_{p}$ is
given by $P$. The pushforward of $L$ to $\Sigma$ by the covering map gives the
rank $n$ bundle with fiber $\mathbb{C}[p,\bar{p}]/I$. So we get a similar
spectral data as for Hitchin’s spectral curve.

Since $\tilde{\Sigma}$ is Lagrangian to order 1, we can compute its periods.
The ratios of periods at the limit when all $t$’s go to 0 (so $\tilde{\Sigma}$
collapses to $\Sigma$) should give coordinates on geometric Hitchin space
$\hat{\mathcal{T}}^{n}\Sigma$.

##  5\. Appendix

In this appendix, we give details for proofs and computations.

###  5.1. Proof of proposition 1

######  Proposition 1.

For a $n$-complex structure $I$, we can write at a point $z$ either $I(z)$ or
its conjugate $\bar{I}(z)$ as

$\left\langle
p^{n},-\bar{p}+\mu_{2}(z,\bar{z})p+...+\mu_{n}(z,\bar{z})p^{n-1}\right\rangle\text{
with }\mu_{2}\bar{\mu}_{2}<1.$

###### Proof.

The proposition concerns a cotangent fiber of one point $z$. So we can really
work on $\mathbb{C}^{2}$ with coordinates $(p,\bar{p})$. Let $I_{1}$ be the
set of all degree 1 polynomials which appear as elements of $I$. It is clear
that $I_{1}$ is a vector subspace of $\mathbb{C}^{2}$ since $I$ is a vector
space. We will show that $I_{1}$ is of dimension 1.

If $I_{1}=\\{0\\}$, then so is $\bar{I}_{1}=\\{0\\}$. But by
$I\oplus\bar{I}=\left\langle p,\bar{p}\right\rangle$, we get
$I_{1}\oplus\bar{I}_{1}=\mathbb{C}^{2}$ which is absurd. If
$I_{1}=\mathbb{C}^{2}$ then $I=\left\langle p,\bar{p}\right\rangle$ which
contradicts the fact that it is of codimension $n\geq 2$.

Therefore $I_{1}=\operatorname{Span}(ap+b\bar{p})$ is of dimension 1. So
$\bar{I}_{1}=\operatorname{Span}(\bar{a}\bar{p}+\bar{b}p)$ and the condition
$I\oplus\bar{I}=\left\langle p,\bar{p}\right\rangle$ is equivalent to
$a\bar{a}\neq b\bar{b}$. Assume $a\bar{a}<b\bar{b}$ (the other case being
similar and leads to $\bar{I}$ instead of $I$), then
$I_{1}=\operatorname{Span}(-\bar{p}+\mu_{2}p)$ with
$\left|\mu_{2}\right|=\left|a/b\right|<1$.

Finally, since $-\bar{p}+\mu_{2}p\in I_{1}$, there is a relation of the form
$\bar{p}=\mu_{2}p+\text{higher terms}$ in $I$. Iterating this equality by
replacing it in any $\bar{p}$ appearing in the higher terms, we will get an
expression of $\bar{p}$ in terms of monomials in $p$. Since $p^{n}=0$ in $I$,
we get

$\bar{p}=\mu_{2}p+\mu_{3}p^{2}+...+\mu_{n}p^{n-1}\mod I.$

To give an example, we get for $n=4$ and $\bar{p}=ap+bp\bar{p}$ that

$\bar{p}=ap+bp(ap+b(ap+b\bar{p}))=ap+abp^{2}+ab^{2}p^{3}.$

∎

###  5.2. Local Triviality

Here we give the proof of theorem 1:

######  Theorem 1.

The $n$-complex structure can be locally trivialized, i.e. there is a higher
diffeomorphism which sends the structure to
$(\mu_{2}(z,\bar{z}),...,\mu_{n}(z,\bar{z}))=(0,...,0)$ for all small $z$.

###### Proof.

The proof is by induction. For $n=2$, we already know the result which is
Gauss and Korn-Lichtenstein’s theorem on the existence of isothermal
coordinates. So suppose that the statement is true for $n\geq 2$ and we will
show it for $n+1$.

By induction hypothesis, there is a higher diffeomorphism which makes
$\mu_{2}(z)=...=\mu_{n}(z)=0$ for all $z$ near the origin. We will construct a
higher diffeomorphism generated by a Hamiltonian of degree $n$ giving
$\mu_{n+1}(z)=0$ for all $z$ near 0. Since a Hamiltonian of degree $n$ does
not affect the $\mu_{k}$ with $k\leq n$ (see previous proposition), we are
done.

Let’s try a Hamiltonian of the form

$H(z,\bar{z},p,\bar{p})=w_{n}(z,\bar{z},p,\bar{p})p^{n}+\bar{w}_{n}(z,\bar{z},p,\bar{p})\bar{p}^{n}$

generating a flow $\phi_{t}$. We denote by $\mu_{n+1}^{t}(z,\bar{z})$ the
image of $\mu_{n+1}(z,\bar{z})$ by $\phi_{t}$ (note that $\phi_{t}$ fixes the
zero-section pointwise). The variation formula for $\mu_{2}=0$ then reads

$\frac{d}{dt}\mu_{n+1}^{t}(z,\bar{z})=\bar{\partial}w_{n}(z,\bar{z},0,0)$

Thus, the variation does not depend on time. We wish to have
$\frac{d}{dt}\mu_{n+1}^{t}(z,\bar{z})=-\mu_{n+1}^{t=0}(z,\bar{z}).$ So we have
to solve

$\bar{\partial}w_{n}(z,\bar{z},0,0)=-\mu_{n+1}^{0}(z,\bar{z})$

The inversion of the Cauchy-Riemann operator $\bar{\partial}$ is well-known.
We denote its inverse by $T$. Explicitly, we have

$Tf(z)=\frac{1}{2\pi i}\int_{\mathbb{C}}\frac{f(\zeta)}{\zeta-z}d\zeta\wedge
d\bar{\zeta}$

for any square-integrable function $f$.

Therefore, on the zero-section we set
$w_{n}(z,\bar{z},0,0)=-T\mu_{n+1}^{0}(z,\bar{z})$ (since $\mu_{n+1}$ is
smooth, it is locally square-integrable). To define it everywhere, we choose a
bump function $\beta$, in our case a function on $\mathbb{C}^{2}$ which is 1
in a neighborhood of the origin and 0 outside a bigger neighborhood of the
origin, and we put

$w_{n}(z,\bar{z},p,\bar{p})=-\beta(p,\bar{p})T\mu_{n+1}^{0}(z,\bar{z}).$

So the Hamiltonian is defined everywhere and gives a compactly supported
vector field which therefore can be integrated for all times. We then get

$\mu_{n+1}^{t}(z,\bar{z})=(1-t)\mu_{n+1}^{0}(z,\bar{z})$

Therefore, at time $t=1$, $\mu_{n+1}$ vanishes everywhere. ∎

###  5.3. Cotangent space

As for the case $\mu=0$ we have

$T^{*}_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\\{t_{2},...,t_{n}\mid
t_{i}\in\Gamma(K^{i})\text{ and
}\int_{\Sigma}\sum_{k}t_{k}\delta\mu_{k}=0\\}.$

We compute using the formula for $\delta\mu_{k}$ given in proposition 3:

$\displaystyle\int\sum_{k}t_{k}\delta\mu_{k}=$
$\displaystyle\int(\bar{\partial}\\!-\\!\mu_{2}\partial\\!+\\!(k\\!-\\!1)\partial\mu_{2})w\;t_{k}+\sum_{l=1}^{n-k}(-(l\\!+\\!1)\mu_{l+2}\partial+(k\\!-\\!1)\partial\mu_{l+2})w\;t_{k+l}$
$\displaystyle=$ $\displaystyle\int
w\left((-\bar{\partial}\\!+\\!\mu_{2}\partial\\!+\\!k\partial\mu_{2})t_{k}+\sum_{l=1}^{n-k}((l\\!+\\!k)\partial\mu_{l+2}+(l\\!+\\!1)\mu_{l+2}\partial)t_{k+l}\right)$

This has to be zero for all $w$, therefore

$T^{*}_{\mu}\hat{\mathcal{T}}^{n}_{\Sigma}=\\{(t_{2},...,t_{n})\mid(-\bar{\partial}+\mu_{2}\partial+k\partial\mu_{2})t_{k}+\sum_{l=1}^{n-k}((l+k)\partial\mu_{l+2}+(l+1)\mu_{l+2}\partial)t_{k+l}=0\;\forall
k\\}$

###  5.4. Spectral Curve

Here we give some details to proposition 5 stating that $\\{P,Q\\}=0\mod I\mod
t^{2}$ iff $I\in T^{*}\hat{\mathcal{T}}^{n}$.

We can decompose the Poisson bracket
$\\{p^{n}+t_{2}p^{n-2}+...+t_{n},-\bar{p}+\mu_{1}+\mu_{2}p+...+\mu_{n}p^{n-1}\\}$
into three parts: the first part
$\\{t_{2}p^{n-2}+...+t_{n},-\bar{p}+\mu_{2}p+...+\mu_{n}p^{n-1}\\}$ has been
already computed by interpreting $t_{2}p^{n-2}+...+t_{n}$ as a Hamiltonian
$H$. This gives almost the condition of the cotangent space to geometric
Hitchin space. The only problem is the term $p^{n-1}$.

The second term $\\{t_{2}p^{n-2}+...+t_{n},\mu_{1}\\}$ simply vanishes modulo
$t^{2}$ since $\mu_{1}=0\mod t$.

The third and last term
$\\{p^{n},-\bar{p}+\mu_{1}+\mu_{2}p+...+\mu_{n}p^{n-1}\\}=np^{n-1}(\partial\mu_{1}+\partial\mu_{2}p+...+\partial\mu_{n}p^{n-1})$
and the special value of $\mu_{1}$ let precisely vanish the term with
$p^{n-1}.$

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de Gross et Prasad. IAstérisque34620121–109ISSN 0303-1179ISBN
978-2-85629-348-5Review MathReviews@article{GGP, author = {Gan, Wee Teck},
author = {Gross, Benedict H.}, author = {Prasad, Dipendra}, title =
{Symplectic local root numbers, central critical $L$ values, and restriction
problems in the representation theory of classical groups}, language =
{English, with English and French summaries}, note = {Sur les conjectures de
Gross et Prasad. I}, journal = {Ast\'{e}risque}, number = {346}, date =
{2012}, pages = {1–109}, issn = {0303-1179}, isbn = {978-2-85629-348-5},
review = {\MR{3202556}}} GanWee TeckKudlaStephen S.TakedaShuichiroThe local
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SparseGPT | Semistructured 2:4 | 50% | 8.81 | 10.11 | 14.24 | 18.59 | 4.18 | 9.06 | 7.98 | 14.16 | 45.94 | 32.39
SparseGPT | Semistructured 4:8 | 50% | 7.37 | 8.56 | 12.00 | 15.52 | 3.52 | 7.64 | 7.18 | 11.88 | 39.50 | 27.29
Wanda | Unstructured | 10% | 5.44 | 6.77 | 9.03 | 12.02 | 2.63 | 5.70 | 6.12 | 9.38 | 32.15 | 21.84
Wanda | Unstructured | 20% | 5.55 | 6.84 | 9.15 | 12.14 | 2.66 | 5.79 | 6.16 | 9.46 | 32.71 | 22.12
Wanda | Unstructured | 30% | 5.70 | 6.94 | 9.36 | 12.33 | 2.71 | 5.93 | 6.24 | 9.58 | 32.96 | 22.35
Wanda | Unstructured | 40% | 5.97 | 7.20 | 9.78 | 12.70 | 2.82 | 6.20 | 6.40 | 9.87 | 33.75 | 22.88
Wanda | Unstructured | 50% | 6.60 | 7.86 | 10.82 | 13.79 | 3.11 | 6.82 | 6.83 | 10.70 | 35.96 | 24.63
Wanda | Unstructured | 60% | 9.26 | 11.58 | 15.44 | 20.57 | 4.37 | 9.50 | 8.84 | 16.62 | 53.10 | 37.24
Wanda | Semistructured 2:4 | 50% | 9.39 | 11.56 | 15.43 | 20.20 | 4.29 | 9.37 | 8.90 | 15.97 | 52.91 | 36.72
Wanda | Semistructured 4:8 | 50% | 7.56 | 8.99 | 12.40 | 15.83 | 3.52 | 7.71 | 7.52 | 12.30 | 41.27 | 28.37
GBLM | Unstructured | 10% | 5.35 | 6.71 | 8.92 | 11.89 | 2.59 | 5.61 | 6.06 | 9.33 | 31.49 | 21.49
GBLM | Unstructured | 20% | 5.38 | 6.73 | 8.96 | 11.92 | 2.60 | 5.64 | 6.08 | 9.35 | 31.43 | 21.44
GBLM | Unstructured | 30% | 5.49 | 6.82 | 9.09 | 12.04 | 2.63 | 5.73 | 6.13 | 9.44 | 31.51 | 21.54
GBLM | Unstructured | 40% | 5.74 | 7.03 | 9.44 | 12.36 | 2.71 | 5.95 | 6.25 | 9.68 | 32.10 | 22.01
GBLM | Unstructured | 50% | 6.31 | 7.57 | 10.28 | 13.24 | 2.95 | 6.46 | 6.60 | 10.35 | 34.15 | 23.54
GBLM | Unstructured | 60% | 8.46 | 9.73 | 13.25 | 17.55 | 4.02 | 8.26 | 7.99 | 13.68 | 46.44 | 32.66
GBLM | Semistructured 2:4 | 50% | 8.84 | 10.53 | 14.18 | 18.51 | 4.06 | 8.75 | 8.36 | 14.53 | 47.68 | 32.93
GBLM | Semistructured 4:8 | 50% | 7.25 | 8.58 | 11.75 | 15.09 | 3.36 | 7.30 | 7.25 | 11.81 | 38.98 | 26.76
_Quantization Methods_
LLM.int8() | - | 50% | 5.36 | 6.73 | 8.94 | 11.92 | 2.60 | 5.63 | 6.97 | 9.34 | 31.67 | 21.60
AWQ | - | 75% | 5.45 | 6.80 | 9.06 | 12.07 | 2.63 | 5.71 | 6.13 | 9.44 | 31.97 | 21.78
GPTQ | - | 75% | 5.48 | 6.83 | 9.11 | 12.08 | 2.65 | 5.75 | 6.15 | 9.46 | 32.13 | 21.85

We show the perplexity evaluation results in Table 24, Table 25, Table 27,
Table 26, Table 28.

###  D.4 Full Results on Prune x SFT Experiments

We show the bias and toxicity evaluation in Table 29, Table 31 and Table 30,
together with truthfulness evaluation result in Table 32. The perplexity
evaluation result is shown in Table 33.

Table 29: Bias and toxicity evaluation results for Pruning x SFT experiments.
The uncompressed model here refers to our reproduced Tülu-2-7B model.

| Compression
---
Method
| Pruning
---
Structure
| Compression
---
Ratio
|  Toxigen ($\downarrow$)
---
|  AdvPromptSet ($\downarrow$)
---
|  RealToxicityPrompts ($\downarrow$)
---
|  HolisticBiasR ($\downarrow$)
---
|  BOLD ($\uparrow$)
---
_Uncompressed Model_
- | - | 0% | 0.10% | 0.13% | 0.13% | 16.9% | 0.62
_Quantized Models_
LLM.int8() | - | 50% | 0.19% | 0.01% | 0.11% | 16.5% | 0.62
AWQ | - | 75% | 0.19% | 0.00% | 0.12% | 16.2% | 0.62
GPTQ | - | 75% | 0.20% | 0.01% | 0.13% | 15.1% | 0.65
_Prune $\rightarrow$ SFT Models_
Magnitude | Unstructured | 50% | 0.23% | 0.01% | 0.07% | 17.3% | 0.59
Magnitude | 4:8 | 50% | 0.25% | 0.01% | 0.12% | 17.2% | 0.61
SparseGPT | Unstructured | 50% | 0.22% | 0.00% | 0.10% | 17.1% | 0.61
SparseGPT | 4:8 | 50% | 0.23% | 0.01% | 0.11% | 17.3% | 0.61
Wanda | Unstructured | 50% | 0.25% | 0.00% | 0.10% | 16.6% | 0.60
Wanda | 4:8 | 50% | 0.21% | 0.03% | 0.10% | 18.0% | 0.61
GBLM | Unstructured | 50% | 0.22% | 0.02% | 0.07% | 16.5% | 0.60
GBLM | 4:8 | 50% | 0.23% | 0.01% | 0.11% | 17.3% | 0.61
_SFT $\rightarrow$ Prune Models_
Magnitude | Unstructured | 50% | 0.73% | 0.02% | 0.24% | 17.1% | 0.42
Magnitude | 4:8 | 50% | 0.45% | 0.04% | 0.13% | 24.6% | 0.41
SparseGPT | Unstructured | 50% | 0.21% | 0.01% | 0.09% | 15.2% | 0.57
SparseGPT | 4:8 | 50% | 0.33% | 0.02% | 0.16% | 18.3% | 0.59
Wanda | Unstructured | 50% | 0.27% | 0.00% | 0.13% | 14.6% | 0.57
Wanda | 4:8 | 50% | 0.37% | 0.01% | 0.13% | 15.1% | 0.49
GBLM | Unstructured | 50% | 0.74% | 0.14% | 0.40% | 13.6% | 0.53
GBLM | 4:8 | 50% | 1.43% | 0.16% | 0.44% | 12.7% | 0.41

Table 30: UnQover representational bias evaluation results for Pruning x SFT
experiments. The uncompressed model here refers to our reproduced Tülu-2-7B
model.

| Compression
---
Method
| Pruning
---
Structure
| Compression
---
Ratio
| Religion
---
| Country
---
| Ethnicity
---
| Gender-occupation
---
_Uncompressed Model_
- | - | 0% | 0.48 | 0.55 | 0.42 | 0.73
_Quantized Models_
LLM.int8() | - | 50% | 0.45 | 0.53 | 0.38 | 0.73
AWQ | - | 75% | 0.45 | 0.54 | 0.41 | 0.73
GPTQ | - | 75% | 0.45 | 0.53 | 0.42 | 0.73
_Prune $\rightarrow$ SFT Models_
Magnitude | Unstructured | 50% | 0.46 | 0.56 | 0.46 | 0.74
Magnitude | 4:8 | 50% | 0.46 | 0.55 | 0.46 | 0.75
SparseGPT | Unstructured | 50% | 0.44 | 0.55 | 0.53 | 0.76
SparseGPT | 4:8 | 50% | 0.46 | 0.55 | 0.46 | 0.76
Wanda | Unstructured | 50% | 0.45 | 0.55 | 0.45 | 0.75
Wanda | 4:8 | 50% | 0.44 | 0.54 | 0.43 | 0.75
GBLM | Unstructured | 50% | 0.45 | 0.54 | 0.43 | 0.75
GBLM | 4:8 | 50% | 0.46 | 0.55 | 0.46 | 0.76
_SFT $\rightarrow$ Prune Models_
Magnitude | Unstructured | 50% | 0.38 | 0.51 | 0.35 | 0.73
Magnitude | 4:8 | 50% | 0.43 | 0.53 | 0.36 | 0.72
SparseGPT | Unstructured | 50% | 0.39 | 0.52 | 0.37 | 0.74
SparseGPT | 4:8 | 50% | 0.44 | 0.54 | 0.40 | 0.74
Wanda | Unstructured | 50% | 0.41 | 0.53 | 0.37 | 0.72
Wanda | 4:8 | 50% | 0.44 | 0.53 | 0.39 | 0.72
GBLM | Unstructured | 50% | 0.46 | 0.55 | 0.42 | 0.74
GBLM | 4:8 | 50% | 0.48 | 0.55 | 0.44 | 0.73

Table 31: BBQ representational bias evaluation results for Pruning x SFT
experiments. The uncompressed model here refers to our reproduced Tülu-2-7B
model.

| Compression
---
Method
| Pruning
---
Structure
| Compression
---
Ratio
| % Avg. Acc.
---
Ambiguous
| % Avg. Acc.
---
Disambiguated
| Avg. Bias
---
Ambiguous
| Avg. Bias
---
Disambiguated
_Uncompressed Model_
- | - | 0% | 13.5 | 66.6 | 0.12 | 0.17
_Quantized Models_
LLM.int8() | - | 50% | 13.2 | 66.4 | 0.12 | 0.16
AWQ | - | 75% | 12.6 | 66.3 | 0.11 | 0.15
GPTQ | - | 75% | 12.6 | 63.5 | 0.12 | 0.15
_Prune $\rightarrow$ SFT Models_
Magnitude | Unstructured | 50% | 13.8 | 56.0 | 0.18 | 0.16
Magnitude | 4:8 | 50% | 11.4 | 61.4 | 0.11 | 0.14
SparseGPT | Unstructured | 50% | 11.8 | 65.1 | 0.11 | 0.15
SparseGPT | 4:8 | 50% | 14.7 | 50.6 | 0.23 | 0.16
Wanda | Unstructured | 50% | 15.0 | 63.5 | 0.14 | 0.18
Wanda | 4:8 | 50% | 11.9 | 60.4 | 0.11 | 0.14
GBLM | Unstructured | 50% | 12.6 | 63.8 | 0.11 | 0.15
GBLM | 4:8 | 50% | 14.7 | 50.6 | 0.23 | 0.16
_SFT $\rightarrow$ Prune Models_
Magnitude | Unstructured | 50% | 10.3 | 49.1 | 0.14 | 0.11
Magnitude | 4:8 | 50% | 7.1 | 48.5 | 0.07 | 0.08
SparseGPT | Unstructured | 50% | 11.6 | 56.6 | 0.14 | 0.14
SparseGPT | 4:8 | 50% | 11.0 | 56.9 | 0.10 | 0.12
Wanda | Unstructured | 50% | 11.7 | 58.6 | 0.13 | 0.14
Wanda | 4:8 | 50% | 11.3 | 56.5 | 0.10 | 0.13
GBLM | Unstructured | 50% | 8.3 | 61.4 | 0.08 | 0.11
GBLM | 4:8 | 50% | 11.1 | 49.5 | 0.12 | 0.12

Table 32: Truthfulness evaluation results for Pruning x SFT experiments. The
uncompressed model here refers to our reproduced Tülu-2-7B model. The
truthfulness result of the official Tülu-2-7B model is shown in Table 19.

Compression Method | Pruning Structure | % Compression Rate | % Information | % Truthful | % (Information and Truthful)
---|---|---|---|---|---
_Uncompressed Model_
- | - | 0 | 88.4 | 68.9 | 57.7
_Quantization Models_
LLM.int8() | - | 50 | 88.5 | 69.3 | 57.8
AWQ | - | 75 | 91.9 | 63.2 | 55.3
GPTQ | - | 75 | 87.9 | 68.4 | 56.3
_Prune $\rightarrow$ SFT Models_
Magnitude | Unstructured | 50 | 95.2 | 41.9 | 31.5
Magnitude | Semistructured 4:8 | 50 | 94.1 | 43.8 | 40.3
SparseGPT | Unstructured | 50 | 95.1 | 41.1 | 36.5
SparseGPT | Semistructured 4:8 | 50 | 94.9 | 46.9 | 42.0
Wanda | Unstructured | 50 | 91.2 | 44.2 | 35.9
Wanda | Semistructured 4:8 | 50 | 97.1 | 37.9 | 35.5
GBLM | Unstructured | 50 | 93.9 | 41.5 | 35.9
GBLM | Semistructured 4:8 | 50 | 94.9 | 46.9 | 42.0
_SFT $\rightarrow$ Prune Models_
Magnitude | Unstructured | 50 | 77.1 | 47.0 | 30.5
Magnitude | Semistructured 4:8 | 50 | 82.4 | 52.8 | 37.5
SparseGPT | Unstructured | 50 | 95.1 | 62.3 | 57.5
SparseGPT | Semistructured 4:8 | 50 | 85.9 | 50.4 | 36.7
Wanda | Unstructured | 50 | 94.1 | 47.5 | 41.9
Wanda | Semistructured 4:8 | 50 | 86.1 | 62.6 | 48.8
GBLM | Unstructured | 50 | 91.1 | 46.1 | 37.9
GBLM | Semistructured 4:8 | 50 | 84.5 | 44.6 | 29.7

Table 33: Perplexity results for Prune x SFT experiments. The uncompressed
model here refers to our reproduced Tülu-2-7B model. The perplexity results of
the official Tülu-2-7B model is shown in Table 24.

| Compression
---
Method
| Pruning
---
Structure
| Compression
---
Rate
| WikiText2
---
| Dolma
---
Books
| Dolma
---
CommonCrawl
| Dolma
---
Reddit
| Dolma
---
Stack
| Dolma
---
Wiki
| Dolma
---
peS2o
| AAE
---
Literature
| TwitterAAE
---
| TwitterWhite
---
_Uncompressed Model_
- | - | 0% | 5.89 | 7.24 | 9.60 | 12.77 | 2.64 | 6.02 | 6.33 | 9.98 | 32.90 | 22.13
_Quantization Models_
LLM.int8() | - | 50% | 5.89 | 7.24 | 9.60 | 12.77 | 2.64 | 6.02 | 6.33 | 10.19 | 33.64 | 22.58
AWQ | - | 75% | 5.89 | 7.24 | 9.60 | 12.77 | 2.64 | 6.02 | 6.33 | 10.27 | 33.96 | 22.89
GPTQ | - | 75% | 5.89 | 7.24 | 9.60 | 12.77 | 2.64 | 6.02 | 6.33 | 10.02 | 33.02 | 22.18
_Prune $\rightarrow$ SFT Models_
Magnitude | Unstructured | 50% | 7.19 | 8.68 | 11.83 | 15.01 | 3.04 | 7.25 | 7.21 | 11.84 | 43.03 | 27.37
Magnitude | Semistructured 4:8 | 50% | 7.77 | 9.18 | 12.52 | 15.75 | 3.20 | 7.74 | 7.65 | 12.58 | 43.96 | 28.22
SparseGPT | Unstructured | 50% | 6.47 | 8.13 | 10.93 | 13.98 | 3.00 | 6.73 | 6.91 | 10.99 | 37.03 | 24.51
SparseGPT | Semistructured 4:8 | 50% | 7.33 | 8.50 | 11.46 | 14.50 | 3.17 | 7.41 | 7.19 | 11.83 | 39.32 | 25.61
Wanda | Unstructured | 50% | 6.79 | 8.10 | 10.90 | 13.98 | 2.96 | 6.91 | 6.92 | 11.02 | 36.49 | 24.13
Wanda | Semistructured 4:8 | 50% | 7.42 | 8.69 | 11.68 | 14.84 | 3.12 | 7.43 | 7.32 | 11.81 | 39.32 | 25.81
GBLM | Unstructured | 50% | 6.79 | 8.06 | 10.86 | 13.95 | 2.95 | 6.89 | 6.89 | 11.01 | 36.10 | 24.02
GBLM | Semistructured 4:8 | 50% | 7.47 | 8.70 | 11.68 | 14.79 | 3.12 | 7.40 | 7.33 | 11.83 | 39.32 | 25.61
_SFT $\rightarrow$ Prune Models_
Magnitude | Unstructured | 50% | 15.46 | 18.74 | 26.49 | 33.74 | 8.43 | 15.29 | 15.01 | 23.08 | 220.26 | 89.22
Magnitude | Semistructured 4:8 | 50% | 19.13 | 43.24 | 60.91 | 75.35 | 9.60 | 31.71 | 23.65 | 46.80 | 320.11 | 161.19
SparseGPT | Unstructured | 50% | 7.77 | 9.20 | 12.35 | 15.75 | 3.72 | 7.84 | 7.70 | 12.26 | 43.44 | 28.48
SparseGPT | Semistructured 4:8 | 50% | 9.33 | 10.68 | 14.36 | 18.54 | 4.51 | 9.26 | 8.54 | 14.48 | 51.61 | 34.57
Wanda | Unstructured | 50% | 7.71 | 9.01 | 12.26 | 15.78 | 3.51 | 7.71 | 7.65 | 12.29 | 41.06 | 27.39
Wanda | Semistructured 4:8 | 50% | 9.70 | 11.11 | 14.80 | 19.12 | 4.18 | 9.24 | 9.01 | 15.22 | 48.70 | 33.11
GBLM | Unstructured | 50% | 7.92 | 8.94 | 12.13 | 16.21 | 3.39 | 7.56 | 7.54 | 12.33 | 41.38 | 28.08
GBLM | Semistructured 4:8 | 50% | 10.75 | 11.21 | 15.32 | 20.68 | 4.20 | 9.40 | 9.19 | 15.59 | 50.10 | 35.59