# Vector bundles without intermediate cohomology and the trichotomy result

Giorgio Ottaviani  Department of Mathematics and Computer Science “Ulisse
Dini”, University of Florence, Italy<EMAIL_ADDRESS>
###### Abstract.

Horrocks proved in 1964 [Hor] that vector bundles on ${\mathbb{P}}^{n}$
without intermediate cohomology split as direct sum of line bundles. This
result has been the starting point of a great research activity on other
varieties, showing interesting connections with derived categories and other
areas. We follow some paths into this fascinating story, which has classical
roots. The story has a culmination with the trichotomy result
(finite/tame/wild) for arithmetically Cohen Macaulay (ACM) varieties obtained
by Faenzi and Pons-Llopis in 2021. This is an expanded version of the talk
given at the conference ”Homemade Algebraic Geometry” in July 2023 at Alcalá
de Henares celebrating Enrique Arrondo’s 60th birthday.

###### 2020 Mathematics Subject Classification:

14-02, 14F17, 14F08, 14M15, 14M17

The author is member of Italian GNSAGA-INDAM. Partially supported by the PRIN
project “Multilinear Algebraic Geometry” of MUR.

_Dedicated to Enrique Arrondo on the occasion of his 60-th birthday._

##  0\. Introduction

A vector bundle $E$ on a variety $X$ polarized by ${\mathcal{O}}(1)$ has no
intermediate cohomology if $H^{i}(X,E\otimes{\mathcal{O}}(t))=0$ $\forall
t\in{\mathbb{Z}}$, $0<i<\dim X$. This brief survey about bundles without
intermediate cohomology is divided into three sections.

In the first one we recall cohomological criteria for a vector bundle on
projective space ${\mathbb{P}}^{n}$ and on smooth quadrics $Q_{n}$ to split as
a sum of line bundles. Bundles on ${\mathbb{P}}^{n}$ or $Q_{n}$ without
intermediate cohomology are finitely many, up to twist with line bundles. This
means that ${\mathbb{P}}^{n}$ and $Q_{n}$ are finite.

In the second section we consider the approach to these topics with derived
categories. The analogous facts are the Beilinson description od the derived
category of ${\mathbb{P}}^{n}$ and the Kapranov description of the derived
category of ${\mathbb{P}}^{n}$. The description of the derived category of
Grassmannians by Kapranov is a breakthrough in the story. Bundles on
Grassmannians without intermediate cohomology make families of arbitrarily
large dimension, this means that any Grassmannian different from
${\mathbb{P}}^{n}$ or $Gr({\mathbb{P}}^{1},{\mathbb{P}}^{3})$ is wild.

A variety polarized with a very ample line bundle ${\mathcal{O}}(1)$ is called
arithmetically Cohen-Macaulay (ACM) when ${\mathcal{O}}(1)$ has no
intermediate cohomology In the third section we expose the classification of
bundles on ACM varieties into the three classes finite/tame/wild, the
trichotomy.

The style of the exposition is informal and concerns the development of main
ideas through examples. There are three open problems left respectively as
questions 1.10, 1.17, 2.9. I thank Daniele Faenzi for several useful
discussions.

##  1\. The Horrocks criterion, the spinor bundles, classical roots on ACM
bundles

###  1.1. The Horrocks criterion on ${\mathbb{P}}^{n}$ and its classical roots

The starting point of this story is the remarkable

######  Theorem 1.1.

Horrocks criterion (1964)[Hor] Let $E$ be a vector bundle on
${\mathbb{P}}^{n}({\mathbb{C}})$.

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ for some integers $a_{i}$
$\Longleftrightarrow$ $H^{i}(E(t))=0,\quad 1\leq i\leq n-1,\forall
t\in{\mathbb{Z}}$.

This result characterizes bundles on ${\mathbb{P}}^{n}$ without intermediate
cohomology , called also arithmetically Cohen-Macaulay(ACM) bundles.

Note for $n=1$ the condition in the criterion is empty and we get

######  Corollary 1.2.

(Segre-Grothendieck) Any vector bundles $E$ on ${\mathbb{P}}^{1}$ split as the
sum of line bundles, that is

$E=\oplus_{i=1}^{r}{\mathcal{O}}_{{\mathbb{P}}^{1}}(a_{i})$

for some integers $a_{i}$.

Corollary 1.2 is commonly attributed to Grothendieck although, as often
happens in Mathematics, Grothendieck was not the first to prove it. A short
history is contained in [OSS, §2.4].

We list of some proofs in reverse chronological order, revising slightly the
list in [OSS, §2.4], where most of the original references are quoted.

  * •

Grauert-Remmert (1975), the most popular proof, available also in [OSS,
Theorem 2.1.1],

  * •

Grothendieck (1956),

  * •

Hilbert (1906), in the language of matrices on ${\mathbb{C}}^{*}$,

  * •

Bellatalla (1901), his diploma thesis with C. Segre supervision [Bel],

  * •

C. Segre, for rank $2$ (1883), a geometric argument [S1],

  * •

Dedekind-Weber (1882), in algebraic setting.

For the convenience of the reader we recall the sketch of the Grauert-Remmert
proof of Segre-Grothendieck Cor. 1.2. Let $E$ be a vector bundle on
${\mathbb{P}}^{1}$.

$\bullet$ Step 1, up to twist by a line bundle, we may assume $h^{0}(E)\neq
0$, $h^{0}(E(-1))=0$. Let
${\mathcal{O}}\smash{\mathop{\longrightarrow}\limits^{s}}E$ be a section, it
has no zeroes, otherwise if $s(x)=0$ then $s$ is a section of $E\otimes
I_{x}=E(-1)$.

$\bullet$ Step 2, we get the exact sequence

(1.1)
$0\smash{\mathop{\longrightarrow}\limits}{\mathcal{O}}\smash{\mathop{\longrightarrow}\limits^{s}}E\smash{\mathop{\longrightarrow}\limits}F\smash{\mathop{\longrightarrow}\limits}0$

where $F=\oplus_{i=1}^{r-1}{\mathcal{O}}(b_{i})$ splits by induction on the
rank.

$\bullet$ Step 3, tensoring (1.1) by ${\mathcal{O}}(-1)$ we get

$\underset{\displaystyle\overset{\displaystyle\shortparallel}{0}}{H^{0}(E(-1))}\smash{\mathop{\longrightarrow}\limits}H^{0}(\oplus_{i=1}^{r-1}{\mathcal{O}}(b_{i}-1))\smash{\mathop{\longrightarrow}\limits}\underset{\displaystyle\overset{\displaystyle\shortparallel}{0}}{H^{1}({\mathcal{O}}(-1))}$

hence $b_{i}\leq 0$.

$\bullet$ Step 4, the sequence (1.1) splits since
$Ext^{1}(F,{\mathcal{O}})=H^{1}(F^{\vee})=0$.

For a modern exposition of Segre paper[S1] we refer to [GO, §2, 3, 4]. Segre
worked in the setting of ruled varieties. In modern notations, for a
$2$-bundle $E$ on ${\mathbb{P}}^{1}$ he considered the projection
${\mathbb{P}}(E)\smash{\mathop{\longrightarrow}\limits^{\pi}}{\mathbb{P}}^{1}$
where ${\mathbb{P}}(E)$ is a ruled surface. The goal was to construct a
section of $\pi$ such that its image make a subbundle of $E$. Segre’s proof
resembles steps 1, 2 and 3 of the Grauert-Remmert proof and anticipates them.
Step 4 needed a different approach. Barth and Hulek had the smart idea in [BH]
to prove Theorem 1.1 by induction on $n$ by taking Corollary 1.2 as starting
point of the induction. The bundle $E$ restricted to a line splits as
$\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$, and one lifts the isomorphism
$E\to\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ from the line to the whole
${\mathbb{P}}^{n}$, by the cohomological vanishing that we have in the
assumptions. This approach has been nicely exposed in [OSS, Theorem 2.3.1].

Giuseppe Gherardelli wrote in 1943 the following remarkable result, under
Severi advice. It can be considered the ancestor of Horrocks criterion.

######  Theorem 1.3 (G. Gherardelli).

[Gh] Let $C\subset{\mathbb{P}}^{3}$ be smooth. Then $C$ it is a complete
intersection if and only if the following two conditions hold

  * •

(i) $C$ is subcanonical, that is $K_{C}={\mathcal{O}}(e)_{|C}$ for some
$e\in{\mathbb{Z}}$

  * •

(ii) $C$ is projectively normal, that is the restriction map

$H^{0}({\mathbb{P}}^{3},{\mathcal{O}}(k))\to H^{0}(C,{\mathcal{O}}(k))$ is
surjective.

Actually the assumptions in [Gh] on $C$ were weaker, but Gherardelli can
conclude that $C$ is smooth. In the setting of ACM bundles, the assumption
that makes Theorem 1.3 to work is that $C$ is a locally CM curve . After Serre
correspondence ([OSS, chap. 1 §5] and [Hart, Theorem 8.21A (e)] ), any locally
CM subcanonical curve in ${\mathbb{P}}^{3}$ is obtained as the zero locus of a
section of a rank $2$ bundle $E$, which splits as the sum of two line bundles
if and only if $C$ is a complete intersection. In this optic, Gherardelli
Theorem for curves in ${\mathbb{P}}^{3}$ is equivalent to Horrocks criterion
for $2$-bundles on ${\mathbb{P}}^{3}$, indeed the condition (ii) is equivalent
to the vanishing $H^{1}(I_{C,{\mathbb{P}}^{3}}(k))=0$ which from the sequence

$0\smash{\mathop{\longrightarrow}\limits}{\mathcal{O}}_{{\mathbb{P}}^{3}}\smash{\mathop{\longrightarrow}\limits}E\smash{\mathop{\longrightarrow}\limits}I_{C,{\mathbb{P}}^{3}}(c_{1}(E))\smash{\mathop{\longrightarrow}\limits}0$

is equivalent to $H^{1}(E(*))=0$, and the second vanishing $H^{2}(E(*))=0$
follows from Serre duality.

###  1.2. Spinor bundles and the splitting criterion on quadrics

On quadrics there is a notable class of bundles without intermediate
cohomology beyond line bundles, the spinor bundles.

The linear spaces contained in a smooth quadric
$Q_{n}\subset{\mathbb{P}}^{n+1}$ have maximal dimension $\lfloor n/2\rfloor$.
Linear spaces contained in a quadric are called isotropic, since a linear
space ${\mathbb{P}}(L)$ is contained in $Q_{n}$ if and only if $Q(v,w)=0$
$\forall v,w\in L$ where $Q(-,-)$ is the bilinear form associated to the
quadric.

Let for simplicity $n=2k+1$ ($n$ odd). The spinor variety $S_{k+1}$
parametrizes the isotropic linear spaces of maximal dimension

(1.2) $S_{k+1}=\left\\{{\mathbb{P}}^{k}\in
Gr({\mathbb{P}}^{k},{\mathbb{P}}^{2k+2})|{\mathbb{P}}^{k}\subset
Q_{2k+1}\right\\}$

and it has easily seen to have dimension $\frac{(k+1)(k+2)}{2}$. The first
cases are $S_{1}={\mathbb{P}}^{1}$ and $S_{2}={\mathbb{P}}^{3}$. By the
Plücker embedding, $S_{k+1}$ is embedded in the large projective space
${\mathbb{P}}^{N}$ with $N={{2k+3}\choose{k+1}}-1$. The striking fact is that
${\mathrm{P}ic}(S_{k+1})={\mathbb{Z}}$ and

${\mathcal{O}}_{Gr}(1)_{|S_{k+1}}={\mathcal{O}}_{S_{k+1}}(2)$

so that the spinor variety $S_{k+1}$ has another embedding corresponding to
the “half” line bundle ${\mathcal{O}}(1))$ (replacing ${\mathcal{O}}(2)$) in
the projective space of a vector space of smaller dimension $2^{k+1}$, the
famous spin representation $(\mathrm{Spin}{\mathbb{C}}^{2k+1})$, that we
recall briefly in Remark 1.4.

We have the incidence variety

$\mathbb{F}\colon=\left\\{(x,{\mathbb{P}}^{k})\in Q_{2k+1}\times
S_{k+1}|x\in{\mathbb{P}}^{k}\right\\}$

with the two projections $p_{1}$ and $p_{2}$ and
$S^{\vee}=(p_{1})_{*}(p_{2})^{*}{\mathcal{O}}(1)$ is the dual of the spinor
bundle. Since the tangent space $T_{x}Q_{2k+1}$ meets $Q_{2k+1}$ in a cone
with vertex $x$ over a smaller quadric $Q_{2k-1}$, the fibers of $p_{1}$ are
isomorphic to a smaller spinor variety $S_{k}$. This proves that
${\mathrm{r}ank}S=h^{0}(S_{k},{\mathcal{O}}(1))=2^{k}$.

On even dimensional quadrics $Q_{2k}$ there are two families of maximal
isotropic subspaces ${\mathbb{P}}^{k}$, both parametrized by the same spinor
variety $S_{k}$ encountered in the odd case, and accordingly two spinor
bundles $S^{\prime}$, $S^{\prime\prime}$ of rank $2^{k-1}$. In [Ott88] it is
described how the spinor bundles restrict to hyperplane quadrics, this
argument allows to prove directly by induction that ${\mathrm{r}ank}S=2^{k}$.

It is useful to compare the following facts.

  * •

The Grassmannian $Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$ is the unique closed
$SL(n+1)$-orbit in ${\mathbb{P}}(\wedge^{k+1}{\mathbb{C}}^{n+1})$.

  * •

The Veronese variety $v_{d}({\mathbb{P}}^{n})$ is the unique closed
$SL(n+1)$-orbit in ${\mathbb{P}}(\mathrm{Sym}^{d}{\mathbb{C}}^{n+1})$.

  * •

The spinor variety $S_{k}$ is the unique closed $Spin(2k+1)$-orbit in
${\mathbb{P}}(\mathrm{Spin}{\mathbb{C}}^{2k+1})$.

######  Remark 1.4.

We take the opportunity to expose the spin representation in a nutshell, as
can be taught at undergraduate level. In an algebra course, ${\mathbb{C}}$ is
commonly introduced as ${\mathbb{R}}\oplus{\mathbb{R}}i$ where you declare
$i^{2}=-1$.

In a linear algebra course, ${\mathbb{C}}$ may be introduced as the
${\mathbb{R}}$-algebra

${\mathbb{C}}=\left\\{\begin{bmatrix}a&-b\\\
b&a\end{bmatrix}|a,b\in{\mathbb{R}}\right\\}$

, where $i=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ satisfies $i^{2}=-1$. This
has the advantage that the imaginary unit is a concrete real $2\times 2$
matrix.

In a second course, you may introduce the quaternion as the
${\mathbb{C}}$-algebra

${\mathbb{H}}=\left\\{\begin{bmatrix}z&-w\\\
\overline{w}&\overline{z}\end{bmatrix}|z,w\in{\mathbb{C}}\right\\}$

and now you pose

${\mathbf{i}}=\begin{bmatrix}i&0\\\
0&-i\end{bmatrix},{\mathbf{j}}=\begin{bmatrix}0&-1\\\
1&0\end{bmatrix},{\mathbf{k}}=\begin{bmatrix}0&-i\\\ i&0\end{bmatrix}$

they satisfy the usual rules, and they are a realization of the Pauli matrices
(with a different normalization). The quaternion conjugate corresponds to the
operation ${\overline{A}}^{t}$ on the matrix $A$ and this allows to define
quaternion norm as $|q|^{2}=q\overline{q}$ for $q\in{\mathbb{H}}$.

Now ${\mathbb{H}}={\mathbb{R}}\oplus{\mathbb{I}}$ contains
$S^{3}=\\{q||q|=1\\}$ which shows the remarkable fact that $S^{3}$ has a group
structure, indeed it is a real Spin group.

Construct the map

$\begin{array}[]{ccc}S^{3}&\to&SO(3)=SO({\mathbb{I}})\\\ q&\mapsto&(x\mapsto
q^{-1}xq)\end{array}$

which is $2:1$ group homomorphism, since the fibers consist of $\\{-q,q\\}$.

It follows that $S^{3}$ is the universal covering of $SO(3)$, we pose
$S^{3}=Spin(3)$ and the natural inclusion $S^{3}\subset GL(2,{\mathbb{C}})$ as
quaternions of norm $1$ identifies $S^{3}=SU(2)$ and is the $2$-dimensional
real spin representation of $Spin(3)$. On complex numbers we have,
analogously, $Spin(3)=SL(2)$.

The general $Spin$ representation follows this path by replacing the algebra
of quaternions ${\mathbb{H}}$ with the more general Clifford algebra.

The spinor bundle is ACM, so that to get a splitting criterion analog to
Horrocks criterion, it is necessary to add a further condition, as in the
following.

######  Theorem 1.5.

Splitting Criterion on Quadrics, [Ott89] Let $E$ be a vector bundle on
$Q_{n}$, let $S$ be a spinor bundle. Then

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ $\Longleftrightarrow$
$\left\\{\begin{array}[]{lr}H^{i}(E(*))=0,&1\leq i\leq n-1\\\ H^{n-1}(E\otimes
S(*))=0,&\end{array}\right.$

Nevertheless, Horrocks criterion can be generalized in a second way, by
describing the class of ACM bundles on quadrics.

######  Theorem 1.6.

Characterization of ACM bundles on Quadrics, Knörrer, [Kn] Let $E$ be an
indecomposable vector bundle on $Q_{n}$. Then

$E={\mathcal{O}}(a)$ or $E=S(b)$ $\Longleftrightarrow$ $H^{i}(E(*))=0,1\leq
i\leq n-1$

It is easy to check that

$\textrm{Splitting Criterion on\ }Q_{n}\Longleftarrow\textrm{Kn\"{o}rrer
Characterization}$

Enrique patiently explained to me in 1989, at the conference ”Projective
varieties” in Trieste, that also the converse implication holds !

His technique, by successive “killing of $H^{1}$”, will be exposed later in
[AG] that we recall in a while, see Theorem 1.12.

###  1.3. Improvements of Horrocks criterion for small rank

The results in this section improve the Horrocks criterion when the rank of
$E$ is relatively small.

######  Theorem 1.7 (Evans-Griffith criterion (1981)).

[EG] Let $E$ be a vector bundle on ${\mathbb{P}}^{n}({\mathbb{C}})$ of rank
$r\leq n$.

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ $\Longleftrightarrow$
$H^{i}(E(*))=0,\quad 1\leq i\leq r-1$

Note for $2$-bundles only the vanishing of $H^{1}(E(*))$ characterizes the
splitting of $E$. A simple proof of Theorem 1.7 as a consequence of Le Potier
Vanishing Theorem has been given by L. Ein in [E].

######  Theorem 1.8 (Kumar-Peterson-Rao (2003)).

[KPR] Let $\mathrm{rk}(E)\leq n-2$, $n\geq 3$. Then $E$ splits if and only if
$H^{i}(E(*))=0\quad 2\leq i\leq n-2$.

The cohomological bound in Theorem 1.8 has been called inner cohomology so
that this Theorem characterizes bundles without inner cohomology. Note in
particular a $2$-bundle $E$ on ${\mathbb{P}}^{4}$ splits if and only if
$H^{2}(E(*))=0$.

The following result generalizes Theorem 1.8 to quadrics.

######  Theorem 1.9 (Ancona-Peternell-Wisniewski 1994, Malaspina 2009).

[APW, Mal09] Let $E$ be a rank $2$ bundle on $X={\mathbb{P}}^{n}$ or
$X=Q_{n}$, $n\geq 4$. TFAE

  1. (1)

$H^{i}(E(*))=0$ for $2\leq i\leq n-2$ (without inner cohomology)

  2. (2)

${\mathbb{P}}(E)$ is Fano

  3. (3)

$E={\mathcal{O}}(a)\oplus{\mathcal{O}}(b)$, $E$ is a spinor bundle on $Q_{4}$,
and two further examples.

######  Question 1.10.

Are the first two items equivalent on other Fano varieties of dimension $\geq
4$ ? Note that the proof of the equivalence on ${\mathbb{P}}^{n}$ and $Q_{n}$
is indirect, through the third item.

Enrique Arrondo and Laura Costa[AC] classified rank-2 indecomposable ACM
bundles over Fano $3$-folds of index 2 by Hartshorne–Serre correspondence.

######  Theorem 1.11 (Arrondo-Costa).

[AC] There are three families of rank $2$ indecomposable ACM bundles on the
smooth Fano $3$-fold of index $2$ ($V_{3}$, $V_{4}$ and $V_{5}$), having a
section of a suitable twist vanishing respectively on

  1. (1)

a line

  2. (2)

a conic

  3. (3)

a projectively normal elliptic curve (in the threefold).

This result has been generalized to very Fano $3$-fold by Brambilla and Faenzi
in [BF] and to $V_{22}$ by Arrondo and Faenzi in [AF],

###  1.4. Grassmannians and beyond, the appearance of many ACM bundles

It remained an open problem to characterize ACM bundles on Grassmannians.
Since the universal and the quotient bundle are ACM one could at first hope
that they are the only ones or just a few ones. The following result by
Arrondo, Graña in 1999 destroyed this hope.

Arrondo - Grana (1999) prove that ACM bundles on
$Gr({\mathbb{P}}^{1},{\mathbb{P}}^{4})$ make big families, in particular there
exist non homogeneous ACM bundles.

######  Theorem 1.12 (Arrondo, Graña).

[AG] There are families of positive dimension of vector bundles $E$ on ACM
bundles on $Gr({\mathbb{P}}^{1},{\mathbb{P}}^{4})$ . Some of these examples
are non homogeneous under $SL(5)$-action.

This result is a turning point in the story since shows that the structure of
ACM bundles can be in general quite complicated. It is worth that the class of
ACM bundles on $Gr({\mathbb{P}}^{1},{\mathbb{P}}^{4})$ is explicitly
described, up to extensions of known bundles.

What is in embryo in [AG] is the concept that
$Gr({\mathbb{P}}^{1},{\mathbb{P}}^{4})$ is wild, that is, it carries families
of ACM bundles with arbitrarily large dimension. This was understood later and
popularized by Hartshorne, see the section 3. Wilderness of any Grassmannian
different from projective spaces and $Gr({\mathbb{P}}^{1},{\mathbb{P}}^{3})$
was definively settled by Costa and Mirò-Roig in [CMR16].

######  Remark 1.13.

I recall several discussions with Enrique Arrondo at the time of [AG], when we
did not yet image the wilderness of most Grassmannians, and we were already
impressed by the non homogeneous examples.

I have started this subsection with the results of [AG], since I wished to
show immediately the appearance of many ACM bundles. The chronological order
is a bit different, since 10 years before it was proved the following
splitting criterion. Using the language of Schur functors, for a bundle $E$
and a Young diagram $\alpha$ it is defined the bundle $S^{\alpha}E$. When
$\alpha$ is a horizontal Young diagram like $\alpha=\yng(4)$ then
$S^{\alpha}E={\mathrm{S}ym}^{4}E$, while if $\alpha$ is a vertical Young
diagram like $\alpha=\yng(1,1,1)$ then $S^{\alpha}E=\wedge^{3}E$. General
Young diagrams act by “mixing” the symmetric and skew-symmetric actions.

It turns out, by using Bott Theorem, that the universal bundle $U$ is ACM and
$S^{\alpha}U$ is ACM if and only if $\alpha$ has at most $n-k-1$ columns.

######  Theorem 1.14 (O, 1989).

Let $E$ be a vector bundle on $G=Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$.

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ $\Longleftrightarrow$
$H^{i}(E(*)\otimes S^{\alpha}U)=0,\quad 1\leq i\leq\dim G-1$

where $\alpha$ is a Young diagram with at most $n-k-1$ columns, ${U}$ is the
universal bundle of rank $k+1$.

The proof is by induction on $k$, in Grauert-Remmert style.

Arrondo-Malaspina [AM] and Arrondo-Tocino [AT] find a characterization of
symmetric powers of quotient bundle on
$Gr({\mathbb{P}}^{1},{\mathbb{P}}^{n})$. We state explicitly a result from the
second paper.

######  Theorem 1.15 (Arrondo-Tocino).

[AT] Let $E$ be a bundle on $G=Gr({\mathbb{P}}^{1},{\mathbb{P}}^{n})$.

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ $\Longleftrightarrow$
$\left\\{\begin{array}[]{cc}H^{i}(E\otimes\mathrm{Sym}^{i}Q(*))=0&1\leq i\leq
n-2\\\ H^{i}(E\otimes\mathrm{Sym}^{2n-3-i}Q(*))=0&n-1\leq i\leq
2n-3\end{array}\right.$

The main result of [AT] is the determination of a finite subset $S_{k}$ of
pairs such that, given $0\leq k\leq n-2$, $E$ is a direct sum of twists of
${\mathcal{O}},Q,\ldots,\mathrm{Sym}^{k}Q$ if and only if
$H^{j}(E\otimes\mathrm{Sym}^{i}Q(*))=0$ for $(i,j)\in S_{k}$.

We end this section with the open case of Lagrangian Grassmannians Let
${\mathbb{C}}^{2k+2}$ be equipped with a nondegenerate skew form $J$. Let
$LG(k)$ be the Lagrangian Grassmannian of maximal isotropic
${\mathbb{P}}^{k}\subset{\mathbb{P}}^{2k+1}$, that is

(1.3) $LG(k)=\left\\{L\subset{\mathbb{C}}^{2k+2}|\dim
L=k+1,J(v,w)=0\quad\forall v,w\in L\right\\}.$

Let $E$ be a vector bundle on $LG(k)$. A naive attempt to generalize the
splitting criterion on Grassmannian to $LG(k)$ is the following.

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})$ $\overbrace{\Longleftrightarrow}^{?}$
$H^{i}(E(*)\otimes S^{\alpha}U)=0,1\leq i\leq dimLG(k)-1$ where $\alpha$ has
at most $k$ columns.

While $\Longleftarrow$ is true, the implication $\Longrightarrow$ is true on
$LG(1)=Q_{3}$ but it is false for $k=2$.

Oeding and Macias Marques have made, during 2009 Pragmatic, an attempt which
works, but only partially.

######  Theorem 1.16 (Oeding-Macias Marques).

[OM] Let $E$ be a vector bundle on $LG(k)$. Let $C_{k}(E)$ be the vanishing
condition

$H^{i}(LG(k),\wedge^{j_{1}}U\otimes\ldots\wedge^{j_{k}}U\otimes E(*))=0,0\leq
j_{q}\leq q+1,i=\sum_{q=1}^{k}j_{q}$

  1. (1)

If $C_{k}(E)$ holds then $E$ splits as a sum of line bundles.

  2. (2)

If $E$ splits as a sum of line bundles and $k\leq 6$ then $C_{k}(E)$ holds.

  3. (3)

Let $k=7$ and $E={\mathcal{O}}$. Then $C_{7}(E)$ does NOT hold.

######  Question 1.17.

Can be found a cohomological splitting criterion on $LG(k)$ for any $k$ ? One
should likely use irreducible homogeneous bundles (with the language of
maximal weights) and not just products of wedge powers.

##  2\. The approach with the derived category

###  2.1. The Grothendieck groups and a bit of K-theory

Let $X$ be a smooth variety. Let $K^{0}(X)$ (resp. $K_{0}(X)$) be the free
group generated by isomorphism classes of vector bundles (resp. coherent
${\mathcal{O}}_{X}$-modules) modulo the relation $[A]-[B]+[C]=0$ when we have
the exact sequence

$0\smash{\mathop{\longrightarrow}\limits}A\smash{\mathop{\longrightarrow}\limits}B\smash{\mathop{\longrightarrow}\limits}C\smash{\mathop{\longrightarrow}\limits}0$

.

$K^{0}(X)$ is called the Grothendieck group of vector bundles on $X$ and
$K_{0}(X)$ is called the Grothendieck group of coherent sheaves on $X$.

The following is one of the main results by Grothendieck in his SGA6, we
recommend the readable proof by Murre in [Mur, §2.10].

######  Theorem 2.1 (Grothendieck).

  1. (1)

$K^{0}(X)=K_{0}(X)$

  2. (2)

the Chern character gives a ring isomorphism

$ch\colon K^{0}(X)\otimes{\mathbb{Q}}\to CH(X)_{{\mathbb{Q}}}$

where $CH(X)_{{\mathbb{Q}}}$ is the rational Chow ring.

For any rational homogeneous variety $X=G/P$, every element of the Lie algebra
${\mathfrak{g}}$ gives an element of the tangent space at $[P]$ which, with
$G$-action, defines a section of $TX$. It is a well known fact that all
sections are defined in this way, that is $H^{0}(TX)={\mathfrak{g}}$.

This fact is important since it allows to compute the zero locus of a general
section of $TX$, hence, by Gauss-Bonnet Theorem, the topological Euler
characteristic $\chi(X)$. It is well known that the category of $G$-bundles on
$G/P$ is equivalent to the category of $P$-modules. The tangent bundle
$T(G/P)$ is associated to an irreducible representation of $P$ just in a few
cases which are called irreducible Hermitian symmetric. They are the three
classical cases listed in Proposition 2.2, smooth quadrics (where $\chi Q_{n}$
is equal to $n+1$ if $n$ is odd and to $n+2$ if $n$ is even) and just two
exceptional cases for the groups $E_{6}$ and $E_{7}$ that we skip for
simplicity. The following result is well known (indeed all Betti numbers are
known), but we think it is worth to expose the simple and unified proof for
these three cases.

######  Proposition 2.2.

  1. (1)

$\mathrm{rank}K_{0}\left(Gr(k,n)\right)=\chi\left(Gr(k,n)\right)={n\choose k}$

  2. (2)

$\mathrm{rank}K_{0}\left(LG(n+1)\right)=\chi
LG(n+1)=2^{n}\left(\textrm{see}(\ref{eq:LGk})\right)$

  3. (3)

$\mathrm{rank}K_{0}\left(Spin_{n}\right)=\chi
Spin_{n}=2^{n}\left(\textrm{see}(\ref{eq:spinor})\right)$

Proof:  In all three cases a section is given by a map $A\colon V\to V$. In
case (i) it is general, in case (ii) it preserves the quadric $Q$ (that is
$AQ+QA^{t}=0$), in case (iii) it preserves the skew-symmetric form $J$ (that
is $AJ+JA^{t}=0$). We may assume, by genericity, that $A$ has $n$ distinct
eigenvectors with nonzero eigenvalues. In case (i), the section $A$ vanishes
on the linear spaces ${\mathbb{C}}^{k}$ which are span of $k$ among the $n$
eigenvalues, they are ${n\choose k}$. In case (ii), we have the further
restriction that the subspace is isotropic. The eigenvectors of $A$ satisfy
$vQv^{t}=0$. We may assume we have an orthonormal basis
$\left\\{e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}\right\\}$ with
$Q(e_{i},f_{j})=\delta_{ij}$, $Q(e_{i},e_{j})=Q(f_{i},f_{j})=0$. The isotropic
subspaces are generated by choosing exactly one between $e_{i}$ and $f_{i}$,
for $i=1,\ldots n$, so we have $n$ binary choices and $2^{n}$ subspaces. Hence
the isotropic $C^{n}$ are $2^{n}$ ($2^{n-1}$ in one family) and the isotropic
$C^{k}$ are ${n\choose k}2^{k}$. In case (iii) the construction is analogous.
Now all vectors are isotropic, in particular the eigenvectors are isotropic.

###  2.2. The derived category of ${\mathbb{P}}^{n}$ and $Q_{n}$

We refer to [GM] for basics on derived categories. See also [AO89] for a short
introduction. The derived category $D_{b}(X)$ of bounded complexes of sheaves
on $X$ can be thought as an enhancement of $K_{0}(X)$. Recall that a sequence
of bundles $(E_{0},\ldots,E_{n})$ on a variety $X$ is called exceptional if
$Hom(E_{i},E_{i})={\mathbb{C}}$ and $Ext^{m}(E_{i},E_{j})=0$ for $i\geq j$,
$m>0$ or $i>j$, $m\geq 0$ and it is called full if it generates $D_{b}(X)$.

######  Theorem 2.3 (Beilinson, 1978).

[Bei, AO89] The derived category $D_{b}({\mathbb{P}}^{n})$ has a full
exceptional sequence (can be understood as a basis)
$\langle{\mathcal{O}}(-n),\ldots,{\mathcal{O}}\rangle$

######  Corollary 2.4.

$K_{0}({\mathbb{P}}^{n})$ is freely generated by the classes

$[{\mathcal{O}}(-n)],\ldots,[{\mathcal{O}}]$

Note the Chow ring of ${\mathbb{P}}^{6}$ has the following graphical
representation

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$1$$2$$3$$4$$5$$6$

Beilinson in [Bei] considers the Segre product
${\mathbb{P}}(V)\times{\mathbb{P}}(V)$ with the two projections on the factors
$p_{1}$, $p_{2}$. Let

(2.1) $E=p_{1}^{*}Q\otimes p_{2}^{*}{\mathcal{O}}(1)$

where $Q$ is the quotient bundle. We have the identification $H^{0}(E)=End(V)$
and the section $s$ corresponding to the identity endomorphism vanishes on the
diagonal $\Delta\subset{\mathbb{P}}(V)\times{\mathbb{P}}(V)$. The sheaf
${\mathcal{O}}_{\Delta}$ is equivalent in the derived category to its
resolution given by the Koszul complex of $s$. Beilinson smart idea is to
compare $E={Rp_{1}}_{*}(p_{2}^{*}E_{|\Delta}$ with the derived functor
$R{p_{1}}_{*}$ applied to the Koszul complex of $s$ tensored by $p_{2}^{*}E$.

Ten years later, Kapranov extends this construction to quadrics and
Grassmannians in [Kap]. Let state first his main result for quadrics.

######  Theorem 2.5 (Kapranov, 1988).

[Kap] The derived category $D_{b}(Q_{n})$ has a full exceptional sequence
$\langle{\mathcal{O}}(-n+1),\ldots,{\mathcal{O}},S\rangle$ for $n$ odd and by
$\langle{\mathcal{O}}(-n+1),\ldots,{\mathcal{O}},S^{\prime},S^{\prime\prime}\rangle$
for $n$ even, where $S$, $S^{\prime}$, $S^{\prime\prime}$ are the spinor
bundles introduced in §1.2.

######  Corollary 2.6.

$K_{0}(Q_{n})$ is freely generated by the classes

$[{\mathcal{O}}(-n+1)],\ldots,[{\mathcal{O}}],[S]$

for $n$ odd, by the classes

$[{\mathcal{O}}(-n+1)],\ldots,[{\mathcal{O}}],[S^{\prime}],[S^{\prime\prime}]$

for $n$ even.

Note the Chow ring of $Q_{n}$ is analogous to the one of ${\mathbb{P}}^{n}$
when $n$ is odd, in the sense that the Betti numbers are the same, but the
powers $H^{i}$, with $H$ the hyperplane divisor, generate $CH^{i}$ for
$i<\lfloor n/2\rfloor$ and are equal to twice the generator for $i\geq\lfloor
n/2\rfloor$. When $n$ is even the middle part has dimension $2$, as it is
shown in the following diagram of $Q_{6}$. It is interesting to remark that
the splitting of the middle cohomology testifies the presence of the two
spinor bundles, replacing the single spinor bundle for the odd case.

$\bullet$$\bullet$$\bullet$$0$$1$$2$$3$$3$$4$$5$$6$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

###  2.3. Cohomological characterizations via derived category

In [AO91] we found an alternative proof of Horrocks criterion (Theorem 1.1)
based on Beilinson description of $D_{b}({\mathbb{P}}^{n})$. The proof allows
to extend the characterization to any coherent sheaf as in the following.

######  Theorem 2.7 (Ancona-O, 1991).

[AO91, Theorem 2.2] Let $E$ be a coherent sheaf on
${\mathbb{P}}^{n}({\mathbb{C}})$.

$E=\oplus_{i=1}^{r}{\mathcal{O}}(a_{i})\oplus{\mathcal{S}}$
$\Longleftrightarrow$ $H^{i}\left(E(*)\right)=0,\quad 1\leq i\leq n-1$

where ${\mathcal{S}}$ is supported on a $0$-dimensional scheme.

An analogous statement[AO91, Corollary 6.2] holds for quadrics, using Kapranov
description of $D_{b}(Q_{n})$.

It is now time to state the remarkable description of the derived category of
Grassmannians given by Kapranov. Kapranov replaces the bundle $E$ in (2.1)
with the tensor product $p_{1}^{*}Q\otimes p_{2}^{*}U^{\vee}$ on
$Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})\times
Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$ where $Q$ is the quotient bundle and
$U$ is the universal bundle. Again there is a section vanishing on the
diagonal and the whole construction of [Bei] can be repeated at the price of
more technical stuff involving Schur functors. The final result is the
following.

######  Theorem 2.8 (Kapranov, 1988).

[Kap] The derived category $D_{b}(Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$ has
the full exceptional sequence $\langle S^{\alpha}U\rangle$ for $\alpha$ Young
diagram with at most $n-k-1$ columns, where $U$ is the universal bundle. In
particular such $S^{\alpha}U$ generate
$K_{0}(Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$.

The Hasse diagram of Schubert varieties in
$Gr({\mathbb{P}}^{1},{\mathbb{P}}^{4})$, where the vertices generate the Chow
ring $K_{0}(Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$ is the following (see also
[OR, §4]):

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$1$$2$$2$$3$$3$$4$$4$$5$$6$

The corresponding partitions $\alpha$ corresponding to the generators
$S^{\alpha}U$ of $K_{0}(Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$ are in the
following table

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr 0&1&2&3&4&5&6\\\
\hline\cr&&\yng(2)&&\yng(2,2)&&\\\
&\yng(1)&&\yng(2,1)&&\yng(2,2,1)&\yng(2,2,2)\\\ &&\yng(1,1)&&\yng(2,1,1)&&\\\
&&&\yng(1,1,1)&&&\\\ \hline\cr\end{array}$

We warn the reader that the Hasse diagram of Schubert varieties in
$Gr({\mathbb{C}}^{3},{\mathbb{C}}^{6})$ is no more planar.

There is an evident analogy between the generators of
$D_{b}(Gr({\mathbb{P}}^{k},{\mathbb{P}}^{n})$ in Theorem 2.8 and the splitting
criterion on Grassmannians in Theorem 1.14.

######  Question 2.9.

Is there a proof of the splitting criterion Theorem 1.14 by using the derived
category approach, analog to [AO91] for the case of projective spaces and
quadrics ? Some results for Segre products are in [Mal08, CMR05].

######  Remark 2.10.

After the work of Kapranov on Grassmannians, there is a natural folklore
conjecture (see [Fon]) that the bounded derived category of a rational
homogeneous variety $G/P$ admits a full exceptional collection of
$G$-homogeneous bundles.

The conjecture of previous remark is open even for irreducible Hermitian
symmetric spaces, but the case of isotropic Lagrangian Grassmannian $LG(n)$
has been recently solved. Kuznetsov and Polishchuk constructed collections on
$LG(n+1)$ of the right length $2^{n}$ in [KP].

Fonarev in [Fon] shows the fullness of the exceptional collections constructed
by Kuznetsov and Polishchuk and gives a geometric description. The full
collection consists of $G$-equivariant vector bundles.

Beyond the rational homogeneous varieties $G/P$, we conclude this section with
some of the many results on the Fano case. Recall the Fano $3$-fold $V_{5}$ (
already met exposing [AC] in Theorem 1.11) is $SL(2)$-quasi homogeneous. The
derived category of $V_{5}$ has been found by Orlov in [Orl91]. Faenzi in
[Fa05] found more mutations and applied them to classify the ACM bundles.

######  Theorem 2.11 (Faenzi, Orlov).

The Fano $3$-fold of index $2$ $V_{5}$ has the full exceptional sequence

$\langle{\mathcal{O}}(-1),U,Q^{\vee},{\mathcal{O}}\rangle$

where $U$ is the rank $2$ universal bundle and $Q$ is the rank $3$ quotient
bundle, both $SL(2)$-invariant.

Kuznetsov in [Kuz96] have found a similar result for the Fano $3$-fold
$V_{22}$, which has index $1$, but it has a member of the deformation class
which is $SL(2)$-quasi homogeneous. Faenzi in [Fa07] found again more
mutations and applied them to classify the ACM bundles.

Kuznetsov found also the decomposition of the derived category of cubic
$3$-fold.

######  Theorem 2.12 (Bondal-Orlov, Kuznetsov).

[Kuz04] Let $X\subset{\mathbb{P}}^{n+1}$ be a smooth cubic hypersurface.
$D_{b}(X)$ has a semiorthogonal decomposition

$\langle{\bf
T}_{X},{\mathcal{O}},{\mathcal{O}}(1),\ldots,{\mathcal{O}}(n-2)\rangle$

######  Theorem 2.13 (Lahoz-Macrì-Stellari, 2015).

[LMS] Let $X\subset{\mathbb{P}}^{4}$ be a smooth cubic $3$-fold. Any ACM
stable bundle on $X$ has a twist which belongs to ${\bf T}_{X}$.

This result is applied to show non-emptiness and smoothness of moduli space of
Ulrich bundles (of rank $\geq 2$) on a smooth cubic $3$-fold.

##  3\. The trichotomy for ACM varieties

Buchweitz, Greuel and Schreyer proved in [BGS, Theorem C] that hypersurfaces
in ${\mathbb{P}}^{n}$, $n\geq 2$ of degree $d\geq 3$ have infinite non
isomorphic ACM bundles. This is the first step toward the wilderness of such
hypersurfaces as defined below. I heard the first suggestions of the
trichotomy finite/tame/wild in a memorable talk by Hartshorne in Turin in
2005. In that conference Hartshorne suggested to study ACM bundles of rank two
on cubic surfaces, which was carried out by Faenzi in [Fa08]. Then Casanellas
and Hartshorne proved that cubic surfaces are wild in [CH], where they
introduced the terminology finite/tame/wild for ACM sheaves. It was discovered
already in [BGS] that the quadric cone in ${\mathbb{P}}^{3}$ has a countable
discrete set of indecomposable ACM bundles of rank $2$, so making the tame
case for singular varieties quite subtle.

######  Theorem 3.1 (Faenzi - Pons Llopis, 2019, Trichotomy [FP].).

Any reduced ACM closed subscheme $X\subset{\mathbb{P}}^{n}$ of positive
dimension falls in exactly one of the following classes:

  * •

Finite: there are only finitely many indecomposable ACM sheaves over $X$ up to
isomorphism.

  * •

Tame: for any given rank $r$, the moduli space of indecomposable non-
isomorphic ACM sheaves of rank $r$ is a finite or countable union of points or
curves.

  * •

Wild: $X$ supports families of arbitrarily large dimension of indecomposable
non-isomorphic ACM sheaves.

Finite and tame cases are completely classified. Precisely we have from [FP]
the following list.

Finite connected $X\subset{\mathbb{P}}^{m}$ are

  * •

(i) ${\mathbb{P}}^{m}$

  * •

(ii) a smooth quadric hypersurface;

  * •

(iii) a smooth rational curve

  * •

(iv) the smooth cubic scroll in ${\mathbb{P}}^{4}$, isomorphic to
${\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^{1}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(1))$,
also to ${\mathbb{P}}^{2}$ blown up in a point,

  * •

(v) the Veronese surface in ${\mathbb{P}}^{5}$.

Tame connected $X\subset{\mathbb{P}}^{m}$ are

  * •

(i) quadric hypersurface of corank one;

  * •

(ii) a reducible tree of rational curves;

in cases (i), (ii) the parameter space of indecomposable non isomorphic ACM
sheaves is a countable set of points,

  * •

(iii) a smooth elliptic curve or a cycle of rational curves;

  * •

(iv) the smooth quartic scroll in ${\mathbb{P}}^{5}$, there are two kinds of
them, isomorphic to ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ embedded with
${\mathcal{O}}(1,2)$, or isomorphic to
${\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^{1}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(2))$,
see [Hart, Ex. 2.19.1].

in cases (iii), (iv) the parameter space of indecomposable non isomorphic ACM
sheaves is a curve.

Note that the Del Pezzo surface which is the complete intersection of two
quadrics in ${\mathbb{P}}^{4}$ is wild, since it does not appear in the above
list. Also the Segre $3$-fold ${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ (a
variety of minimal degree) and
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ in their
minimal equivariant embedding are wild, see [CFM] for the latter.

The paper [FP] by Faenzi and Pons-Llopis is the culmination of a large amount
of work, beyond the two authors, by Buchweitz, Casnati, Casanellas, Costa,
Drozd, Eisenbud, Greuel, Hartshorne, Herzog, Malaspina, Mirò-Roig, Schreyer
and other people that it is used in the paper and it cannot be resumed here,
and we refer to the bibliography in [FP] for further informations.

Also the trichotomy has classical roots, C. Segre proved in 1886 [S2] that
elliptic curves are tame, a result commonly attributed to Atiyah. For a modern
exposition of this Segre result we quote [GO, §5].

######  Remark 3.2.

An updated reference which explores the related topic of Ulrich bundles is the
book [CMP] by Laura Costa , Rosa María Miró-Roig and Joan Pons-Llopis. See the
nice survey [Beau]. Ulrich bundles are special ACM bundles. The conjecture
that every projective variety carries a Ulrich bundle is the next challenge.
This is another interesting story beyond the limits of the present paper.

## References

  * [AO89] V. Ancona, G. Ottaviani, An introduction to the derived categories and the theorem of Beilinson, Atti Accad. Peloritana Pericolanti. Cl. Sci. Fis. Mat. Natur.67(1989), 99–110.
  * [AO91] V. Ancona, G. Ottaviani, Some applications of Beilinson’s theorem to projective spaces and quadrics, Forum Math. 3 (1991), no. 2, 157–176.
  * [APW] V. Ancona, Th. Peternell, J. Wiśniewski, Fano bundles and splitting theorems on projective spaces and quadrics, Pacific J. Math. 163 (1994), no.1, 17–42.
  * [AC] E. Arrondo, L. Costa, Vector bundles on Fano 3-folds without intermediate cohomology, Comm. Algebra 28 (2000), no.8, 3899–3911.
  * [AF] E. Arrondo, D. Faenzi, Vector bundles with no intermediate cohomology on Fano threefolds of type $V_{22}$ , Pacific J. Math. 225 (2006) No. 2, 201–220.
  * [AG] E. Arrondo, B. Graña, Vector bundles on $G(1,4)$ without intermediate cohomology, J. Algebra 214(1999), no.1, 128–142.
  * [AM] E. Arrondo, F. Malaspina, Cohomological characterization of vector bundles on Grassmannians of lines J. Algebra 323 (2010), no.4, 1098–1106.
  * [AT] E. Arrondo, A. Tocino, Cohomological characterization of universal bundles of $G(1,n)$, J. Algebra 540 (2019), 206–233.
  * [BH] W. Barth, K. Hulek, Monads and moduli of vector bundles, Manuscripta Math. 25 (1978), no. 4, 323–347.
  * [Beau] A. Beauville, An introduction to Ulrich bundles, Eur. J. Math. 4 (2018), no. 1, 26–36.
  * [Bei] A. A. Beĭlinson, Coherent sheaves on ${\mathbb{P}}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69.
  * [Bel] A. Bellatalla, Sulle varietà razionali normali composte di $\infty^{1}$ spazi lineari, Atti R. Accad. Torino, 36 (1901), 803–833.
  * [BFS] V. Benedetti, D. Faenzi, M. Smirnov, Derived category of the spinor $15$-fold, arXiv:2310.01090
  * [BF] M.C. Brambilla, D. Faenzi, Moduli spaces of rank-2 ACM bundles on prime Fano threefolds, Michigan Math. J. 60 (2011), no.1, 113–148.
  * [BGS] R.O. Buchweitz, G.M. Greuel, F.O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), no.1, 165–182.
  * [CH] M. Casanellas, R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. 13(2011), no.3, 709–731.
  * [CFM] G. Casnati, D. Faenzi, F. Malaspina, Moduli spaces of rank two aCM bundles on the Segre product of three projective lines, J. Pure Appl. Algebra 220 (2016), no.4, 1554–1575.
  * [CMR05] L. Costa , R. M. Miró-Roig, Cohomological characterization of vector bundles on multiprojective spaces, J. Algebra 294, 73–96 (2005), corrigendum J. Algebra 319 (2008), no.3, 1336–1338.
  * [CMR16] L. Costa , R. M. Miró-Roig, Homogeneous ACM bundles on a Grassmannian, Adv. Math. 289 (2016), 95–113.
  * [CMP] L. Costa , R. M. Miró-Roig and Joan Pons-Llopis, Ulrich bundles, Studies in Math. 77, De Gruyter, Berlin 2021
  * [E] L. Ein, An analogue of Max Noether’s theorem, Duke Math. J.52(1985), no.3, 689–706.
  * [EG] E.G. Evans, Ph. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333.
  * [Fa05] D. Faenzi, Bundles over the Fano threefold $V_{5}$, Comm. Algebra 33 (2005), no. 9, 3061–3080.
  * [Fa07] D. Faenzi, Bundles over Fano threefolds of type $V_{22}$, Ann. Mat. Pura Appl. (4) 186 (2007), no. 1, 1–24.
  * [Fa08] D. Faenzi, Rank 2 arithmetically Cohen-Macaulay bundles on a nonsingular cubic surface, J. Algebra 319 (2008), 143–186.
  * [FP] D. Faenzi, J. Pons-Llopis, The Cohen-Macaulay representation type of projective arithmetically Cohen-Macaulay varieties Épijournal Géom. Algébrique 5 (2021), Art. 8, 37 pp.
  * [Fon] A. Fonarev, Full exceptional collections on Lagrangian Grassmannians, Int. Math. Res. Not. IMRN (2022), no.2, 1081–1122.
  * [GM] S. Gelfand, Y. Manin, Methods of homological algebra, Springer Monogr. Math., 2003
  * [Gh] G. Gherardelli, Sulle curve sghembe algebriche intersezioni complete di due superfici, Rend. Atti Accad. d’Italia s. VII, IV (1942), 128–132.
  * [GO] F. Ghione, G. Ottaviani, A tribute to Corrado Segre, in Complex Projective Geometry, London Math. Soc. Lect. Notes Series 179, 175–188, Cambridge 1992.
  * [Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52. Springer, 1997.
  * [Hor] G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3)14(1964), 689–713.
  * [Kap] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508.
  * [Kn] H. Knörrer, Cohen-Macaulay modules on hypersurface singularities. I Invent. Math. 88 (1987), no. 1, 153–164.
  * [KPR] M. Kumar, C. Peterson, A.P. Rao, Monads on projective spaces, Manuscripta Math. 112 (2003), no.2, 183–189.
  * [Kuz96] A. G. Kuznetsov, An exceptional set of vector bundles on the varieties $V_{22}$, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (3):41–44, 92, 1996.
  * [Kuz04] A. G. Kuznetsov, Derived category of a cubic threefold and the variety $V_{14}$, English translation in Proc. Steklov Inst. Math.(2004), no.3, 171–194.
  * [KP] A. Kuznetsov, A. Polishchuk, Exceptional collections on isotropic Grassmannians, Journal of the European Mathematical Society, 18(3):507–574, 2016
  * [LMS] M. Lahoz, E. Macrì, P. Stellari, Arithmetically Cohen-Macaulay bundles on cubic threefolds, Algebr. Geom. 2 (2015), no.2, 231–269.
  * [Mal08] F. Malaspina, A few splitting criteria for vector bundles, Ric. Mat. 57 (2008), no.1, 55–64.
  * [Mal09] F. Malaspina, Monads and rank 3 vector bundles on quadrics, Ann. Mat. Pura Appl. (4) 188 (2009), no. 3, 455–465.
  * [Mur] J.P. Murre, Algebraic cycles and algebraic aspects of cohomology and K-theory, Algebraic cycles and Hodge theory (Torino, 1993), 93–152. Lecture Notes in Math., 1594 Springer, Berlin, 1994.
  * [OM] L. Oeding, P. Macias-Marques, Splitting criteria for vector bundles on the symplectic isotropic Grassmannian, Matematiche (Catania) 64 (2009), no.2, 155–176.
  * [OSS] C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Corrected reprint of the 1988 edition. With an appendix by S. I. Gelfand, Birkhäuser/Springer, Basel, 2011.
  * [Orl91] D.D. Orlov, Exceptional set of vector bundles on the variety $V_{5}$, English translation in Moscow Univ. Math. Bull.46(1991), no.5, 48–50.
  * [Ott88] G. Ottaviani, Spinor bundles on quadrics, Trans. Amer. Math. Soc. 307(1988), no.1, 301–316.
  * [Ott89] G. Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. (4) 155 (1989), 317–341.
  * [OR] G. Ottaviani, E. Rubei, Quivers and the cohomology of homogeneous vector bundles, Duke Math. J. 132 (2006), no. 3, 459–508.
  * [S1] C. Segre, Sulle rigate razionali in uno spazio lineare qualunque  Atti R. Accad. Torino 19 (1883-84), 265–282.
  * [S2] C. Segre, Ricerche sulla rigate ellittiche di qualunque ordine Atti R. Accad. Torino 21 (1885-86), 628–651.


# detailing the natural creation process of a single molecule MCB junction in
the partially wet phase

C.J. Muller

This research has, in its entirety, been privately conducted and funded by the
author Correspondence: PO Box 60, 6580 AB Malden, The Netherlands

###### Abstract

A novel way to arrive at a single molecule junction in the partially wet phase
is described. The fabrication of this device makes use of a natural drying
process. Details are provided concerning the fabrication of the device and the
related material choices. In addition a process description is provided with a
step by step approach for creating the partially wet phase molecular MCB
junction.

##  1 Introduction

The wetting of surfaces is an interesting and current field where chemistry,
physics and engineering come together [1]. Chemistry ensures interaction at
the molecular scale. Van der Waal forces extend over tens of molecules
providing a larger interaction scale. The engineering comes in by creating for
example repetitive structures thus providing additional length scales as a
means to enhance our understanding of specific wetting situations.

The partially wet phase layer of tetrahydrofuran fluid is a requirement for
observing a quantum effect related to continuous Q-charging at ambient
conditions muller2021source2,*muller2021source2b. This is only adding to the
interest in the field of wetting, in this case at the microscopic level. The
fundamental question is how a microscopic layer of fluid will wet a single
molecule anchored between two electrodes and how this layer behaves over time.
This system is in analogy to the terminology in the field of wetting [1]
called a partially wet phase molecular MCB junction.

Due to a large number of, only in part controlled, degrees of freedom
associated with the partially wet phase molecular MCB junction a number of
critical material decisions need to be made. Each of these decisions can lead
to failure when not carefully investigated. The experimenter will be
confronted with many unknowns and uncertainties, to mention a few:

  * •

How to end up with one molecule bridging the electrodes without any positional
dependence of the IV curve on the gap within a fluid.

  * •

Is the solvent used in the cell pure enough and not contaminating the
electrodes.

  * •

Are the molecules used in the solution modified at ambient conditions.

  * •

Is the seal-material or any other material inside the cell dissolving in the
solution and perhaps contaminating the electrode material.

  * •

Is any of the material inside the cell which is in contact with the solution,
incompatible with or modifying the dissolved molecules.

To find a suitable process-window and way of work in the above mentioned
parameter space can be tedious and laborious. It is the aim of this paper to
enable those who are trying to replicate and build on the earlier reported
results in this field [2]. The limitation will be that the MCB assembly as
well as the solution and molecule choice is restricted to what the author has
been using: a solution of anhydrous tetrahydrofuran (THF) with dissolved
benzene di-thiol (BDT) molecules or a pure THF fluid in combination with a
notched filament MCB assembly. In addition H2O can be used in combination with
the aforementioned possibilities.

Macroscopic wetting layers may affect micro fabricated bridges in a different
manner as compared to their notched filament MCB counterparts, simply because
the gaps where fluid can be trapped are smaller. Trapped fluid may
influence/exert forces on free hanging bridges. In order to be able to mimic
the experimental situation in reference [2], the entire notched filament
device fabrication is described in detail. For operating the device the use of
a pivoting MCB setup [2] can be controlled in case a $4.5\text{\,}\mathrm{mm}$
inner diameter cell ($\phi_{\text{in}}^{4.5}$) is used. This cell will release
the fluid to the cell-opening under pivoting angles where it can be absorbed.
Also a cell with a smaller diameter can be used in a static MCB setup in this
case the draining will take place via a paper tip. Many devices with a cell
inner diameter of $2\text{\,}\mathrm{mm}$ ($\phi_{\text{in}}^{2}$) have
successfully been operated.

Since the number of junction types as well as partially wet phase liquids will
be limitless the following naming convention is proposed. The barrier molecule
forms the base and the lining partially wet phase forms the exponent. For
example BDTTHF for a BDT barrier junction in the THF partially wet phase and
BDTH2O for a BDT barrier junction in the H2O partially wet phase.

##  2 Device fabrication

The basis for the device is a $1\times 5\times 24$ $\mathrm{mm}$ phosphor
bronze bending beam. The bending beams have been laser cut, are wet sanded and
aceton ultrasonic cleaned prior to a furnace step of
$130\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 3 hours. This will discolor
the bending beams from a shiny metallic color to an oily film like color. This
step enhances the adhesion of the Stycast glue layer used to surface the
bending beam with Kapton. The Kapton foil of
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick is glued on the bending
beam, using Stycast 2850 with $3.5\text{\,}\mathrm{\char 37\relax}$ CAT9
catalyst. This combination is favored over other Stycast catalyst
combinations, as after ambient curing it results in the most chemical
resistant epoxy (amongst the Stycast family). In addition it caters for a
relative high temperature furnace cure, which further enhances chemical
resistance.

For the electrode material a filament of hard temper Au
($25\text{\,}\mathrm{\SIUnitSymbolMicro m}$) or a filament of annealed soft
temper Au ($75\text{\,}\mathrm{\SIUnitSymbolMicro m}$) has been used. The
filament is notched by a scalpel on a lathed stainless steel surface finish
(e.g. bottom of a pan), making use of the minute grooves which cater for not
fully cutting through the filament. Subsequently half of the filament is
fixated on this surface by covering it with a piece of metal; the notch is
"fatigued" by moving the other half up and down until it almost breaks. The
weakened notched filament is positioned on the Kapton and fixated at the outer
ends using Stycast. For the fixation of the center notched section again the
same Stycast catalyst combination is used. The creation of the central Stycast
contacts is performed with half cured Stycast, 2 hours and 45 minutes after
mixing of the compound at room-temperature. About $2030$ minutes are available
for creating the central Stycast contacts on a hot plate at approximately
$50\text{\,}\mathrm{\SIUnitSymbolCelsius}$, under a microscope. The Stycast
has hardened considerable, however can still be picked up. Holding it in close
proximity to the hot plate will make it malleable again and allows it to be
used for the center epoxy contacts. The central unglued section can be fine-
tuned by the use of heat to let the Stycast contacts flow wider or by taking
the assembly off the hot plate to stop the slowly flowing Stycast contacts.
Alternatively the epoxy contacts can be pushed closer together while the
assembly rests on the hot plate. After the MCB assembly, including the
filament fixture, is cured at ambient conditions it will undergo a 3 hour
$130\text{\,}\mathrm{\SIUnitSymbolCelsius}$ furnace cure in order to enhance
chemical resistance of the Stycast contacts. Ramp up and down should be slow,
15 minutes, in order to prevent breaking filaments.

After the creation of the bending beam construction, the liquid cell needs to
be assembled. A glass cell is used, which is simply a piece of glass tube cut
to the right size, approximately $25\text{\,}\mathrm{mm}$ has been used. One
opening of the cell should encircle the notched filament. The liquid rubber
will after curing ensure a leak tight electrical feedthrough. Furthermore the
elastomeric seal provides a leak tight connection between the Kapton layer and
the glass cell; it also caters for flexing of the bending beam. Ideally the
liquid rubber seal does not extend higher than
$1.5\text{\,}\mathrm{mm}2\text{\,}\mathrm{mm}$ above the bending beam for the
$\phi_{\text{in}}^{2}$ cell. This is to ensure the possibility during the
draining process to look from the side of the cell to the inside. The low seal
height can be achieved as follows:

  * •

Fixate the cell to the bending beam using an extremely thin ribbon of liquid
rubber at the outer diameter bottom side. Liquid rubber preferably does not
enter the inside of the cell. This is only for fixation, this step will in
general not deliver a leak tight seal.

  * •

Let the assembly cure for approximately an hour.

  * •

Apply some tape circling the cell, exposing the bottom 1.5-2 $\mathrm{mm}$.

  * •

Apply a big blob of liquid rubber circling the cell within 5 seconds in one go
making sure all exposed surfaces are well contacted.

  * •

Tearing off the tape leaves a low height seal.

  * •

The assembly needs to cure again, this time following the specifications
related to the liquid rubber.

After application of the glass cell and curing of the liquid rubber at ambient
conditions, the entire assembly gets another furnace cure for 1 hour at
$110\text{\,}\mathrm{\SIUnitSymbolCelsius}$, in this case to enhance the
chemical resistance of the seal. For the seal Bison Rubber Repair/liquid
rubber is used, which is a transparent, solvent free rubber paste from a tube.
It cures under the influence of air humidity and is chemically resistant. Fig.
1 shows a photo of the $\phi_{\text{in}}^{2}$ cell. The THF/BDT solution in
the cell is in contact with the following materials:

  * •

Gold electrode and filament material

  * •

Stycast, the center notched filament epoxy contacts

  * •

Kapton, covering and insulating the bending beam

  * •

Glass, the capillary used as a cell

  * •

Liquid rubber, in use as an elastomeric seal

Figure 1: A photo of the bending beam assembly. Visible are the notched
section with Stycast contacts, the elastomeric seal, the cell and the
electrical feedthroughs. The notched gold wire is connected to two copper
wires which are connected to the wiring of the MCB setup.

It can never be excluded that some traces of the used materials get dissolved
in the THF fluid, however reproducible measurement differences with respect to
THFTHF and BDTTHF junctions have been obtained with this material combination
[2].

##  3 The formation process of a molecular junction

Draining and the subsequent drying process is critical in obtaining the
partially wet phase junction, be it the BDT barrier or the THF barrier
junction. The formation of the molecular junction is initiated by the drying
process. In addition the right timing is required for starting the
measurements in the partially wet phase. In reference [2] is has been phrased
as follows:

"A law of nature seems to hold for two electrodes spaced at nanometers
distance during the drying process. Be it soft temper Au or hard temper Au
used as the MCB electrode material, be it broken in air and after that
immersed in fluid or broken directly in a THF environment, the shown electrode
behavior upon the drying process is the same. Once the excess fluid has been
drained, during the drying process the conductivity continuously increases.
This is attributed to a continuously decreasing distance between the two
closest points on the opposing electrodes, due to the lasting evaporation of
the fluid on and between the electrodes. Apparently there is enough elasticity
in the nanometers separated electrodes such that the exact initial separation
does not matter. The continuously decreasing amount of fluid between the
electrodes pulls the electrodes ever closer together. This process endures
until the fluid is getting depleted, and a molecule is squeezed between the
electrodes, blocking any further movement. The adhesion forces of the
molecules (either BDT or THF) to the gold electrode surface, together with the
partially wet phase layer which is lining the entire electrode-molecule-
electrode system, are such that a metal-molecule-metal junction is favored
over a metallic junction."

Although in the completely wet phase it is not possible to derive any
positional dependence of the electrodes by electrical measurements during the
drying process the situation is better. Since the conductance increases
continuously during the drying process it is possible to monitor the
conductivity over time. Also IV curves can be continuously recorded. It is
however a possibility that a measurement influences the creation of a
partially wet phase layer. Ideally in the end a partially wet phase layer of a
uniform thickness is lining the molecular device. Current or electrical fields
may exert forces on fluid molecules which may preferably occupy available
space near the junction and get trapped there because of the geometry. Most of
the data in reference [2] has been obtained by performing single point current
measurements during the drying process at $-5\text{\,}\mathrm{V}$ every few
minutes or so and only start continuous IV scan measurements once the
partially wet phase has formed. The exact trigger when to start the continuous
IV measurements relies on both the current value at $-5\text{\,}\mathrm{V}$
and a visible check of the cell.

##  4 Draining and relevant current values

Just after breaking in either pure THF fluid or in a
$1\text{\,}\mathrm{m\textsc{m}}$ BDT/THF solution any current value at
$-5\text{\,}\mathrm{V}$ between $-5\text{\,}\mathrm{nA}$ and
$-50\text{\,}\mathrm{nA}$ is the norm. Occasionally values of
$-1\text{\,}\mathrm{nA}$ or smaller are measured in the completely wet phase.
Although there is no evidence it is suspected to be due to H2O molecules
adhering to the surface of the electrodes. This cannot be undone. Continuing
with the drying process will lead to a reduced end value of the current and
degraded results as compared to a junction that showed normal start current
values.

Draining of the $\phi_{\text{in}}^{4.5}$ cell is simply performed by pivoting
the setup [2] absorbing the fluid at the cell opening and reverting back to
the vertical cell position. For a $\phi_{\text{in}}^{2}$ cell the draining has
to be performed by a paper tip and is more laborious. The last 3-4
$\mathrm{m}\mathrm{m}$ of the tip needs to be sharply bent in order to
eliminate the possibility of touching the junction in the center. In this case
the end of the tip will touch the glass wall on the inside. During draining of
the last $2\text{\,}\mathrm{mm}$ fluid the inside of the cell needs to be
monitored, as soon as the paper tip attracts fluid this will be visible and
the tip needs to be retracted immediately. This is the reason for having a
1.5-2 $\mathrm{m}\mathrm{m}$ seal-height it allows monitoring of the cell
bottom from the outside during draining. Every time the tip has been retracted
a visual inspection by a magnifying glass of the cell bottom is required in
order to understand the exact phase of the draining process the cell is in.
Once the desired phase in the drain process has been reached (see below) the
cell needs to remain untouched.

After draining the cell starts to gradually dry. It is important to note that
the cell and the junction are coupled systems with respect to wetting. As long
as the cell is wet, so is the junction. The cell acts as a buffer and the
junction can only follow. This also implies that if it is possible to observe
the last bit of fluid in the cell that this provides a good trigger for the
start of the continuous IV curve measurements. This trigger becomes even more
secure if it is possible to also tie it to a certain current value. The norm
of the trigger value is $-100\text{\,}\mathrm{nA}$ at the
$-5\text{\,}\mathrm{V}$ value. At this value the current value in the IV curve
measures 300-500 $\mathrm{nA}$ at $10\text{\,}\mathrm{V}$. It is expected to
represent the completion of the formation process of a single molecular
junction.

##  5 Optical contour graphs as a means of determining the stage in the drain
and drying process

How to know what to look for during the draining process for the
$\phi_{\text{in}}^{2}$ cell is indicated in Fig. 2. The draining via a paper
tip should progress the experimenter from a situation depicted in Fig. 2a to
Fig. 2c. The cross section of the fluid profiles is indicated below the top
down contour graphs.

Figure 2: Optical contour graphs detailing the wetting geometry inside a
$2\text{\,}\mathrm{mm}$ inner diameter cell as a function of time. The THF
contour in picture a may still last for hours. Paper tip draining should
progress towards the contour in picture c from this stage it will take about
30 minutes to the completely dry phase. Picture d shows a THF liner around the
Stycast contacts, once this liner is evaporated the partially wet phase is
entered. Picture e provides the completely dry phase.

Initially when the bulk of the fluid is removed a situation indicated in Fig.
2a is encountered. It may still take hours to evaporate this amount, depending
on ambient conditions. Carefully continuing draining leads at some point to a
donut shaped THF ring pinned to the Stycast contacts, Fig. 2b. This ring is no
longer touching the cell-wall. Continuing even further with the drain process
will lead to an “outer ring” touching the cell-wall, Fig. 2c. This is the
situation to stop draining and let the drying process progress to the
partially wet phase in half an hour or so.

For both the $\phi_{\text{in}}^{2}$ cell and the $\phi_{\text{in}}^{4.5}$ cell
the stage in the drying process just prior to the partially wet phase is as
indicated in Fig. 2d. A liner of moisture encircles the Stycast contacts. This
liner reduces in time until the cell is completely dry as in Fig. 2e. In
practice the trigger to start continuous IV measurements is when the liner is
almost invisible at the junction location. It is possible that fluid is still
trapped at the edge where the inner cell wall contacts the Kapton. This will
normally show as glare, which slowly degrades and may lengthen the life of the
liner of moisture. In practice either the current value of
$-100\text{\,}\mathrm{nA}$ at $-5\text{\,}\mathrm{V}$ or the visual
disappearance of the liner of moisture are used as a trigger to start the
continuous IV scans. Using them both provides enhanced control.

##  6 The BDTH2O device as an enduring molecular junction at ambient
conditions

Enduring molecular junctions at ambient conditions are inherently lined with
an H2O partially wet phase layer [1]. In ref [2] it was reported that these
types of junctions can be created starting off with the BDT/THF solution
catering for a BDTTHF junction. The procedure to arrive at these interesting
structures is detailed below. For all BDTH2O junctions the
$\phi_{\text{in}}^{2}$ cell has been used.

The start to arrive at a BDTH2O junction is by using the BDTTHF junction
formation process. Towards the end of the formation process the humidity is
enhanced to about $90\text{\,}\mathrm{\char 37\relax}95\text{\,}\mathrm{\char
37\relax}$ by a humidifier in an enclosed space also containing the MCB setup.
This is done in the $-80\text{\,}\mathrm{nA}-100\text{\,}\mathrm{nA}$ range at
$-5\text{\,}\mathrm{V}$ in the BDTTHF formation process. Subsequently
continuous IV curve scanning needs to start from
$-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ and reverse. Fig. 3a shows the
initial IV curve of this process just after the humidity was raised. The
current start level is about $-400\text{\,}\mathrm{nA}$ at
$-10\text{\,}\mathrm{V}$ and reaches values beyond
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro A}$ at $10\text{\,}\mathrm{V}$ in a
$45\text{\,}\mathrm{s}$ scan. From this stage onward the junction still
modifies over the next half hour under continuous IV scanning
($20\text{\,}\mathrm{V}$/$45\text{\,}\mathrm{s}$) to arrive at the final
enduring IV curve which is shown in Fig. 3b.

The exact point in time when to switch the humidifier on is not critical. Also
a BDTH2O junction has been created by increasing the humidity to
$95\text{\,}\mathrm{\char 37\relax}$ after the cell dried and the conductivity
was reduced to $30\text{\,}\mathrm{nA}$ at $10\text{\,}\mathrm{V}$. The
subsequent scanning with a rate of
$20\text{\,}\mathrm{V}$/$5\text{\,}\mathrm{s}$ lasted for approximately half
an hour. At that point in time the enduring IV curve similar as in Fig. 3b was
reached.

Figure 3: Progression to an enduring BDTH2O junction by switching to
$95\text{\,}\mathrm{\char 37\relax}$ humidity towards the end of the BDTTHF
formation process. The first resulting asymmetric IV curve is shown in panel
a. Under successive IV trace recording after approximately half an hour the
enduring IV curve as shown in panel b has been obtained.

The scanning is required for settling of the H2O partially wet phase molecules
in the direct vicinity of the junction. The high electric fields exert
changing forces on the H2O dipoles building the partially wet phase layer at
the junction. This is shown in Fig. 4 which displays successive stages of the
junction in panel a to c. Panel a shows an IV curve which is not at the final
enduring state. Panel b shows that the outer legs of the IV curve have moved
apart as compared to panel a. After switching to a scan-rate of
$4\text{\,}\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$ between panel b and c for
several minutes the two legs gradually pulled together to create the enduring
IV curve as shown in panel c.

This is evidence that the IV measurements have an impact on the formation of a
partially wet phase. The sub-gap structure was shown to be scan-speed
dependent and is expected to result from H2O dipoles in the partially wet
phase layer [2]. Scan speed values for the displayed curves in panel a, b and
c are $4\text{\,}\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$,
$0.44\text{\,}\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$ and
$0.44\text{\,}\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$ respectively over the
entire $-10\text{\,}\mathrm{V}10\text{\,}\mathrm{V}$ range.

The remarkable aspect about this junction is the stability over hours, after
the initial high humidity is gradually reduced to ambient values. Multiple
attempts on one assembly have been executed as long as the conduction remained
above $-5\text{\,}\mathrm{nA}$ at $-5\text{\,}\mathrm{V}$.

Figure 4: Progress of the formation of an enduring BDTH2O junction from panel
a to c. Panel a shows a junction which almost attains the enduring form. It
has been achieved by continuous IV scanning. After stopping the continuous IV
scans panel b showed the degradation of the junction as the outer legs are
drifting apart. After a couple of minutes of stressing scans with a scan speed
of $4\text{\,}\mathrm{V}\text{\,}{\mathrm{s}}^{-1}$ the final enduring
situation has been reached in panel c.

Four out of eight assemblies led to the same IV characteristic shown in Fig.
3b and Fig. 4c after the BDTH2O junction was completed. In analogy to the
reasoning in the early 90’s why single atom point contacts exist, the
consistent measured value close to $2e^{2}/h$ of the last conductance plateau
has been a convincing argument see for example [4]. A similar reasoning holds
for molecular junctions. Using the same type of molecules ending up in the
same reproducible and enduring IV curves is a very strong indication that the
same single molecular junction is created over and over again.

In the cases where the IV curve does not end up as shown in Fig. 3b or Fig. 4c
the conductance may show a substantial initial increase in the IV curve.
However at some point the conductance collapses to sub $\mathrm{nA}$ values,
as in the completely dry phase. This is shown in Fig. 5. The current
compliance has been set at $3\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and
$-3\text{\,}\mathrm{\SIUnitSymbolMicro A}$ for Fig. 3-5.

Figure 5: An IV trace showing the failure to arrive at an enduring BDTH2O
junction.

The possibility to compare same barrier different partially wet phase
junctions or different barrier same partially wet phase junctions may provide
an additional angle to increase our understanding of the physics of these
systems. In comparing the IV curves from the BDTH2O junction against a BDTTHF
junction it should be noted that the BDTH2O junction obtains a fixed and
reproducible enduring state/IV-curve. For the BDTTHF junction this is not the
case at ambient conditions. For the THFTHF or BDTTHF junction it is simple, at
some point the THF microscopic layer will be evaporated. At ambient conditions
there will not be an enduring BDTTHF or THFTHF junction. For H2O this is
different as it is the microscopic layer at ambient conditions that lines
everything [1]. That it is possible to create an enduring BDTH2O junction is
fascinating. It implies that von Hippel’s vision is actually starting to
become a reality vonhippel1956source3a,*vonhippel1959source3b. We now have the
tools to create reproducible enduring single molecular junctions lined with a
microscopic partially wet phase layer. The partially wet phase layer is an
integral part of the total mechanism ensuring the stability of the entire
system.

##  7 Handling of the chemicals

The molecular deposition on the electrodes of a freshly broken MCB junction
starts by preparing the approximately $1\text{\,}\mathrm{m\textsc{m}}$ BDT/THF
solution. The BDT molecules ($99\text{\,}\mathrm{\char 37\relax}$ pure) are
commercially available. For the THF solvent anhydrous BHT stabilized THF of
$99.85\text{\,}\mathrm{\char 37\relax}$ purity with less than
$50\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ of water is used. The BDT
molecules are handled in a home build miniature glove-box continuously flowed
with Argon gas. This to ensure that the remaining BDT in the vial, used in
future experiments, stays stored under Argon as well as the BDT used for the
experiment ends up in a vial under Ar atmosphere. One vial of
$1\text{\,}\mathrm{m\textsc{m}}$ BDT/THF solution per MCB assembly is prepared
under Ar, using standard cannulation techniques for air sensitive reagents.
The anhydrous THF is handled with a syringe, making use of a septum sealed THF
bottle and septum sealed vials. Only at injecting the solution into the cell,
by a glass syringe with stainless needle the solution is exposed to air.

##  8 General discussion

Weather can have an influence on the measurements. A low humidity
($40\text{\,}\mathrm{\char 37\relax}$) leads to considerable shorter
measurement times. At low temperatures water may condensate for example on the
cold cell wall which will lead to a higher number of unsuccessful junctions
with a too low current start value in solution. The majority of measurements
in ref [2] have been performed at temperatures around
$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and at humidity’s in the
$50\text{\,}\mathrm{\char 37\relax}$ and $60\text{\,}\mathrm{\char 37\relax}$.

The use of a $\phi_{\text{in}}^{2}$ cell provides a more robust option against
weather influences as compared to the $\phi_{\text{in}}^{4.5}$ cell. Just
prior to the partially wet phase when there is only a minute amount of fluid
in the cell left the evaporation rate in both cell types is of a similar order
of magnitude. This is related to a small amount of fluid trapped somewhere,
the amount being similar for both cell types. THF vapor outweighs air, so the
$\phi_{\text{in}}^{2}$ cell with a 5 times smaller surface as compared to the
$\phi_{\text{in}}^{4.5}$ cell is more robust against diffusion processes
required for air to get to the bottom of the cell.

On rare occasions an air bubble remains at the bottom of the cell, below the
filled THF. Draining the last $\mathrm{mm}$ is in this case impossible as the
entire THF amount will be gone once the bubble bursts. There is not a lot that
can be done in this case. The junction is still in the partially wet phase
once the bubble bursts. If it concerns a BDT barrier junction there is still a
possibility to create a BDTH2O junction by immediately raising the humidity to
$95\text{\,}\mathrm{\char 37\relax}$ followed by continuous IV scanning as
described above.

Successive use of an assembly for two or more times is recommended although
often times not successful. As long as the current at $-5\text{\,}\mathrm{V}$
has not been reduced below approximately $-5\text{\,}\mathrm{nA}$ a re-fill
and re-drain procedure can be tried. If the current has fallen towards sub
$1\text{\,}\mathrm{nA}$ levels there is no use in retrying.

##  9 Conclusions

A breakdown of fabrication and process details has been provided for the
creation and operation of a single molecule junction in the partially wet
phase. Fabrication details describe material choices in combination with the
used solvent and molecule. Process details can be used to arrive at the
partially wet phase in a structured manner.

## References

  * [1] Daniel Bonn, Jens Eggers, Joseph Indekeu, Jacques Meunier, and Etienne Rolley.  Wetting and spreading.  Reviews of modern physics, 81(2):739, 2009.
  * [2] C. J. Muller.  Decoupling environmental modes from tunneling electrons in a partially wet phase molecular mechanically contrallable break junction, 2021.
  * [3] C. J. Muller.  A pivoting mechanically controllable break junction setup enabling partially wet phase mcb-junctions, 2021.
  * [4] C Zhou, CJ Muller, MR Deshpande, JW Sleight, and MA Reed.  Microfabrication of a mechanically controllable break junction in silicon.  Applied Physics Letters, 67(8):1160–1162, 1995.
  * [5] Arthur Von Hippel.  Molecular engineering.  Science, 123(3191):315–317, 1956.
  * [6] Arthur Von Hippel et al.  Molecular science and molecular engineering.  Published jointly by the Technology Press of MIT and J. Wiley, 1959.