# Berglund-Hübsch Transpose Rule and Sasakian Geometry

Ralph R. Gomez  Department of Mathematics and Statistics
Swarthmore College
Swarthmore, PA 19081

###### Abstract.

We apply the Berglund-Hübsch transpose rule from BHK mirror symmetry to show
that to an $n-1$-dimensional Calabi-Yau orbifold in weighted projective space
defined by an invertible polynomial, we can associate four (possibly) distinct
Sasaki manifolds of dimension $2n+1$ which are $n-1$-connected and admit a
metric of positive Ricci curvature. We apply this theorem to show that for a
given K3 orbifold, there exists four seven dimensional Sasakian manifolds of
positive Ricci curvature, two of which are actually Sasaki-Einstein.

##  1\. Introduction

In this article, we use the Berglund-Hübsch transposition rule that is
involved in Berglund-Hübsch-Krawitz (BHK) mirror symmetry to show that to a
compact $n-1$-dimensional Calabi-Yau hypersurface defined by an invertible
polynomial in weighted projective space one can associate four (possibly)
topologically distinct $2n+1$ dimensional Sasaki manifolds
$M_{i},i=1,\ldots,4$ of positive Ricci curvature for which
$H_{n}(M_{i},\mathbb{Z})$ is completely determined. Furthermore, we give an
explicit example in which some of these manifolds are Sasaki-Einstein of
positive scalar curvature. As we discuss, the construction proposed is
motivated by a new relationship we observe between two specific Fano Kähler-
Einstein orbifolds used in [6] to prove the existence of Sasaki-Einstein
metrics on $S^{2}\times S^{3}$. The key insight we notice is that these two
orbifolds are related to one another via the Berglund-Hübsch transpose rule,
which is a key rule in the so-called BHK construction of mirror Calabi-Yau
pairs.
Briefly, mirror symmetry is a conjectured phenomenon that occurs between
Calabi-Yau varieties. BHK mirror symmetry, in particular, is a construction
developed by P. Berglund and T. Hübsch in the 1990s [3], and later expanded
upon and generalized by Krawitz in 2009 [18]. In their construction one begins
by associating the $n\times n$ matrix of exponents from the invertible
polynomial which defines a Calabi-Yau orbifold $X$ in weighted projective
space. Then by taking the transpose of the matrix, called the Berglund-Hübsch
transpose, it is possible to construct the associated polynomial of a certain
degree from the transpose matrix, which generates another Calabi-Yau orbifold
hypersurface $X^{\wedge}$ in a typically different weighted projective space.
Then by taking certain quotients of $X$ and $X^{\wedge}$, one can create
Calabi-Yau mirror pairs.
Now, many examples of Sasaki manifolds of positive Ricci curvature (and
Sasaki-Einstein) come from the the $S^{1}$-orbibundle construction developed
in [6] (see also [7] for comprehensive review) which is of the form
$S^{1}\rightarrow L_{f}\rightarrow X_{f}$ where $L_{f}$ is the link of an
isolated hypersurface singularity defined by the quasi-homogeneous polynomials
$f$ and $X_{f}$ is a Fano orbifold hypersurface in weighted projective space.
The main goal of this paper is to apply the Berglund-Hübsch transpose rule to
the Fano orbifold hypersurface $X_{f}$ and explore its implications for the
Sasakian geometry and topology on $L_{f}.$
One particularly interesting feature of the construction which we give as an
example in Section 5 is that if one starts with a K3 orbifold mirror pair (in
the sense of Dolgachev [11] and Voisin [23]), then the construction can give
rise to Sasaki-Einstein structures in dimension seven and therefore by the
AdS/CFT correspondence there should be a related superconformal theory in
dimension three. Thus the construction described in this article provides a
framework by which one can go from a mirror symmetry setting to an AdS/CFT
setting.

This paper is organized as follows: in Section 2 we review some preliminaries
needed for the main theorem. In Section 3, we review the BHK mirror symmetry
construction and in Section 4 we state and prove our main theorem. In Section
5 we present an application of the main theorem.

## Acknowledgements

I would like to thank Marco Aldi for suggesting to me the idea of looking into
how ideas from BHK mirror symmetry might be used in Sasakian geometry. I also
would like to thank Christina Tønneson-Friedman for useful conversations and
encouragement in writing this article. Parts of this article were written
while attending the Union College Mathematics Conference 2022 so I want to
thank the organizers for creating such a hospitable place to share ideas. I
would also like to thank Jaime Cuadros Valle and Tyler Kelly for useful
conversations. Finally, I would like to that the Lang Faculty fellowship from
Swarthmore College.

##  2\. Background

###  2.1. Links of Isolated Hypersurface Singularities

Let us consider the affine space $\mathbb{C}^{n+1}$ in conjunction with a
weighted $\mathbb{C}^{*}$ action, denoted by

$(z_{0},\dots,z_{n})\rightarrow(\lambda^{w_{0}}z_{0},\ldots,\lambda^{w_{n}}z_{n})$

where $\lambda\in\mathbb{C}^{*}.$ Throughout this paper, we assume
$gcd(w_{0},\ldots,w_{n})=1.$ The positive integral weights $w_{i}$ are viewed
as the components of a weight vector $\mathbf{w}\in(\mathbb{Z}^{+})^{n+1}.$ We
will also use the notation $|\mathbf{w}|$ to denote the sum of the weights in
the weight vector. Now, let $f\in\mathbb{C}[z_{0},\ldots,z_{n}]$. We say that
$f$ is a quasi-homogeneous polynomial of degree $d$ if

$f(\lambda^{w_{0}}z_{0},\ldots,\lambda^{w_{n}}z_{n})=\lambda^{d}f(z_{0},\ldots,z_{n})$

such that $d\in\mathbb{Z}^{+}.$ Note that the terminology “weighted
homogeneous” is also used in the literature. The weighted affine cone denoted
by $C_{f}$ is the zero set of $f$ and we will assume throughout the paper that
the only singularity of $f$ is at the origin. We are interested in special
manifolds that arise from this cone, called the link $L_{f}$, defined by

$L_{f}=C_{f}\cap S^{2n+1}$

where we view the odd dimensional unit sphere in $\mathbb{C}^{n+1}.$ It was
proven by Milnor in [20] that $L_{f}$ is a $(n-2)$-connected, compact smooth
manifold of dimension $2n-1.$ It is now well-known that the link $L_{f}$ and
weighted projective space $\mathbb{P}(\mathbf{w})$ are part of the following
commutative diagram. See for example [6] or [7] for a review.

${L_{f}}$${S^{2n+1}_{\mathbf{w}}}$${X_{f}}$${\mathbb{P}(\mathbf{w})\par}$$\scriptstyle{\pi}$

We think of the weighted projective space
$\mathbb{P}(\mathbf{w})=\mathbb{C}^{n+1}\setminus\\{0\\}/\mathbb{C}^{*}(\mathbf{w})$
where $\mathbb{C}^{*}(\mathbf{w})$ is the weighted $\mathbb{C}^{*}$ discussed
above.
Let us briefly explain the above commutative diagram. The manifold
$S^{2n+1}_{\mathbf{w}}$ admits a weighted Sasakian structure and this induces
an Sasakian structure on $L_{f}$. A Sasakian structure on an odd-dimensional
manifold is a quadruple of structure tensors $\mathcal{S}=(\xi,\eta,\Phi,g)$
where $g$ is a Riemannian metric, $\xi$ is a unit length Killing vector field
called the Reeb vector field, $\eta$ a contact 1-form where $\xi$ is the
associated Reeb vector field, the endomorphism of the tangent bundle $\Phi$ is
a $(1,1)$ tensor field which annihilates $\xi.$ Moreover, $\Phi$, when
restricted to the contact sub-bundle $\mathcal{D}=\text{ker}\eta$, is an
integrable complex structure. The odd-dimensional sphere $S^{2n+1}$ admits a
so-called weighted Sasakian structure defined by inserting weights into the
structure tensors. We do not need this explicitly so we will omit the precise
formulas but the interested reader should consult [7] for further details.
The flow of the weighted Reeb vector field $\xi_{\mathbf{w}}$ generates a
locally free $S^{1}$ action on $S^{2n+1}$ as well as $L_{f}.$ Taking
quotients, we obtain the weighted hypersurface $X_{f}$ and
$\mathbb{P}(\mathbf{w}).$ Note that $X_{f}$ is a compact Kähler orbifold which
cuts out a hypersurface in $\mathbb{P}(\mathbf{w})$. Finally, the horizontal
arrows in the commutative diagram above are Sasaki and Kähler embeddings while
the vertical arrows comprise $S^{1}$ orbibundles and which are orbifold
Riemannian submersions.
We now state a key theorem which provides the differential geometric
relationship between links, positive Ricci curvature, and weighted
hypersurfaces.

######  Theorem 1.

_(_[5]_)_ Let $L_{f}$ be a link of an isolated hypersurface singularity
defined by a quasi-homogeneous polynomial $f$ of degree $d$ with a Sasakian
structure so that we have the $S^{1}$ orbibundle $S^{1}\rightarrow
L_{f}\rightarrow X_{f}.$ If $X_{f}$ is a Fano hypersurface of degree $d$
($|\mathbf{w}|-d>0)$ in weighted projective space, then $L_{f}$ admits a
Sasakian structure such that the Sasakian metric has positive Ricci curvature.
Moreover, if $X_{f}$ is a Fano Kähler-Einstein orbifold hypersurface in
$\mathbb{P}(\mathbf{w})$ such that

$Id<\frac{n}{n-1}\emph{min}_{i,j}(w_{i}w_{j})$

where $I=|\mathbf{w}|-d$ then $L_{f}$ is Sasaki-Einstein of positive scalar
curvature.

Thus, links of isolated hypersurface singularities provides a setting for
constructing Sasakian metrics of positive Ricci curvature as well as Sasaki-
Einstein metrics on certain odd dimensional manifolds via the existence of
special hypersurfaces in weighted projective space. It is through this setting
of investigating hypersurfaces in weighted projective space that we turn now
to a mirror symmetry construction which relies upon hypersurfaces in weighted
projective space. This mirror symmetry construction is called BHK mirror
symmetry.

###  2.2. The BHK Mirror Symmetry Construction

Mirror symmetry is a conjectured duality between Calabi-Yau varieties which
was initially discovered by theoretical physicists Greene and Plesser [14].
This duality originally arose in the context of string theory in the 1990s. In
its classical formulation the main idea of mirror symmetry is the following:
if $X$ is a Calabi-Yau variety of dimension $n$, then there exists a Calabi-
Yau variety $X^{\wedge}$ such that

$H^{p,q}(X,\mathbb{C})\cong H^{n-p,q}(X^{\wedge},\mathbb{C})$

where $X$ and $X^{\wedge}$ are called a mirror pair. Since then there have
been many mathematical constructions of mirror symmetry in a wide variety of
contexts. Shortly after mirror symmetry emerged, the physicists P. Berglund
and T. Hübsch developed a very fascinating constructing that allowed one to
construct many examples of mirror pairs [3]. This method was overshadowed by
the powerful tools of toric constructions of mirror pairs developed by Batyrev
and Borisov [1], [2]. A few years later, Krawitz [18] revived the construction
by making it more mathematically precise and at the same time generalizing the
construction.
Consider the quasi-homogeneous polynomial $f$ of degree $d$ with weight vector
$\mathbf{w}$ whose zero locus defines a hypersurface in weighted projective
space $\mathbb{P}(\mathbf{w}).$ We shall consider a very special type of
quasi-homogeneous polynomial, called an invertible polynomial, which can be
represented by

(1) $f=\sum_{i=0}^{n}\prod_{j=0}^{n}x_{j}^{a_{ij}}.$

Note that the polynomial has $n$ variables containing $n$ monomials. We must
also assume that $f$ when viewed as a map
$f:\mathbb{A}^{n+1}\rightarrow\mathbb{A}^{1}$ has a single critical point only
at the origin or equivalently its affine cone is smooth outside $(0,...,0).$
This condition is called quasi-smooth. This allows us to state the following:

######  Definition 1.

A quasi-smooth, quasi-homogeneous polynomial, as given above by $(1)$ is
invertible if the corresponding $n\times n$ matrix of exponents $A=(a_{ij})$
is invertible.

An invertible polynomial is a rather restrictive condition. We denote the
inverse matrix by $A^{-1}=(a^{ij})_{i,j=1,...,n}.$ Observe that the matrix
$A$, the weight vector $\mathbf{w}$ and the degree of the polynomial $d$ must
satisfy the equation $A\mathbf{w}=\mathbf{d}$ where we view $\mathbf{d}$ as
the $n\times 1$ column vector with all of its entries the scalar $d.$ Now, we
can define $q_{i}=\sum_{j=1}^{n}a^{ij}$ to mean the sum of the entries of the
$i$-th row of the inverse matrix $A^{-1}.$ Therefore, if we let $d$ be the
least common denominator of the $q_{i}$ then the components of the weight
vector $w_{i}$ are given by $w_{i}=dq_{i}.$
It was shown in [19] that one can classify these polynomials. Such polynomials
come in three types, sometimes called the _atomic types_.

######  Theorem 2.

Any invertible polynomial is a sum of atomic types:

  1. (1)

Fermat: $f=x^{a}$

  2. (2)

Loop:
$f=x_{0}^{a_{0}}x_{1}+x_{1}^{a_{1}}x_{2}+\ldots+x_{n-1}^{a_{n-1}}x_{n}+x_{n}^{a_{n}}x_{0}$

  3. (3)

Chain:
$f=x_{0}^{a_{0}}x_{1}+x_{1}^{a_{1}}x_{2}+\ldots+x_{n-1}^{a_{n-1}}x_{n}+x_{n}^{a_{n}}$

We can now state the BHK mirror symmetry construction. Let $X_{f}$ be a
Calabi-Yau hypersurface in $\mathbb{P}(\mathbf{w})$ defined by the invertible
polynomial $f$ of degree $d$ and form the square matrix $A_{f}.$ Let $T$
denote transpose and form the matrix $A^{T}$. We call this the Berglund-Hübsch
transpose rule. It is now possible to create the new invertible polynomial
$f_{T}$ and so as we discussed above one needs to find $\mathbf{w}_{T}$ and
$d_{T}$ which obey $A^{T}\mathbf{w}_{T}=\mathbf{d}_{T}.$ The new quasi-
homogenous polynomial $f_{T}$ of degree $d_{T}$ in the weighted projective
space $\mathbb{P}(\mathbf{w}_{T})$ defines the orbifold $X_{f_{T}}$ which is
again a Calabi-Yau hypersurface (see next section for proof) where
$A^{T}_{f}=A_{f_{T}}.$ We can represent this process via the diagram:

${f}$${f_{T}}$${A_{f}}$${A_{f}^{T}}$$\scriptstyle{T}$

It is worthwhile to mention that if $f$ is an invertible polynomial then so is
$f_{T}$. This follows from Theorem 1 in [19].
Now, one would hope that $X_{f}$ and $X_{f_{T}}$ are mirror Calabi-Yau pairs
but unfortunately it is not true. The Fermat quintic is a good counterexample.
If $W$ is the Fermat quintic, then its mirror turns out to be the Calabi-Yau
variety obtained after resolving singularities of $W/(5\mathbb{Z})^{3}$. For
there to be a genuine mirror relation coming from the above Calabi-Yau
varieties, one must quotient each Calabi-Yau hypersurface by specially crafted
groups which are related to the automorphsims of $f$ and $f_{T}.$ In fact, in
[9], Chiodo and Ruan were able to prove a mirror symmetry relationship
involving $X_{f}$ and $X_{f_{T}}$ which is stated as

$H^{p,q}_{CR}(X_{f}/\tilde{G},\mathbb{C})\cong
H^{n-1-p,q}_{CR}(X_{f_{T}}/\tilde{G}^{T},\mathbb{C})$

with respect to Chen-Ruan cohomology. Here $\tilde{G}$ and $\tilde{G}^{T}$ are
the carefully crafted groups coming from the automorphism groups. We do not
need the precise formulation of these groups in this paper but their
definitions can be found for example in [18] and [9].

###  2.3. Sasakian Geometry and Topology of Links

To elucidate some of the topology of the link $L_{f}$, we can compute
$b_{n-1}(L_{f})$ and torsion in $H_{n-1}(L_{f},\mathbb{Z}).$ This topological
data is encoded in the degree $d$ and the weight vector $\mathbf{w}$ [21]. In
this section we review the combinatorial formulas for the Betti numbers and
the torsion groups in homology.
Given $f$ of degree $d$ with weight vector
$\mathbf{w}=(w_{0},w_{1},...,w_{n})$, it is useful to first define the
quantities

$u_{i}=\frac{d}{gcd(d,w_{i})},\hskip
42.67912ptv_{i}=\frac{w_{i}}{gcd(d,w_{i})}.$

The formula for the Betti number $b_{n-1}(L_{f})$ is given by:

$b_{n-1}(L_{f})=\sum(-1)^{n+1-s}\frac{u_{i_{1}},\ldots
u_{i_{s}}}{v_{i_{1}}\ldots v_{i_{s}}\text{lcm}(u_{i_{1}},\ldots,u_{i_{s}})}.$

Here the sum is over all possible $2^{n+1}$ subsets $\\{i_{1},\ldots,i_{s}\\}$
of $\\{0,\ldots n\\}.$
For the torsion of the homology group, Orlik conjectured [21] that for a given
link $L_{f}$ of dimension $2n-1$ one has

(2)
$H_{n-1}(L_{f},\mathbb{Z})_{tor}=\mathbb{Z}_{d_{1}}\oplus\mathbb{Z}_{d_{2}}\oplus\cdots\oplus\mathbb{Z}_{d_{r}}$

Let us now discuss how the $d_{i}$ data are given, following closely [7].
Given an index set $\\{i_{1},i_{2},....,i_{s}\\}$, define $I$ to be the set of
all of the $2^{s}$ subsets and let us designate $J$ to be all of the proper
subsets. For each possible subset, inductively define the numbers
$c_{i_{1},...,i_{s}}$ and $k_{i_{1},...,i_{s}}$. For each ordered subset
$\\{i_{1},...,i_{s}\\}\subset\\{0,1,2,...,n\\}$ with
$i_{1}<i_{2}<\cdots<i_{s}$ one defines the set of $2^{s}$ positive integers,
beginning with $c_{\emptyset}=gcd(u_{0},...,u_{n}):$

$c_{i_{1},...,i_{s}}=\frac{gcd(u_{0},\ldots,\widehat{u}_{i_{1}},\ldots,\widehat{u}_{i_{s}},\ldots,u_{n})}{\displaystyle\prod_{J}c_{j_{1},\ldots,j_{t}}}.$

Now, to get the $k^{\prime}s$:

$k_{i_{1},...,i_{s}}=\epsilon_{n-s+1}\kappa_{i_{1},...,i_{s}}=\epsilon_{n-s+1}\displaystyle\sum_{I}(-1)^{s-t}\frac{u_{j_{1}}\cdots
u_{j_{t}}}{v_{j_{1}}\cdots v_{j_{t}}lcm(u_{j_{1}},\ldots,u_{j_{t}})}$

where

$\epsilon_{n-s+1}=\begin{cases}0,&\text{if }n-s+1\text{ is even}\\\
1,&\text{if }n-s+1\text{ is odd.}\end{cases}$

Then for each $1\leq j\leq r=\lfloor max\\{k_{i_{1},\ldots,{i_{s}}}\\}\rfloor$
we put

$d_{j}=\prod_{k_{i_{1},\ldots,i_{s}}\geq j}c_{i_{1},\ldots,i_{s}}.$

Though the full conjecture is still open 45 years later, it is known to hold
in certain cases. However, substantial progress was made on the conjecture by
Hertling and Mase in [15]. In particular, the authors were able to show the
following:

######  Theorem 3.

_(_[15]_)_ Orlik’s conjecture holds for all invertible polynomials.

Actually far more was proved in [15] but for our purposes, this form of the
statement is all that is needed for the paper. Since we are dealing
exclusively with invertible polynomials, this theorem allows to use the above
formulas to explicitly compute $H_{n-1}(L_{f},\mathbb{Z}).$

##  3\. The Berglund-Hübsch Transpose Rule and Sasakian Geometry

Our goal for this paper is to explore the Berglund-Hübsch transpose rule in
the context of Fano orbifolds in weighted projective space and thus generate
Sasakian structures of positive Ricci curvature on the links. To emphasize the
utility of using the rule for links in Sasakian geometry, we will discuss a
novel interpretation of some examples in [6], which yielded new examples of
Sasaki-Einstein metrics of positive scalar curvature in dimension five.

###  3.1. Key Example with Fano Kähler-Einstein Orbifolds

In their paper [6], the authors show the existence of Sasaki-Einstein metrics
on $S^{2}\times S^{3}$ and $2\\#S^{2}\times S^{3}$. This is accomplished by
using three Fano Kähler-Einstein orbifold hypersurfaces in weighted projective
space and then arguing that the links over these hypersurfaces are
diffeomorphic to the desired manifolds. These examples are given by quasi-
homogeneous polynomials of degree 60 and 256.

  1. (1)

$f_{60}=z_{0}^{5}z_{1}+z_{0}z_{2}^{3}+z_{1}^{4}+z_{3}^{3}\subset\mathbb{P}(9,15,17,20)$,

  2. (2)

$f_{256}=z_{0}^{17}z_{2}+z_{0}z_{1}^{5}+z_{1}z_{2}^{3}+z_{3}^{2}\subset\mathbb{P}(11,49,69,128)$,

  3. (3)

$g_{256}=z_{0}^{17}z_{1}+z_{1}^{5}z_{2}+z_{0}z_{2}^{3}+z_{3}^{2}\subset\mathbb{P}(13,35,81,128).$

The hypersurfaces (and hence the links) are said to be _well-formed_ if the
hypersurface contains no codimension $2$ singular stratum of
$\mathbb{P}(w_{0},\ldots,w_{n}).$ It turns out that this condition is
equivalent to

$\text{gcd}(w_{1},\ldots,\hat{w_{i}},\ldots\hat{w_{j}},\ldots w_{n})|d$

for all $i,j=1,\ldots n.$ The hat means delete the element. We will see later
that interesting links also arise when the hypersurface is not well-formed.
Let us focus on the second hypersurface $f_{256}$ in the weighted projective
space $\mathbb{P}(11,49,69,128).$ Observe the quasi-homogenous polynomial is
indeed invertible since it is quasi-smooth and in fact a sum of a loop and
Fermat polynomial. We can then build the associated $4\times 4$ matrix $A$
given by

$A=\begin{bmatrix}17&0&1&0\\\ 1&5&0&0\\\ 0&1&3&0\\\ 0&0&0&2\end{bmatrix}.$

We compute now $A^{T}$ and so the corresponding quasi-homogenous polynomial is
given by

$z_{0}^{17}z_{1}+z_{1}^{5}z_{2}+z_{0}z_{2}^{3}+z_{3}^{2}.$

The observation to make here is that this polynomial is exactly $g_{256}$.
Furthermore, to find the corresponding weight vector $\mathbf{w}$ and the
degree we must solve $A^{T}\mathbf{w}=\mathbf{d}$ and we follow the procedure
discussed in Section 2.2. Since $A^{T}$ is invertible, to ensure
$\mathbf{w}\in(\mathbb{Z}^{+})^{4}$ we get that $d=256$ and
$\mathbf{w}=(13,35,81,128).$ We therefore have discovered the Fano Kähler-
Einstein oribifolds defined by $f_{256}$ and $g_{256}$ are actually related
via the Berglund-Hübsch transpose rule. This example reveals that the Fano
Kähler-Einstein condition in this example was preserved under the Berglund-
Hübsch transpose rule.
Let’s interpret this in terms of the Sasaki-Einstein structures on the links
over the orbifolds. As described in [6], it was further proved that the links
over these two orbifolds are diffeomorphic to $S^{2}\times S^{3}.$ Thus, the
Berglund-Hübsch transpose rule has allowed us, in this case, to transform from
one Sasaki-Einstein structure on $S^{2}\times S^{3}$ to another distinct
Sasaki-Einstein structure on $S^{2}\times S^{3}.$

###  3.2. The invertible polynomial $f_{60}$

One may ask what happens if we use the Berglund-Hübsch rule for the remaining
polynomial $f_{60}.$

In [6], the authors studied the following hypersurface given by

$f_{60}=z_{0}^{5}z_{1}+z_{0}z_{2}^{3}+z_{1}^{4}+z_{3}^{3}$

of degree $d=60$ in the weighted projective space $\mathbf{w}=(9,15,17,20).$
It is known that this Fano orbifold admits an orbifold Kähler-Einstein metric
[10]. By a change of variables, it is easy to see $f_{60}$ is a sum of a chain
polynomial and Fermat monomial and so is invertible. Now let’s use the
Berglund-Hübsch transpose rule. The exponential matrix for $f$ is given by

$A=\begin{bmatrix}5&1&0&0\\\ 1&0&3&0\\\ 0&4&0&0\\\ 0&0&0&3\end{bmatrix}$

Using the Berglünd-Hübsch transpose rule, we compute $A^{T}$. The
corresponding invertible polynomial is therefore

$\tilde{f}_{T}=z_{0}^{5}z_{1}+z_{0}z_{2}^{4}+z_{1}^{3}+z_{3}^{3}.$

(We can rewrite this as ($z_{0}^{3}+z_{0}z_{1}^{5}+z_{1}z_{2}^{4})+z_{3}^{3}$
which is again a sum of a chain and Fermat.) It is easy to determine
$\mathbf{w}_{T}$ and $d$ which obey $A\mathbf{w}_{T}=\mathbf{d}_{T}.$ We find
that $\mathbf{w}_{T}=(8,20,13,20)$ and $d_{T}=d=60$ works. It is interesting
to note that the corresponding link $L_{\tilde{f}}$ admits a Sasaki-Einstein
metric since the inequality $Id<\frac{n}{n-1}\text{min}_{i,j}(w_{i}w_{j})$ is
satisfied. Furthermore, it should be noted that the hypersurface does have a
branch divisor for $C=\\{z_{0}=0\\}$ in the singular locus. By Theorem 5.7 of
[17], the genus of this branch divisor determines the topology of $L_{f}.$
Using the genus formula given by

$g(C)=\frac{1}{2}\left(\frac{d^{2}}{w_{1}w_{2}w_{3}}-d\sum_{i<j}\frac{\text{gcd}(w_{i},w_{j})}{w_{i}w_{j}}+\sum\frac{\text{gcd}(d,w_{i})}{w_{i}}-1\right)$

we find $g(C)=0$ which implies that $H_{2}(L_{f},\mathbb{Z})$ has no torsion.
Moreover, calculation of the second Betti number as discussed in Section 2.3,
we find $b_{2}(L_{\tilde{f}})=2$ and so $L_{f}$ is diffeomorphic to
$2\\#S^{2}\times S^{3}.$ (Alternatively, we could have also used the ideas
from Section 2.3.) The Milnor number of the link is $94$ and is therefore
distinct from other known Sasaki-Einstein structures on $2\\#S^{2}\times
S^{3}$.
These examples emphasize and highlight the utility of the Berglund-Hübsch
transpose rule in relation to Sasakian geometry. In the next section, we
establish the general connection between the two ideas.

##  4\. Main results

In this section, we prove our main theorem. But first, we establish a simple
lemma. The Calabi-Yau case is already well-known but we include a proof here
for reference.

######  Lemma 1.

Let $f$ be an invertible polynomial defining a hypersurface $X_{f}$ in
weighted projective space $\mathbb{P}(\mathbf{w})$ and let
$A=(a_{ij})_{i,j=1,...,n}$ be the associated invertible $n\times n$ matrix of
exponents. Further let $X_{f_{T}}$ be the associated hypersurface in
$\mathbb{P}(\mathbf{w}_{T})$ defined by $f_{T}$ of degree $d_{T}.$ Then we
have:

  1. (1)

If $X_{f}$ is a Calabi-Yau hypersurface, then so is $X_{f_{T}}.$

  2. (2)

If $X_{f}$ is a Fano hypersurface, then so is $X_{f_{T}}$

###### Proof.

First, note that in either case, if $f$ is invertible then so is $f_{T}$ and
this follows from Theorem 1 of [19]. Since $f$ is quasi-homogenous of degree
$d$, we must have that $A\mathbf{w}=\mathbf{d}$ where $\mathbf{d}$ is the
$n\times 1$ column vector with entries $d$. The matrix $A$ is invertible with
entries of $A^{-1}$ denoted by $a^{ij}.$ Then the components of $\mathbf{w}$
which are positive integers denoted by $w_{i}$ are given by
$w_{i}=\sum_{j}a^{ij}d$. Moreover, since $X_{f}$ is Calabi-Yau, the condition
$|\mathbf{w}|=d$ can be rewritten as

$\sum_{i,j}a^{ij}=1.$

Now let $A^{T}$ denote the transpose of the exponential matrix. From this
matrix, we would like to construct an associated invertible polynomial $f_{T}$
of degree $d_{T}$ in weighted projective space $\mathbb{P}(\mathbf{w}_{T})$.
To find the weights for $f_{T}$ we must solve
$A^{T}\mathbf{w}_{T}=\mathbf{d}_{T}.$ Since $A$ is invertible we have

$\mathbf{w}_{T}=(A^{T})^{-1}\mathbf{d}_{T}=(A^{-1})^{T}\mathbf{d}_{T}$

where $\mathbf{d}_{T}$ is an $n\times 1$ column vector with entries $d_{T}$.
The components of $\mathbf{w}_{T}$, which we will denote by $w^{T}_{i}$ to
avoid cluttering of indices, are given by

$w^{T}_{i}=\sum_{i}a^{ji}d_{T}.$

So to establish the first statement that $X_{f_{T}}$ is Calabi-Yau, we compute
using the hypothesis $X_{f}$ is Calabi-Yau to obtain

$|\mathbf{w}_{T}|=\sum_{i,j}a^{ji}d_{T}=d_{T}.$

Hence $|\mathbf{w}_{T}|=d_{T}$ and $X_{f_{T}}$ is Calabi-Yau.

To establish the second statement, assume $X_{f}$ is Fano and hence the Fano
condition is $|\mathbf{w}|-d>0$ i.e.

$\displaystyle\sum_{i,j}a^{ij}>1.$

So, to establish that $X_{f_{T}}$ is Fano, we compute

$|\mathbf{w}_{T}|-d_{T}=\left(\sum_{i,j}a^{ij}-1\right)d_{T}.$

Since $\sum_{i,j}a^{ij}-1>0$ this implies $|\mathbf{w}_{T}|-d_{T}>0.$
Therefore, $X_{f_{T}}$ is Fano.

∎

We are now ready to state and prove the main theorem of this article.

######  Theorem 4.

Let $X_{f}$ be an $n-1$ dimensional Calabi-Yau hypersurface defined by an
invertible polynomial $f(x_{0},...,x_{n})$, not of Fermat type, in weighted
projective space $\mathbb{P}(w_{0},...,w_{n})$. Then there exists four
(possibly) distinct $n-1$-connected Sasaki manifolds $M_{i}$, $i=1,2,3,4$ of
dimension $2n+1$ of positive Ricci curvature such that
$H_{n}(M_{i},\mathbb{Z})$ is completely determined.

###### Proof.

Let $f$ be an invertible polynomial of degree in $d$ that cuts out a Calabi-
Yau orbifold hypersurface $X_{f}$ in weighted projective space
$\mathbb{P}(\mathbf{w}).$ Define the new quasi-homogenous polynomial of the
same degree $d$ given by $\tilde{f}=z^{d}+f$ which generates the new
hypersurface $Y_{\tilde{f}}$ in the weighted projective space
$\mathbb{P}(1,\mathbf{w}).$ It is clear that $\tilde{f}$ is invertible since
it is a sum of a Fermat monomial $z^{d}$ and the invertible polynomial $f.$
Note that $1+|\mathbf{w}|-d=1+d-d=1>0$, where we have used $|\mathbf{w}|=d$.
Thus $Y_{\tilde{f}}$ is an orbifold Fano hypersurface. By Lemma 1 above we can
construct $Y_{\tilde{f}_{T}}$ which is also an orbifold Fano hypersurface. Now
by Theorem 1, we have $S^{1}$-orbibundles $M_{1}$ and $M_{2}$ over
$Y_{\tilde{f}}$ and $Y_{\tilde{f}_{T}}$ respectively. Let us represent what we
have so far in a diagram:

${M_{1}}$${M_{2}}$${X_{f}}$${Y_{\tilde{f}}}$${Y_{\tilde{f}_{T}}.}$$\scriptstyle{\theta}$$\scriptstyle{T}$

We are abusing notation here slightly for the sake of illustrating how the
different spaces are related to one another by the construction. The map
$\theta$ takes the given Calabi-Yau hypersurface and assigns to it the Fano
orbifold $Y_{\tilde{f}}$ by mapping $f$ to $\tilde{f}=z^{d}+f$ where $z^{d}$
is of weight $1.$ The $T$ maps one Fano hypersurface to another via the
Berglund-Hübsch transpose rule.
Now let’s return to our initial Calab-Yau hypersurface $X_{f}.$ We can apply
the Berglund-Hübsch transpose rule to it obtaining $X_{f_{T}}$ where $f_{T}$
is of degree $d^{\prime}$. Using the same process as discussed above, we can
use the $\theta$ map and the Berglund-Hübsch transpose rule $T$ to obtain the
following sequence:

${M_{3}}$${M_{4}}$${X_{f_{T}}}$${Y_{\tilde{g}}}$${Y_{\tilde{g_{T}}}}$$\scriptstyle{\theta}$$\scriptstyle{T}$

where we view $M_{3}$ and $M_{4}$ as the total spaces arising from Theorem 1.
The polynomial $\tilde{g}=z^{d^{\prime}}+f_{T}$ is the orbifold Fano
hypersurface of degree $d^{\prime}$ in some weighted projective space We can
put the two diagrams together getting a summary of the entire process:

${M_{1}}$${{M_{2}}}$${X_{f}}$${Y_{\tilde{f}}}$${Y_{\tilde{f}_{T}}}$${X_{f_{T}}}$${Y_{\tilde{g}}}$${Y_{\tilde{g}_{T}}}$${M_{3}}$${M_{4}}$$\scriptstyle{\theta}$$\scriptstyle{T}$$\scriptstyle{T}$$\scriptstyle{\theta}$$\scriptstyle{T}$

To conclude the proof that $M_{i},i=1,2,3,4$ are Sasakian manifolds admitting
metrics of positive Ricci curvature such that $H_{n}(M_{i},\mathbb{Z})$ can be
explicitly determined we now apply Theorem 1 and Theorem 3. The fact that
$M_{i}$ are $n-1$-connected follows from the Milnor fibration theorem. This
completes the proof. ∎

######  Remark 1.

In the hypothesis of Theorem 1 we exclude the case in which the initial
Calabi-Yau orbifold defined by the invertible polynomial $f$ is Fermat since
in this case $A$ is diagonal and so $A^{T}=A$ and so one gets the same Calabi-
Yau orbifold upon application of the Berglund-Hübsch rule.

We have shown that if one starts with an input Calabi-Yau orbifold $X_{f}$
defined by an invertible polynomial, then it is possible to construct four
Sasakian manifolds $M_{i}$ of positive Ricci curvature from it.

##  5\. Application

In this section, we apply the main theorem to the case in which the initial
data is a K3 orbifold hypersurface coming from Reid’s list [22] (see also
[16]). By Theorem 4 this will yield four topologically distinct Sasakian
7-manifolds of positive Ricci curvature. However, we can use results of I.
Cheltsov [8] to conclude that at least two of the four Sasaki 7-manifolds are
actually Sasaki-Einstein. More precisely, in [8] it was shown that from Reid’s
famous list of orbifold K3 surfaces in weighted projective space
$\mathbb{P}(w_{0},w_{1},w_{2},w_{3})$, the Fano hypersurfaces in
$\mathbb{P}(1,w_{0},w_{1},w_{2},w_{3})$ are also Kähler-Einstein (with the
exception of four cases.) These hypersurface Fano 3-folds are shown to be
Kähler-Einstein by analyzing the so-called log canonical threshold.

Example 1.  Consider the $K3$ hypersurface
$X_{f}=\\{f=z_{1}^{2}z_{4}+z_{2}^{4}+z_{1}z_{3}^{3}+z_{4}^{8}=0\\}$ of degree
$d=16$ in weighted projective space $\mathbb{P}(7,4,3,2).$ Using the map
$\theta$, we can introduce a Fermat term $z^{16}_{0}$ of weight one to obtain
the invertible polynomial $\tilde{f}=z_{0}^{16}+f$ and so this gives a Fano
orbifold hypersurface $Y_{\tilde{f}}$ in $\mathbb{P}(1,7,4,3,2).$ By Corollary
1.4 in [8], $Y_{\tilde{f}}$ is a Fano Kähler-Einstein orbifold and thus by
Theorem 1 is a Sasaki-Einstein $7$-manifold $M_{1}$. From the invertible
polynomial $\tilde{f}$ we can associate its exponent matrix $A$ given by

$A=\begin{bmatrix}16&0&0&0&0\\\ 0&2&0&0&1\\\ 0&0&4&0&0\\\ 0&1&0&3&0\\\
0&0&0&0&8\\\ \end{bmatrix}.$

We apply the Berglund-Hübsch transpose rule to obtain $A^{T}$ and thus
determine the Fano orbifold $Y_{\tilde{f}_{T}}$ defined by
$\tilde{f}_{T}=z_{0}^{16}+z_{1}^{2}z_{3}+z_{2}^{4}+z_{3}^{3}+z_{1}z_{4}^{8}.$
A straightforward calculation determines $\mathbf{w}=(3,16,12,16,4)$ and the
degree $d_{\tilde{f}_{T}}=48.$ By Theorem 1, $M_{2}$ is indeed a Sasakian
manifold of positive Ricci curvature. It is unknown if $M_{2}$ is actually
Sasaki-Einstein. The link $M_{2}$ does not satisfy any of the known
obstructions [13] to admit a Sasaki-Einstein metric so it is still possible
the link $M_{2}$ is Sasaki-Einstein. We have so far been unable to prove this.
Note that $Y_{\tilde{f}_{T}}$ is not well-formed, i.e. has a branch divisor so
that complicates the matter.
Now going back to our input K3 orbifold hypersurface $X_{f}$ we can compute
the exponential matrix $B$ given by

$B=\begin{bmatrix}2&0&0&1\\\ 0&4&0&0\\\ 1&0&3&0\\\ 0&0&0&8\\\ \end{bmatrix}.$

This yields the $K3$ surface $X_{f_{T}}$ defined by
$f_{T}=z_{1}^{2}z_{3}+z_{2}^{4}+z_{3}^{3}+z_{1}z_{4}^{8}$ of degree
$d^{\prime}=12$ with weight vector $\mathbf{w}=(4,3,4,1).$ As before we apply
the $\theta$ to introduce the Fermat term $z_{0}^{12}$ of weight one to obtain
$\tilde{g}=z_{0}^{12}+f_{T}$ and this gives a Fano orbifold hypersurface
$Y_{\tilde{g}}$ in $\mathbb{P}(1,4,3,4,1)$. Again, by Corollary 1.4 in [8],
this hypersurface is Fano Kähler-Einstein and therefore there exists a Sasaki-
Einstein $7$ manifold $M_{3}$ which fibers over the base $Y_{\tilde{g}}$ by
Theorem 1.
Furthermore, we can now apply the Berglund-Hübsch transpose rule on
$\tilde{g}$ to obtain
$\tilde{g}_{T}=\\{z_{0}^{12}+z_{1}^{2}z_{4}+z_{2}^{4}+z_{1}z_{3}^{3}+z_{4}^{8}=0\\}$
of degree $48$ with weight vector $\mathbf{w}=(4,21,12,9,6).$ Thus we obtain
the Fano orbifold $Y_{\tilde{g}_{T}}$ and so by Theorem 1, there exists a
Sasakian 7-manifold $M_{4}$ with positive Ricci curvature. It is not known
whether this manifold is actually Sasaki-Einstein.
Furthermore, we can explicitly compute $H_{3}(M_{i},\mathbb{Z})$ for
$i=1,2,3,4$ using the methods described in Section 2.3. In fact, a computer
program based on Orlik’s conjecture was written by Evan Thomas which allows us
to quickly compute the homology groups of the links. Therefore, we see that to
the initial $K3$ orbifold hypersurface $X_{f}$, we can associate four
manifolds of dimension $7$ which are $2$-connected and of positive Ricci
curvature. At least two of these manifolds, are actually Sasaki-Einstein. We
summarize the result in Table 1.

$M_{i}$ |  polynomial |  $H_{3}(M_{i},\mathbb{Z})$
---|---|---
$M_{1}$ |  $z^{16}_{0}+z^{2}_{1}z_{4}+z_{2}^{4}+z_{1}z_{3}^{3}+z_{4}^{8}$ |  $\mathbb{Z}^{108}\oplus\mathbb{Z}_{8}\oplus(\mathbb{Z}_{2})^{2}$
$M_{2}$ |  $z^{16}_{0}+z^{2}_{1}z_{3}+z_{2}^{4}+z_{3}^{3}+z_{1}z_{4}^{8}$ |  $\mathbb{Z}^{30}\oplus(\mathbb{Z}_{48})^{2}\oplus\mathbb{Z}_{12}\oplus(\mathbb{Z}_{4})^{9}$
$M_{3}$ |  $z_{0}^{12}+z^{2}_{1}z_{3}+z_{2}^{4}+z_{3}^{3}+z_{1}z_{4}^{8}$ |  $\mathbb{Z}^{120}\oplus(\mathbb{Z}_{4})^{2}$
$M_{4}$ |  $z_{0}^{12}+z_{1}^{2}z_{4}+z_{2}^{4}+z_{1}z_{3}^{3}+z_{4}^{8}$ |  $\mathbb{Z}^{24}\oplus\mathbb{Z}_{24}\oplus(\mathbb{Z}_{6})^{2}\oplus(\mathbb{Z}_{3})^{6}$
Table 1. Sasaki manifolds of positive Ricci curvature from input CY

The above example also illustrates an interesting feature of the construction.
The initial K3 orbifold $X_{f}$ is mirror dual to $X_{f_{T}}$ in the sense of
Dolgachev and Voisin by [4] Table 10. Now because of $M_{1}$ and $M_{3}$ are
Sasaki-Einstein $7$-manifolds, we know that the AdS/CFT correspondence should
give rise to a three dimensional superconformal theory [12]. Therefore, our
construction illustrates a way to move from a mirror symmetry framework to an
AdS/CFT framework.

## References

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# Firearms Law and Fatal Police Shootings:
A Panel Data Analysis

Marco Rogna Correspondence: Marco Rogna, Department of Economics, Bochum
University of Applied Sciences, Am Hochschulcampus 1, 44801 Bochum, Germany.
Email<EMAIL_ADDRESS>Department of Economics, Bochum University of
Applied Sciences Bich Diep Nguyen  Faculty of Political Theory - Civic
Education, Hanoi National University of Education

###### Abstract

Among industrialized countries, U.S. holds two somehow inglorious records: the
highest rate of fatal police shootings and the highest rate of deaths related
to firearms. The latter has been associated with strong diffusion of firearms
ownership largely due to loose legislation in several member states. The
present paper investigates the relation between firearms legislation\diffusion
and the number of fatal police shooting episodes using a seven-year panel
dataset. While our results confirm the negative impact of stricter firearms
regulations found in previous cross-sectional studies, we find that the
diffusion of guns ownership has no statistically significant effect.
Furthermore, regulations pertaining to the sphere of gun owner accountability
seem to be the most effective in reducing fatal police shootings.

Keywords: Police shootings; Firearms law; Guns diffusion; Panel data.
J.E.L. I12; I18.

## Introduction

The Unites States (U.S.) is one of the countries with the highest
firearms–related death rate, exceptionally high compared to other
industrialized countries (Pallin et al., , 2019). Furthermore, it is the
country with the highest firearms possession rate by citizens, thanks to loose
legislation in several of its member states (Hemenway, , 2017). The relation
between firearms–related death rate and firearms possession rate has been
largely investigated and a causal relation going from the latter to the former
seems very plausible, e.g. Krug et al.,  (1998); Bangalore and Messerli,
(2013) and Siegel et al.,  (2013). If this fact has prompted part of the
public opinion to ask for stricter guns laws, the constitutional right to
posses and bear firearms has been strenuously defended by the opposite
faction, rendering this one of the most popular and controversial topic in the
country.

Almost every aspect of the relation between firearms legislation and gun
violence has been extensively researched. Besides the mentioned studies that
have tried to establish a causal link between firearms–related death rate and
firearms possession rate (Krug et al., , 1998; Bangalore and Messerli, , 2013;
Siegel et al., , 2013), others have investigated the effect of firearms
legislation\possession rate on suicide rates (Kellermann et al., , 1992;
Anestis and Houtsma, , 2018), on pediatric firearms–related mortality Goyal et
al.,  (2019), on homicides rate (Duggan, , 2001; Kovandzic et al., , 2013;
Siegel et al., , 2013, 2014), and on the death rate of police officers on duty
(Lester, , 1987; Mustard, , 2001; Swedler et al., , 2015).

With approximately 1000 deaths per year, United States (U.S.) holds another
inglorious record among industrialized countries: the highest rate of
homicides committed by police forces (Hemenway et al., , 2019). This raises
the question of whether the high rate of police fatal shootings results from
relaxed firearms legislation\high possession rate. From a speculative point of
view, one can argue that the diffusion of firearms increases the probability
of police officers to face armed people while on duty, thus increasing the
probability of being involved in potentially dangerous situations that require
the use of guns from their part. Furthermore, the increased probability to
face dangerous situations is a factor of stress that may lead law enforcement
officers to overreact or, more generally, to commit mistakes. A positive
relation between firearms diffusion and deadly assaults to police officers
(Swedler et al., , 2015) corroborates the theory of an increase danger to
officers for higher levels of firearms ownership.

Thanks to the availability of independent datasets that remedy the
underreporting of police fatal use of force in official statistics (Conner et
al., , 2019), this topic has recently been investigated. Hemenway et al.,
(2019) find a positive association between firearms prevalence and fatal
police shooting rates. Kivisto et al.,  (2017) report that U.S. states with
stricter firearms legislation have lower incidence rates of fatal police
shootings. Both studies are, however, cross-sectional and thus, despite the
use of several controls common in the dedicated literature, may suffer from
the omitted variable problem. Kivisto et al.,  (2017), for example, mention,
among the limitations of their paper, exactly this problem, stating that “it
is possible that states with stricter gun legislation also have better
training for police officers and more stringent hiring practices, or that
states that are already safe are more likely to implement stricter gun laws”.

The novelty of the present study is to use a panel dataset to investigate the
relation between firearms legislation\possession rate and fatal police
shootings. This allows to control for unobserved fixed characteristics at
state level that may have biased previous analysis, providing, therefore, more
robust results.

##  1 Literature Review

As discussed before, firearms legislation and ownership is a strongly
investigated topic, particularly in the United States. A number of studies
document negative impacts of firearms legislation and prevalence, such as
increased suicide and homicide rate, although evidence is not unambiguous.
From an extensive literature review, Kleck,  (2015) concludes that guns
diffusion is a positive determinant of crime rate, but this relation looses
statistical significance in the most methodologically rigorous papers. Branas
et al.,  (2009) find that possessing a gun increases the probability of being
shot during an assault, thus dismantling the opinion of weapons having a
protective role. On the other side, Kleck and Patterson,  (1993), by comparing
170 U.S. cities, find scarce evidence that guns restrictions have some
positive role in reducing the rate of violence. Similarly, Altheimer,  (2008)
evidences that gun availability has no effect in determining the number of
total individual assaults and robbery, but only increases the number of the
ones committed with a fire–weapon.

Regarding one of the most serious violent crimes, homicide, the results are
mixed. Duggan,  (2001), Siegel et al.,  (2013) and Siegel et al.,  (2014) find
a significant and positive relation between guns diffusion and the number of
homicides. However, this view is strongly opposed by Kovandzic et al.,
(2013), which, on the contrary, document a negative relation. They cite a
potential deterrent effect of guns as an explanation for their results. A
similar debate has surrounded the permission of carrying concealed weapons,
with Lott and Mustard,  (1997) showing a positive role of such law in reducing
violent crimes whereas Dezhbakhsh and Rubin,  (1998) rejecting this finding
and claiming the opposite. Findings regarding firearms diffusion and
legislation are therefore very discordant even when considering aspects such
as crime and homicide rate that are among the most studied.

Shifting the attention on police forces, there is a paucity of research
regarding the association between firearms diffusion and legislation and the
number of killings of, and by, police officers. When considering the former,
the killing of law enforcement officers, we have again contrasting findings.
Lester,  (1987) and Swedler et al.,  (2015) find that increasing levels of
households gun ownership are a clear factor of risk for police officers,
whereas Mustard,  (2001), limiting the attention to the possibility of
carrying concealed weapons, puts in evidence a potential protective role of
this law.

Kivisto et al.,  (2017) and Hemenway et al.,  (2019) investigate the relation
between firearms and killings committed by police officers. The two studies
both rely on independent and open source databases to retrieve the number of
fatal police shootings: Kivisto et al.,  (2017) on The Counted, maintained by
The Guardian, and Hemenway et al.,  (2019) on Fatal Force, created by The
Washington Post. Furthermore, they both rely on a cross sectional analysis
using the fraction of suicides with a fire–weapon on the total number of
suicides as a proxy for guns diffusion, as previously done in other papers,
e.g. Kleck,  (2004) and Azrael et al.,  (2004). Compared to Hemenway et al.,
(2019), whose focus is exclusively on guns diffusion as a cause of fatal
police shootings, Kivisto et al.,  (2017) further consider firearms
legislation, using the Brady Campaign scorecards as an indicator of law
strength. In both papers, firearms ownership is found to positively affect the
number of fatal police shootings. Even after controlling for firearms
prevalence, firearms regulations on gun trafficking and on child and consumer
safety significantly reduces fatal police shootings (Kivisto et al., , 2017).

Given the contrasting evidence emerged in other topics related to firearms
diffusion and legislation and since both the last mentioned papers rely on a
cross sectional analysis that may be plagued by the omitted variable bias, the
extension to a panel data setting seems a necessary further step. This may
help to strengthen the findings reached so far or to contest their validity as
the result of a biased analysis.

##  2 Data and Methods

Following Kivisto et al.,  (2017) and Hemenway et al.,  (2019), our units of
observation are the 50 U.S. states, with District of Columbia having being
excluded for lack of data in several covariates. The covered time period is
from January 1, 2012, to December 31, 2018, and all variables are expressed as
yearly values, forming a dataset with seven time periods. Different databases
have been consulted and merged in order to have all the variables of interest
and the necessary controls: Fatal Encounters, Giffords scorecards, the U.S.
Census Bureau data portal, the Federal Bureau of Investigation’s (FBI’s) Crime
Data Explorer and the Centers for Disease Control and Prevention’s (CDC’s)
Web-based Injury Statistics Query and Reporting System (WISQARS). Following is
a description of all variables.

###  2.1 Description of Variables

Our dependent variable, the number of deaths caused by police shooting per
million inhabitants (Pol_Shoot), is retrieved from the Fatal Encounters
database. We choose an independent database to alleviate the likely problem of
underreporting of such episodes in the FBI’s official statistics (Williams et
al., , 2019). Compared to other open source repositories, e.g. The Counted,
Fatal Force, Mapping Police Violence and Gun Violence Archive, the Fatal
Encounters database covers the longest time span – from 2000 to present.
Specific cases of police shooting can be retrieved by selecting the category
“Deadly use of force” and the subcategory “Gunshot”.

Our independent variable of interest is the strength of firearms regulations
at state level (Giff_Score). In order to obtain a synthetic measure of the
strength of a state legislation, we rely on the Giffords scorecards, available
for the period 2010-2018, with the exclusion of the year 2011, hence the need
to drop the year 2010. The overall score is an aggregation of seven component
scores, namely: background checks and access to firearms (BCAF), other
regulations of sales and transfers (ORST), classes of weapons and
magazines\ammunitions (CWAM), consumers and child safety (CCS), gun owner
accountability (GOA), firearms in public places (FPP) and a residual class
(OTH). Disaggregation of the overall score allows us to test the role of each
component in explaining fatal police shootings, following Kivisto et al.,
(2017). This is helpful in identifying the areas where intervention should be
prioritized to reduce police shooting episodes. Since the scoring system has
been slightly modified several times during the study period, we have
implemented a harmonization procedure, retaining only the sub-indicators that
remained unaltered over time. Kivisto et al.,  (2017), using data from the
same source, eliminate the weighting system in favor of a “1 law = 1 point”
scale. They argue that a weighting system necessarily entails a degree of
arbitrariness. However, we think that the equal weighting implied by the “1
law = 1 point” scale is analogously arbitrary. We, therefore, prefer to rely
on the weights assigned by professional lawyers, thus leaving the Giffords
scorecard scale unaltered.

Stricter legislation on firearms may reduce the quantity of fire–weapons owned
by citizens, but may also promote safer use, e.g. by denying dangerous
subjects access to guns or by increasing the safety of circulating weapons.
Besides examining if laws to promote safe gun use are effective in reducing
fatal police shootings, we can test if the effect of firearms legislations
operate via the former channel by looking at the relation between fatal police
shootings and gun diffusion. Lacking state–level data on gun ownership for our
study period, we rely on a commonly adopted proxy (Suicide) – the percentage
of suicides committed with a fire–weapon over the total of suicides (Kleck, ,
2004; Azrael et al., , 2004; Kivisto et al., , 2017; Hemenway et al., , 2019).
These data are retrieved from the Web-based Injury Statistics Query and
Reporting System (WISQARS).

Table 1: Descriptive Statistics Variable | N. Obs. | Mean | Std. Dev. | Min. | Max.
---|---|---|---|---|---
Pol_Shoot | 350 | 3.51 | 2.11 | 0 | 10.85
Giff_Score | 350 | 31.98 | 24.42 | 4 | 105.50
Crime | 350 | 3627.32 | 1388.63 | 1026.24 | 8849.56
Suicide | 350 | 51.54 | 12.35 | 13.20 | 74.30
PC_Income | 350 | 29712.58 | 4828.93 | 20119 | 52500.00
Urban | 350 | 73.59 | 14.44 | 38.70 | 95
Poverty | 350 | 9.99 | 2.80 | 4 | 19.20
White | 350 | 76.95 | 12.67 | 24.30 | 95.10
Low_Edu | 350 | 11.19 | 2.97 | 6.10 | 18.60
Unemp. | 350 | 3.96 | 1.18 | 1.80 | 7.90
Young | 350 | 23.15 | 1.31 | 19.80 | 27.50
Giff_Score disentangled
BCAF | 350 | 7.19 | 5.51 | 0 | 22
ORST | 350 | 4.12 | 5.89 | 0 | 24
CWAM | 350 | 1.82 | 3.86 | 0 | 14
CCS | 350 | 2.25 | 2.02 | 0 | 9
GOA | 350 | 2.76 | 5.00 | 0 | 17.50
FPP | 350 | 9.16 | 4.36 | 0 | 19
OTH | 350 | 4.67 | 2.18 | 0 | 10

Regarding control variables, we retrieved data on the number of violent crimes
(per million inhabitants, Crime) from the FBI’s Crime Data Explorer, where a
crime is defined as any of the four offenses – murder and non-negligent
manslaughter, rape, robbery, and aggravated assault. All the other controls
are retrieved from the U.S. Census Bureau. Note that all values are
projections on the 2010 census data. These controls include per–capita income
in 2010 inflation–adjusted dollars (PC_Income) and the percentage of people
living in urban areas (Urban). It must be noted that, for this last variable,
only figures for the year 2010 were available, thus it is treated as a
time–invariant covariate. Other socio–economic characteristics, such as
poverty rate (Poverty), unemployment rate (Unemp.), and the percentage of
adults with an education lower than high school diploma (Low_Edu), are also
included. The percentage of young population, aged 18–34, (Young) and the
percentage of white Caucasians (White) over the whole population are
controlled for in the analysis. The last variable is added since several
studies find a racial bias in police shootings (Ross, , 2015; Nix et al., ,
2017; Mesic et al., , 2018). Table 1 reports some key statistics of all the
mentioned variables.

###  2.2 Statistical Analysis

The statistical analysis is divided into two main parts. In the first part, we
focus on the role of the legislative strength as a whole, thus considering the
overall score provided by Giffords for each U.S. state. In the second part, we
analyze the component Giffords scores separately. This analysis should provide
more specific policy indications with regard to the legislative field where
intervention may be more productive in reducing fatal police shootings.

The effects of any changer in legislation may take time to be observed. In all
our analysis, therefore, both the overall and the component Giffords scores
enter in their first lags. The inclusion of lags more distant in time (two or
three years) is precluded by the limited number of time periods at our
disposal. We have actually run regressions with the contemporaneous level of
the Giffords scores, but none of them has resulted in significant coefficients
(results are available from the authors upon request). The possibility to
include both the lag and the current level is prevented by their high
correlation ($\rho=0.99$) that most likely causes a problem of collinearity.

Figure 1 shows the Spearman’s rank correlation coefficients of all variables,
except the component Giffords score. Note Giff_Score_L denotes the lagged
Giffords score. From Figure 1 it is possible to observe that several
covariates have a relatively low correlation coefficient. The exceptions are
the violent crime rate ($\rho=0.49$) and the per–capita income ($\rho=-0.29$).

Figure 1: Correlation Matrix

Despite the low correlation of these covariates with the dependent variable,
our full specifications (specifications (1) and (2) in Table 2) include all of
them. Their coefficients are not significant in these models. Failing to
reject the null hypothesis that the joint significance of the socio–economic
covariates (Poverty, White, Low_Edu, Unemp. and Young) is equal to zero
(p–value = 0.6098 for model (1) and p–value = 0.8595 for model (2)), we omit
these variables in any subsequent analysis. Despite its lack of significance,
Urban is kept in all models because of its high correlation with the lagged
Giffords score ($\rho=0.62$) and because it is a significant control in
previous studies (Hemenway et al., , 2019). Models (3) and (4) in Table 2
report the results with the parsimonious set of explanatory variables. From
Table A1 in the Appendix, it is possible to observe non significant p–values
for the RESET and for the Mundlak test. Compared to models (1) and (2), the
coefficients of models (3) and (4) are negligibly different.

A conditional fixed effect (FE) and a random effect (RE) Poisson regressions
(models (1) and (2) in Table 2) are our main models of interest. The
functional form specification is checked through a RESET test, by adding the
squared residuals of the Poisson regressions (FE and RE) and checking their
significance (Ramsey, , 1974). Results are reported in Table A1 in the
Appendix together with the coefficients of the year dummies (year 2013 as
base) included in all models. The p–value of the squared residuals is far
above the 10% significance level, thus dismissing possible concerns about
misspecification. The choice to report both the FE and RE estimates is due to
the fact that the Mundlak test – also reported in Table A1 – has a
significance level very close to the 5% level (Mundlak, , 1978). Although this
test, chosen compared to the more common Hausman test given the presence of
year dummies and a time invariant covariate (Urban), suggests the use of the
random effect model, the closeness of the p–value to the threshold level and
the concerns about the distributional assumptions of the Poisson RE model lead
us to report the fixed effect estimates as well. However, it can be observed
that the estimated coefficients are very similar in both specifications.

Table 2: Regressions Results: Total Giffords Score (Lagged) |  (1) Poisson FE |  (2) Poisson RE |  (3) Poisson FE |  (4) Poisson RE
---|---|---|---|---
Pol_Shoot | Pol_Shoot | Pol_Shoot | Pol_Shoot
Giff_Score_L | 0.990* | 0.992+ | 0.990* | 0.993+
(-2.05) | (-1.65) | (-2.10) | (-1.82)
Crime | 1.000 | 1.000*** | 1.000 | 1.000***
(1.61) | (4.04) | (1.46) | (4.69)
Suicide | 0.993 | 1.007 | 0.993 | 1.008
(-0.77) | (1.03) | (-0.76) | (1.21)
PC_Income | 1.000* | 1.000* | 1.000 | 1.000+
(-2.15) | (-2.05) | (-1.49) | (-1.94)
Urban |  | 1.004 |  | 1.005
| (0.90) |  | (1.32)
Poverty | 0.958 | 0.965 |  |
(-1.51) | (-1.04) |  |
White | 0.958 | 0.996 |  |
(-1.21) | (-0.66) |  |
Low_Edu | 1.025 | 1.012 |  |
(0.90) | (0.38) |  |
Unemp. | 1.064 | 1.032 |  |
(1.20) | (0.80) |  |
Young | 0.946 | 1.008 |  |
(-1.01) | (0.23) |  |
t-statistics in parentheses. P-value: + p <0.1, * p <0.05, ** p <0.01, *** p
<0.001.

As a robustness check, we also report the results of linear models, both FE
and RE, after that the dependent variable has been transformed with a
Yeo–Johnson power transform in order to render its distribution more
normal–like (Yeo and Johnson, , 2000). The results of the linear models are
shown in Table 3. Although the transformation of the dependent variable
prevents the computation of meaningful marginal effects, the sign and
significance of the coefficients serve to confirm the results of, or to signal
a possible problem in, the Poisson regressions. The distribution of the
dependent variable before and after the Yeo–Johnson transformation is shown in
Figure A1 in the Appendix.

For our second purpose, we regress the rate of fatal police shootings on the
component Giffords scores rather than the overall score. Here we present only
the parsimonious models. Four models have been run, two Poisson – FE and RE –
and two analogous linear regressions. Results are reported in Table 4, models
(5), (6), (7) and (8), and auxiliary information can be found in Table A1 in
the Appendix. From this last table, it is possible to see that the p–values of
the RESET and of the Mundlak test are all above conventional significance
levels. As for the previous models, the difference in the significance of the
coefficients of the Poisson and of the linear regressions are very modest, so
as the difference in the coefficients between the FE and RE models.

Table 3: Regressions Results: Linear Models

|  | (1b)
---
Linear FE
| (2b)
---
Linear RE
| (3b)
---
Linear FE
| (4b)
---
Linear RE
Pol_Shoot | Pol_Shoot | Pol_Shoot | Pol_Shoot
Giff_Score_L | -0.00879* | -0.00924** | -0.00925* | -0.00803*
(-2.32) | (-2.86) | (-2.34) | (-2.52)
Crime | 1.47-E004+ | 1.36-E004*** | 1.60-E004+ | 149-E004***
(1.76) | (3.75) | (1.95) | (4.43)
Suicide | -0.00515 | 0.00193 | -0.00562 | 0.00237
(-0.55) | (0.36) | (-0.59) | (0.45)
PC_Income | -1.83-E005* | -1.95-E005* | -1.44-E005+ | -155-E005*
(-2.07) | (-2.22) | (-1.75) | (-2.12)
Urban |  | 0.00255 |  | 0.00357
| (0.66) |  | (0.92)
Poverty | -0.0248 | -0.0259 |  |
(-0.80) | (-0.87) |  |
White | -0.0159 | -0.00411 |  |
(-0.41) | (-1.03) |  |
Low_Edu | 0.0142 | 0.0124 |  |
(0.51) | (0.52) |  |
Unemp. | 0.0393 | 0.0327 |  |
(0.64) | (0.69) |  |
Young | -0.0476 | -0.0138 |  |
(-0.92) | (-0.47) |  |
Const. | 4.304 | 2.195* | 1.938** | 1.291*
(1.34) | (2.06) | (2.83) | (2.42)
2014 | 0.0538 | 0.0639 | 0.0313 | 0.0418
(0.76) | (0.92) | (0.46) | (0.60)
2015 | 0.0661 | 0.0829 | 0.0359 | 0.0522
(0.84) | (1.15) | (0.52) | (0.76)
2016 | 0.137 | 0.151+ | 0.113* | 0.120*
(1.55) | (1.95) | (2.24) | (2.48)
2017 | 0.163 | 0.189+ | 0.129+ | 0.146*
(1.42) | (1.75) | (1.99) | (2.31)
2018 | 0.255* | 0.282* | 0.220** | 0.236***
(2.11) | (2.40) | (3.10) | (3.36)
t-statistics in parentheses. P-value: + p <0.1, * p <0.05, ** p <0.01, *** p
<0.001.

Table 4: Regressions Results: Giffords Score Disentangled (Lagged) |  | (5)
---
Poisson FE
| (6)
---
Poisson RE
| (7)
---
Linear FE
| (8)
---
Linear RE
| Pol_Shoot | Pol_Shoot | Pol_Shoot | Pol_Shoot
BCAF_L | 1.001 | 1.000 | 0.00131 | -6.63-E004
(0.15) | (0.03) | (0.22) | (-0.12)
ORST_L | 1.032 | 1.008 | 0.0231 | 0.0110
(1.44) | (0.47) | (1.17) | (0.93)
CWAM_L | 0.981 | 1.014 | -0.0168 | 0.00372
(-0.93) | (0.44) | (-1.01) | (0.22)
CCS_L | 0.926 | 0.958 | -0.0732 | -0.0409
(-1.09) | (-1.24) | (-1.19) | (-1.47)
GOA_L | 0.963* | 0.962* | -0.0351* | -0.0352**
(-2.57) | (-2.32) | (-2.58) | (-2.93)
FPP_L | 0.984 | 0.983 | -0.0191 | -0.0193
(-1.25) | (-1.38) | (-1.54) | (-1.63)
Crime | 1.000 | 1.000*** | 1.58-E004+ | 1.57-E004***
(1.45) | (4.96) | (1.90) | (4.64)
Suicide | 0.992 | 1.002 | -0.00597 | -0.00118
(-0.83) | (0.37) | (-0.63) | (-0.21)
PC_Income | 1.000 | 1.000* | -1.39-E005+ | -1.62E-005*
(-1.48) | (-2.18) | (-1.70) | (-2.14)
Urban |  | 1.004 |  | 0.00351
| (1.19) |  | (0.94)
Const. |  |  | 1.955** | 1.503**
|  | (2.72) | (2.66)
t-statistics in parentheses. P-value: + p <0.1, * p <0.05, ** p <0.01, *** p
<0.001.

##  3 Results and Discussion

In the presentation of results, we will focus solely on the Poisson
regressions, given the difficult interpretation of the coefficients of the
linear models discussed earlier. Note that all the reported coefficients
relative to the Poisson models are incidence rate ratios (IRR). From Table 2,
it is possible to observe that the Giffords score (lagged) is always
statistically significant, although only at 10% level in the RE models, while
at 5% in the FE models. An increase of one point in the overall Giffords score
causes an approximately 1% reduction in the number of fatal police shootings
per million of inhabitants if considering the FE model. The percentage
reduction is slightly lower for the RE model: 0.8% when including all controls
– model (3) – and 0.7% when excluding the subset of jointly non-significant
covariates – model (4).

An important point to notice is the lack of significance, in all models of
Table 2, of the proxy for firearms diffusion: Suicide. The strong correlation
with the lag of the Giffords score ($\rho=-0.83$) may suggest that the effect
of this last masks the one of the former. However, if running the same
regressions without the inclusion of the Giffords score, the p–value of
Suicide remains above 0.1 in all models: 0.152, 0.106, 0.140 and 0.118 for,
respectively, models (1), (2), (3) and (4) without Giff_Score_L (full results
available from the authors upon request).

When considering Table 4 and the disentangled categories composing the
Giffords score, only the coefficient related to the lag of one category, gun
owner accountability, is significant (at 5% level). This happens to be true
both in the FE and in the RE Poisson regressions, so as in the linear models.
In particular, an increase of one point in the strength of the gun owner
accountability category is associated with a decrease in per–million
inhabitants fatal police shootings of 3.7% (FE model) or of 3.8% (RE model).

We can further notice that in all models, both in Table 2 and 4, the sign of
the coefficients of the main control variables is as expected. In particular,
the number of violent crimes positively impacts the number of fatal police
shooting episodes whereas per–capita income has the opposite effect. A last
word is dedicated to the significance of the violent crime rate that is very
high (0.1%) in all RE Poisson models but absent in the FE models. This is
possibly due to the persistent nature in time of this phenomenon that, in the
FE model, gets captured by the fixed effect component.

###  3.1 Discussion

The present study shows that increasing levels of firearms regulation are
significantly associated with a lower number of fatal police shooting cases.
In particular, a point increase in the overall Giffords score leads to a
decrease of in fatal police shootings of 0.7%–1%, depending from the model.
When considering separately the various categories composing the Giffords
score, one point increase in the strength of gun owner accountability leads to
a decrease of approximately 3.7%–3.8% in the number of people killed by police
officers. The diffusion of fire–weapons, instead, has no statistically
significant role in determining the considered outcome. This finding has been
achieved through the use of a panel dataset, thus controlling for unobserved
heterogeneity through the use of FE Poisson models.

It is interesting to compare our results with previous findings. Regarding the
effect of firearms diffusion, our results clearly contradicts the previous
findings of Kivisto et al.,  (2017) and of Hemenway et al.,  (2019). In fact,
we do not find a statistically significant effect of guns diffusion in
determining the number of fatal police shootings.

Considering the strength of firearms regulations, our analysis basically
confirms the findings of Kivisto et al.,  (2017). However, this is true only
for the overall score. When evaluating each category separately, significant
differences emerge. First of all, it must be noted that the results are not
easily comparable, given the different scoring system used in Kivisto et al.,
(2017), namely the Brady scorecards, and in the present analysis. However, a
comparison is not impossible. In Kivisto et al.,  (2017), two categories
remained significant after all controls were added, namely promoting safe
storage via child and consumer safety laws and curbing gun trafficking. The
former corresponds to the category consumers and child safety (CSS) in the
Giffords scorecards and the latter is included in other regulation of sales
and transfers (ORST), both not significant in our models. The gun owner
accountability, the category found significant in the present analysis, is
instead composed by three elements: licensing of gun owners and purchasers,
having the highest weights, followed by registration of firearms and reporting
lost or stolen firearms. This is an important difference, with potentially
relevant implications for policy–makers. Furthermore, it is reasonable to
think that the first sub-category, namely the need of gun owners to have a
license, has a great discriminant power in determining the final identity of
gun owners. This further suggests that the qualitative side of gun diffusion
(who owns a gun) is more important in limiting the number of police shootings
than the quantitative side (how many guns are owned).

## Conclusions

The present analysis has shown that police shooting episodes are significantly
reduced by stricter levels of firearms law. While this finding partially
confirms what emerged in previous studies, we also find that the diffusion of
fire–weapons is inconsequential in determining the number of police shootings,
thus contradicting the precedent evidence.

The policy recommendations that can be derived from the present paper are
quite straightforward. Improving the strength of firearms regulations seems an
effective way for reducing the number of people killed by law enforcement
officers. Actually, the policy prescriptions can be even more specific. In
fact, from the analysis it emerges that the most effective intervention for
reducing fatal police shooting episodes would be to strengthen the rules of
gun owner accountability, namely licensing of gun owners and purchasers,
registration of firearms and reporting lost or stolen firearms. These policy
prescriptions are different from the ones provided by previous studies.

Another important lesson, and a departure from previous findings, is the lack
of statistical significance of the diffusion of guns in causing fatal police
shootings. This suggests that the cause may be more qualitative (who owns the
guns) rather than quantitative (how many guns). The fact that the only
significant legislative category emerged from this study is the gun owner
accountability further strengthens this hypothesis.

There are several possible ways in which the analysis could be expanded in
order to have more precise and specific prescriptions. One possibility would
be to further disaggregate each category of the Giffords score into its
subcategories. We have not pursued this road due to the limited number of
observations at our disposal. Another interesting extension would be to
consider the episodes of police shootings directed towards unarmed citizens
(Hemenway et al., , 2019). The lack of this information in the Fatal
Encounters database has prevented us to conduct this analysis.

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## Appendix

Figure A1: Distribution of the Dependent Variable (Pol_shoot) before and after
the Yeo-Johnson Power Transformation ($\lambda=0.1384$)

(a) Before

(b) After

Table A1: Ancillary Information for Table 2 and Table 4

Ancillary Information for Table 2
---
| (1) | (2) | (3) | (4)
2014 | 1.089 | 1.105 | 1.055 | 1.081
2015 | 1.106 | 1.137 | 1.071 | 1.102
2016 | 1.149 | 1.190* | 1.137* | 1.148**
2017 | 1.224 | 1.285* | 1.193* | 1.220**
2018 | 1.363** | 1.439** | 1.308*** | 1.344***
ln($\alpha$) |  | 0.0540 |  | 0.0545
RESET Test1 | 0.316 | 0.864 | 0.859 | 0.941
Mundlak Test2 | 0.0555 | 0.3221
N. Obs. | 300 | 300 | 300 | 300
Ancillary Information for Table 4
| (5) | (6) | (7) | (8)
2014 | 1.164 | 1.137 | 0.129 | 0.0943
2015 | 1.173 | 1.152 | 0.128 | 0.0985
2016 | 1.234** | 1.199*** | 0.188** | 0.159**
2017 | 1.288** | 1.264*** | 0.199* | 0.176**
2018 | 1.411*** | 1.397*** | 0.286** | 0.265***
ln($\alpha$) |  | 0.0463 |  |
RESET Test1 | 0.405 | 0.976 | 0.880 | 0.659
Mundlak Test2 | 0.3885 | 0.2854
N. Obs. | 300 | 300 | 300 | 300

2013 as base year.
1) The RESET Test reports the p–value of the squared residuals.
2) The Mundlak Test reports the p-value of the joint significance test of the
time average of all regressors.