# Holomorphic Yang-Mills fields on $B$-branes

Andrés Viña  Departamento de Física. Universidad de Oviedo. García Lorca 18.
33007 Oviedo. Spain<EMAIL_ADDRESS>
###### Abstract.

Considering the $B$-branes over a complex manifold $Y$ as objects of the
bounded derived category $D^{b}(Y)$, we define holomorphic gauge fields on
$B$-branes and the Yang-Mills functional for these fields. These definitions
are a generalization to $B$-branes of concepts that are well known in the
context of vector bundles. Given ${\mathscr{F}}^{\bullet}\in D^{b}(Y),$ we
show that the Atiyah class $a({\mathscr{F}}^{\bullet})\in{\rm
Ext}^{1}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}))$ is
the obstruction to the existence of gauge fields on ${\mathscr{F}}^{\bullet}.$
When $Y$ be either the projective space ${\mathbb{P}}^{n}$ or the variety of
complete flags in ${\mathbb{C}}^{3},$ we prove that the cardinal of the set of
holomorphic gauge fields on any $B$-brane over $Y$ is $\leq 1.$ We prove that
the set of Yang-Mills fields on the $B$-brane ${\mathscr{F}}^{\bullet},$ if it
is nonempty, is in bijective correspondence with the points of an algebraic
subset of ${\mathbb{C}}^{m}$ defined by $m$ polynomial equations of degree
$\leq 3$, where $m={\rm dim}\,{\rm
Ext}^{0}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet})).$

###### Key words and phrases:

Yang-Mills fields, holomorphic connections, coherent reflexive sheaves

MSC 2020: 53C05, 58E15, 18G10

##  1\. Introduction

In this article, we extend the well-known concepts of gauge field and Yang-
Mills field on vector bundles to $B$-branes. From the mathematical point of
view, a $B$-brane over a complex manifold $Y$ is an object of $D^{b}(Y)$, the
bounded derived category of coherent analytic sheaves over $Y$ [2, Sect. 5.4]
[3, Sect. 5.3].

The simplest $B$-banes are the holomorphic  vector bundles. A gauge field on a
holomorphic vector bundle $V\to M,$ in mathematical terms a connection on $V,$
defines a derivation of sections along vectors of $V$ giving rise to sections
not necessarily holomorphic. A holomorphic gauge field on $V,$ i., e.
holomorphic connection on $V$ [4], allows us to define a derivative of the
holomorphic sections of $V$ along any “direction”, giving rise to holomorphic
sections.

But no every holomorphic vector bundle admits a holomorphic connection. The
obstruction for this existence is the Atiyah class of the bundle [4]. Hence,
the vanishing of this class is a necessary condition for the existence of such
gauge fields on the vector bundle. Assuming that there is no obstruction to
the existence of holomorphic gauge fields, the set of these fields on a vector
bundle over a compact manifold is a finite dimensional affine space, in
contrast to the smooth case. This property is a consequence of the finite
dimensionality of the coherent cohomology groups.

The two features discussed above concerning the holomorphic connections on
vector bundles: the possible obstruction to its existence, and the finite
dimension of the space of those fields, when it is not empty, are also present
in the extension of the holomorphic gauge fields to general branes.

Gauge fields on coherent sheaves. A gauge field on $V$ determines an
isomorphism between the fibers over “close” points of $M$. In fact, the
connection is determined by the identification of the fibers at points
“infinitesimally close” in $M;$ i. e. by the identifications in the directions
defined by the tangent vectors of the base. The idea of being “infinitesimally
close” can be formulated by means of the first infinitesimal neighborhood of
the diagonal of $M$. In this setting, a gauge field on $V$ can be regarded as
a right inverse of the projection $\pi:J^{1}(V)\to V,$ where $J^{1}(V)$ is the
$1$-jet bundle of $V$ [30].

That approach admits a natural translation to the context of the coherent
sheaves over complex manifolds. In this way, if ${\mathscr{F}}$ is a coherent
sheaf over the compact analytic manifold $Y,$ we define a holomorphic gauge
field on ${\mathscr{F}}$ as a right inverse of the natural morphism
${\mathscr{J}}^{1}({\mathscr{F}})\to{\mathscr{F}}$, where
${\mathscr{J}}^{1}({\mathscr{F}})$ is the corresponding $1$-jet sheaf. This
inverse determines a morphism of abelian sheaves

$\nabla:{\mathscr{F}}\to\Omega^{1}({\mathscr{F}})=\Omega^{1}_{Y}\otimes_{{\mathscr{O}}_{Y}}{\mathscr{F}},$

which satisfies the Leibniz’s rule. In this context, the obstruction to the
existence of a holomorphic connection on the sheaf ${\mathscr{F}}$ is an
element $a({\mathscr{F}})$ of the group ${\rm
Ext}^{1}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))$. Furthermore, when the
set of holomorphic gauge fields on ${\mathscr{F}}$ is nonempty, it is an
affine space associated to the finite dimensional vector space ${\rm
Hom}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))$.

Yang-Mills fields on shaves. Setting ${\mathcal{S}}$ for the singularity set
of ${\mathscr{F}}$, let us assume that on the locally free sheaf
${\mathscr{F}}|_{Y\setminus{\mathcal{S}}}$ there is defined a Hermitian
metric, we say that ${\mathscr{F}}$ is a Hermitian sheaf. If $Y$ is a Kähler
manifold, on the set of holomorphic connections over ${\mathscr{F}}$, one
defines the Yang-Mills functional $\mathcal{YM}$. Essentially, the value of
$\mathcal{YM}$ at a connection $\nabla$ is the squared norm $\lVert
K_{\nabla}\rVert^{2}$ of the curvature of that connection [14, page 417], [27,
page 44].

The stationary points of functional $\mathcal{YM}$ are the Yang-Mills gauge
fields on ${\mathscr{F}}$. The set of critical points of this functional will
be denoted by ${\sf YM}({\mathscr{F}})$.

The reflexive sheaves might be thought as “vector bundles with singularities”
[16, page 121], and these singularities can in some way be avoided. From this
property, and denoting $m:={\rm dim}\,{\rm
Hom}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}})),$ we will prove that if the
reflexive sheaf ${\mathscr{F}}$ admits holomorphic gauge fields, then the set
${\sf YM}({\mathscr{F}})$ is in bijective correspondence with the points of an
algebraic subset of ${\mathbb{C}}^{m}$ defined by $m$ algebraic equations of
degree $\leq 3$ (Theorem 1). Thus, in general, the set of Yang-Mills fields on
${\mathscr{F}}$ will be a finite set.

Yang-Mills fields on a $B$-brane. As we said, a $B$-brane on $Y$ is a complex
$({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ of analytic coherent sheaves on
$Y.$ According to the preceding paragraphs, it is reasonable to define a gauge
field on this brane as an element of ${\rm
Hom}_{D^{b}(Y)}\big{(}{\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})\big{)}$
which lifts the identity on ${\mathscr{F}}^{\bullet}.$

Let $X$ be either the projective space ${\mathbb{P}}^{n}$ or the variety of
complete flags in ${\mathbb{C}}^{3}.$ We prove that the cardinal of the set of
holomorphic gauge fields on any $B$-brane over $X$ is $\leq 1$ (Theorems 3 and
4). For the proof of that result we use the existence of finite families of
objects in $D^{b}(X)$ which generate those categories. A well-known theorem of
Beilinson asserts that the set
$\\{{\mathscr{O}}_{{\mathbb{P}}^{n}}(-n),\,{\mathscr{O}}_{{\mathbb{P}}^{n}}(-n+1),\dots,{\mathscr{O}}_{{\mathbb{P}}^{n}}(-1),\,{\mathscr{O}}_{{\mathbb{P}}^{n}}\\}$
is a full exceptional family in $D^{b}({\mathbb{P}}^{n})$ [19, Sect 8.3].
However, for the proof of the above result we need to construct a more
appropriate generator set. For the case when $X$ is the mentioned flag
variety, we determine a family of generators $D^{b}(X)$ following an idea from
[5].

If $Y$ is a Hodge manifold, then it is a projective smooth variety and by the
GAGA correspondence the analytic coherent sheaves on $Y$ can be considered as
algebraic ones. Moreover, the category $D^{b}(Y)$ is equivalent to the
homotopy category $K^{b}(Y)_{\rm coh}$ of complexes of injective sheaves with
coherent and bounded cohomology. In this way, a gauge field $\psi$ on
${\mathscr{F}}^{\bullet}$ can be considered as a homotopy class of morphisms
between the complexes of abelian sheaves
${\mathscr{F}}^{\bullet}\to\widehat{\Omega^{1}({\mathscr{F}}^{\bullet})}$,
where $\widehat{\Omega^{1}({\mathscr{F}}^{\bullet})}$ is a complex in
$K^{b}(Y)_{\rm coh}$ quasi-isomorphic to
$\Omega^{1}({\mathscr{F}}^{\bullet})$. Thus, that homotopy class determines a
unique morphism between each of the cohomology sheaves
$\vartheta^{j}:{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\to{\mathscr{H}}^{j}(\Omega^{1}({\mathscr{F}}^{\bullet})).$
In summary, the gauge field on ${\mathscr{F}}^{\bullet}$ defines a family of
connections $\vartheta^{j}$ on the cohomology sheaves
${\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})$.

When the cohomology sheaves ${\mathscr{H}}^{j}$ are Hermitian, we define the
value of the Yang-Mills functional on the above gauge $\psi$ as
$\sum_{i}(-1)^{i}\lVert K_{\vartheta_{i}}\rVert^{2}.$ Thus, the Yang-Mills
functional is a kind of Euler characteristic of the gauge field. Obviously,
this definition of the Yang-Mills functional on branes generalizes the one
given for coherent sheaves.

The gauge field $\psi$ is a Yang-Mills field if it is a stationary point of
Yang-Mills functional. We will prove that if the $\vartheta^{j}$ are Yang-
Mills fields on the ${\mathscr{H}}^{j}$, so is $\psi$ on the brane
(Proposition 14). Theorem 6 is the converse to Proposition 14. In Theorem 5,
we generalize the result given in Theorem 1, about the cardinal of the set of
Yang-Mills fields on sheaves, to a general brane ${\mathscr{F}}^{\bullet}$.

The article is organized in two sections. In Section 2, we define the gauge
fields on a coherent sheaf. In the first subsections of this section, we
revise the definition of the $1$-jet sheaf of a coherent sheaf ${\mathscr{F}}$
on a complex manifold $Y,$ because, although this is well known in algebraic
geometry, it is not so well known in the community of mathematical physicists.
The Yang-Mills functional is introduced in Subsection 2.2, where Theorem 1 is
proved.

In Section 3 are considered the gauge fields on a general $B$-brane.
Subsection 3.1 defines these fields and discusses the obstruction to their
existence. Subsection 3.2 is devoted to the study of gauge fields on
$B$-branes over ${\mathbb{P}}^{n};$ in this subsection we prove the above
Theorem 3 concerning the cardinal of the set of such fields. Theorem 4, about
the cardinal of the set of fields on $B$-branes over the above flag variety,
is proved in Subsection 3.3. We show, in Subsection 3.4, how a gauge field on
a $B$-brane field gives rise to connections in cohomologies. Finally, we
define the Yang-Mills functional on gauge fields over general $B$-branes,
showing the reasons on which this definition is based. We also prove Theorems
5 and 6 and Proposition 14.

##  2\. Gauge fields on coherent sheaves

###  2.1. Gauge fields on a sheaf.

The definition of connection on a coherent sheaf, as it is introduced for
example in [23], is reformulated in this section, so that it is suitable for
translation to objects in the derived category of coherent sheaves.

As explained in the Introduction, the existence of a gauge field on a coherent
sheaf over a complex manifold $Y$ should define an isomorphism between the
stalks of the sheaf at any “infinitesimally close” points of $Y.$ In case Y is
an algebraic variety the neighborhoods in the Zariski topology are too large
to express the idea of “infinitesimal proximity”. Instead by means of the
first infinitesimal neighborhood $Y^{(1)}$ of $Y,$ [13, page 698], it is
possible to formulate this idea in a suitable way also for algebraic
varieties.

If $R$ is a ${\mathbb{C}}$-algebra, ${\rm Hom}({\rm Spec}\,R,\,X)$ is the set
of points of complex algebraic variety $X$ with values in $R$. Two points
$x,y$ are infinitesimally close if the morphism $(x,\,y):{\rm Spec}\,R\to
X\times X$ factorizes through $X^{(1)}$ [7, page 6]. Therefore, following
Deligne, one defines a gauge field on the coherent ${\mathscr{O}}_{X}$-module
${\mathscr{G}}$ as an element of

(2.1) ${\rm
Hom}_{X^{(1)}}(\pi_{1}^{*}{\mathscr{G}},\,\pi_{2}^{*}{\mathscr{G}}),$

which is the identity on $X$, where $\pi_{1},\pi_{2}:X^{(1)}\to X$ are the
projections and $\pi_{i}^{*}$ the corresponding inverse image functor. By the
adjunction isomorphism, a gauge field on ${\mathscr{G}}$ is a morphism
${\mathscr{G}}\to\pi_{1*}\pi^{*}_{2}{\mathscr{G}}$ which is a right inverse of
the projection $\pi:\pi_{1*}\pi^{*}_{2}{\mathscr{G}}\to{\mathscr{G}}.$

As $\pi_{1*}\pi^{*}_{2}{\mathscr{G}}$ is the first jet sheaf of the coherent
sheaf ${\mathscr{G}}$, in the following paragraphs we review the definition of
the jet sheaf and also that of first neighborhood of the diagonal.

####  2.1.1. Neighborhood of the diagonal

Let $Y$ be an analytic complex compact connected manifold. We denote by
$i:\Delta\hookrightarrow Y\times Y$ the embedding of the diagonal. As a closed
subvariety, $\Delta$ is defined by an ideal $\mathscr{I}$ of
$\hat{\mathscr{O}}:={\mathscr{O}}_{Y\times Y}.$ We will consider the following
ringed spaces

$(Y,\,{\mathscr{O}}),\;\;Y^{(1)}=\Big{(}\Delta,\,{\mathscr{O}}_{Y^{(1)}}:=\big{(}\hat{\mathscr{O}}/{\mathscr{I}}^{2}\big{)}|_{\Delta}\Big{)},\;\;(Y\times
Y,\,\hat{\mathscr{O}}).$

$Y^{(1)}$ is the first infinitesimal neighborhood of $\Delta$.

The ${\mathscr{O}}\mbox{-}{\mathscr{O}}$-bimodule structure in
${\mathscr{O}}_{Y^{(1)}}$. We set $p_{1},p_{2}:Y\times Y\rightrightarrows Y$
for the corresponding projection morphisms. The natural morphisms between the
above topological spaces are shown in the following commutative diagram

(2.2)
$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\scriptstyle{j}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{Y\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{Y}$

Given $f\in{\mathscr{O}}$ and $l\in\hat{\mathscr{O}}$, the product

(2.3) $(l+{\mathscr{I}}^{2})\cdot f=l\,(f\circ p_{2})+{\mathscr{I}}^{2}$

defines a right ${\mathscr{O}}$-module structure on
$\hat{\mathscr{O}}/{\mathscr{I}}^{2}$. More explicitly,
$(l(x,\,y)+{\mathscr{I}}^{2})\cdot f=l(x,\,y)f(y)+{\mathscr{I}}^{2}.$
Analogously, $f\cdot(l+{\mathscr{I}}^{2})=(f\circ p_{1})l+{\mathscr{I}}^{2}$
gives to $\hat{\mathscr{O}}/{\mathscr{I}}^{2}$ a left ${\mathscr{O}}$-module
structure. However, the restrictions to ${\mathscr{I}}/{\mathscr{I}}^{2}$ of
these left and right ${\mathscr{O}}$-module structures are equivalent.

The isomorphism ${\mathscr{O}}\oplus\Omega^{1}\simeq{\mathscr{O}}_{Y^{(1)}}$.
The cotangent sheaf $\Omega^{1}:=\Omega^{1}_{Y}$ can be identified with the
pullback $j^{-1}({\mathscr{I}}/{\mathscr{I}}^{2})$ [8, p. 407]. This
identification is defined by the correspondence

(2.4)
$\xi:j^{-1}({\mathscr{I}}/{\mathscr{I}}^{2})\to\Omega^{1},\;\;\;g+{\mathscr{I}}^{2}\mapsto\big{(}d_{x}g\big{)}|_{Y},$

where $d_{x}g$ is the exterior derivative of $g$ with respect to the variables
$x$; i.e. considering $g$ as function of the variables $x$ and keeping the
variables $y$ constant.

If $l\in\hat{\mathscr{O}}$, the correspondence $l+{\mathscr{I}}^{2}\mapsto
l\circ j\oplus(d_{x}l)|_{Y}$ defines an isomorphism of right
${\mathscr{O}}$-modules

(2.5)
$j^{-1}(\hat{\mathscr{O}}/{\mathscr{I}}^{2})\overset{m}{\simeq}{\mathscr{O}}\oplus\Omega^{1}.$

When the abelian sheaf ${\mathscr{O}}\oplus\Omega^{1}$ is endowed with the
left ${\mathscr{O}}$-action

(2.6) $f\cdot(h\oplus\alpha):=fh\oplus(hdf+f\alpha),$

where $\alpha\in\Omega^{1}$, then $m$ in (2.5) is an isomorphism of left
${\mathscr{O}}$-modules. We summarize those well-known results in the
following proposition.

######  Proposition 1.

With the notations above introduced:

  1. (1)

The ${\mathscr{O}}$-modules $j^{-1}({\mathscr{I}}/{\mathscr{I}}^{2})$ and
$\Omega^{1}$ are canonically isomorphic.

  2. (2)

The correspondence (2.5) defines an isomorphism between the right
${\mathcal{O}}$-modules $j^{-1}(\hat{\mathscr{O}}/{\mathscr{I}}^{2})$ and
${\mathcal{O}}\oplus{\Omega}^{1}$.

  3. (3)

Equipped ${\mathcal{O}}\oplus{\Omega}^{1}$ with the left
${\mathscr{O}}$-structure defined in (2.6), the correspondence (2.5) is also
an isomorphism of left ${\mathscr{O}}$-modules.

The exact sequence of right $\hat{\mathscr{O}}$-modules

$0\to{\mathscr{I}}/{\mathscr{I}}^{2}\to\hat{\mathscr{O}}/{\mathscr{I}}^{2}\to\hat{\mathscr{O}}/{\mathscr{I}}\to
0$

gives rise, by means the functor $j^{-1}$, the exact sequence of right
${\mathscr{O}}$-modules

(2.7) $0\to j^{-1}({\mathscr{I}}/{\mathscr{I}}^{2})\to
j^{-1}(\hat{\mathscr{O}}/{\mathscr{I}}^{2})\to
j^{-1}(\hat{\mathscr{O}}/{\mathscr{I}})\to 0,$

or in other terms
$0\to\Omega^{1}\to{\mathscr{O}}\oplus\Omega^{1}\to{\mathscr{O}}\to 0.$

####  2.1.2. The first jet sheaf

We denote by $\pi_{a}=p_{a}\circ i:\Delta\to Y$, for $a=1,2$. One has the
following morphism of sheaves rings over $\Delta$ (see Subsection 2.1.1)

$\pi_{a}^{-1}{\mathscr{O}}\longrightarrow{\mathscr{O}}_{Y^{(1)}}={\hat{\mathscr{O}}}/{\mathscr{I}^{2}},\;\;\;\;h\mapsto
h\circ\pi_{a}+{\mathscr{I}}^{2}.$

In particular, one can consider ${\mathscr{O}}_{Y^{(1)}}$ as a right
${\pi_{2}^{-1}{\mathscr{O}}}$-module.

Given ${\mathscr{F}}$ a left ${\mathscr{O}}$-module on $Y,$ its inverse image
by $\pi_{2}$ is left ${\mathscr{O}}_{Y^{(1)}}$-module

$\pi_{2}^{*}({\mathscr{F}})={\mathscr{O}}_{Y^{(1)}}\otimes_{\pi_{2}^{-1}{\mathscr{O}}}\pi_{2}^{-1}{\mathscr{F}}.$

And the first jet sheaf ${\mathscr{J}}^{1}({\mathscr{F}})$ of ${\mathscr{F}}$
is left ${\mathscr{O}}$-module defined by

(2.8)
${\mathscr{J}}^{1}({\mathscr{F}})={\pi_{1}}_{*}\pi_{2}^{*}({\mathscr{F}}).$

Since $\pi_{1}^{*}$ is the left adjoint of ${\pi_{1}}_{*}$, one has

(2.9) ${\rm
Hom}\,_{{\mathscr{O}}}\big{(}{\mathscr{F}},\,{\mathscr{J}}^{1}({\mathscr{F}})\big{)}={\rm
Hom}\,_{{\mathscr{O}}_{Y^{(1)}}}\big{(}\pi_{1}^{*}{\mathscr{F}},\,\pi_{2}^{*}{\mathscr{F}}\big{)}.$

If ${\mathscr{H}}$ is a left ${\mathscr{O}}_{Y^{(1)}}$-module, the
${\mathscr{O}}$-structure on ${\pi_{1}}_{*}{\mathscr{H}}$ is defined by
$f\cdot s=(f\circ\pi_{1})s$, where $f\in{\mathscr{O}}$ and
$s\in{\mathscr{H}}$. On the other hand, the ${\mathscr{O}}$ action on

$k^{*}{\mathscr{H}}={\mathscr{O}}\otimes_{k^{-1}{\mathscr{O}}_{\Delta}}k^{-1}{\mathscr{H}}$

is determined by $f\cdot(1\otimes s)=f\otimes s=1\otimes
gs=1\otimes(f\circ\pi_{1})s$, where $f=g\circ k$. Hence, $k^{*}{\mathscr{H}}$
and ${\pi_{1}}_{*}{\mathscr{H}}$ are isomorphic ${\mathscr{O}}$-modules. Thus,
as $\pi_{2}\circ k={\rm id},$

(2.10)
${\mathscr{J}}^{1}({\mathscr{F}})=k^{-1}({\mathscr{O}}_{Y^{(1)}})\otimes_{{\mathscr{O}}}{\mathscr{F}}=j^{-1}(\hat{\mathscr{O}}/{\mathscr{I}}^{2})\otimes_{{\mathscr{O}}}{\mathscr{F}}.$

By Proposition 1, ${\mathscr{J}}^{1}({\mathscr{F}})$ is the abelian sheaf
${\mathscr{F}}\oplus\Omega^{1}({\mathscr{F}})$ endowed with the following left
${\mathscr{O}}$-module structure

(2.11) $f\ast(\sigma\oplus\beta)=f\sigma\oplus(f\beta+df\otimes\sigma).$

One has the morphism of abelian sheaves

$\eta:{\mathscr{F}}\to{\mathscr{J}}^{1}({\mathscr{F}})={\mathscr{F}}\oplus\Omega^{1}({\mathscr{F}}),\;\;\sigma\mapsto\sigma\oplus
0.$

And from (2.11), it follows

(2.12) $f\eta(\sigma)=\eta(f\sigma)+df\otimes\sigma.$

Summarizing, one has the exact sequence of ${\mathscr{O}}$-modules

(2.13)
$0\to\Omega^{1}({\mathscr{F}})\overset{\iota}{\to}{\mathscr{J}}^{1}({\mathscr{F}})\overset{\pi}{\to}{\mathscr{F}}\to
0,$

Thus, taking into account (2.1) and (2.9), a gauge field on ${\mathscr{F}}$
will be an element $\psi\in{\rm
Hom}({\mathscr{F}},\,{\mathscr{J}}^{1}({\mathscr{F}}))$ right inverse of
$\pi$. That is, we give the following definition.

######  Definition 1.

A holomorphic gauge field on the coherent sheaf ${\mathscr{F}}$ is a right
inverse of the ${\mathscr{O}}$-module morphism
$\pi:{\mathscr{J}}^{1}({\mathscr{F}})\to{\mathscr{F}}.$

We introduce a piece of notation that will be used later on. Let ${\sf Forg}$
denote the forgetful functor from the category of ${\mathscr{O}}$-modules to
the category of abelian sheaves over $Y.$ We set

$\boldsymbol{\mathscr{J}}^{1}({\mathscr{F}})={\sf
Forg}\big{(}{\mathscr{F}}\oplus\Omega^{1}({\mathscr{F}})\big{)}.$

In this way ${\mathscr{J}}^{1}({\mathscr{F}})$ is the abelian sheaf
$\boldsymbol{\mathscr{J}}^{1}({\mathscr{F}})$ endowed with the
${\mathscr{O}}$-module structure defined in (2.11).

####  2.1.3. The Atiyah class.

The exact sequence (2.13) defines an element $a({\mathscr{F}})$ of ${\rm
Ext}^{1}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))$, called the Atiyah class
of ${\mathscr{F}}$. The exact sequence (2.13) does not split, in general, in
the category of ${\mathscr{O}}$-modules. Thus, there will be exist gauge
fields on ${\mathscr{F}}$ iff the Atiyah class $a({\mathscr{F}})$ vanishes.

If there exists a right inverse $\psi$ of $\pi$; then $\pi(\psi-\eta)=0$; and
thus $\nabla:=\psi-\eta$ takes its values in $\Omega^{1}({\mathscr{F}})$.
Moreover, by (2.12), for $f\in{\mathscr{O}}$ and $\sigma\in{\mathscr{F}}$

(2.14)
$\nabla(f\sigma)=f\psi(\sigma)-f\eta(\sigma)+df\otimes\sigma=f\nabla(\sigma)+df\otimes\sigma;$

that is, for $\nabla$ holds the Leibniz’s rule satisfied by the connections on
vector bundles.

We denote by ${\mathcal{C}}$ the set consisting of all morphisms of
${\mathbb{C}}_{Y}$-modules $\nabla:{\mathscr{F}}\to\Omega^{1}({\mathscr{F}})$
satisfying (2.14). An element $\nabla\in{\mathcal{C}}$ determines the morphism
of ${\mathscr{O}}$-modules

$\kappa:{\mathscr{F}}\oplus\Omega^{1}({\mathscr{F}})\to\Omega^{1}({\mathscr{F}}),\;\;(\sigma,\beta)\mapsto\beta-\nabla(\sigma).$

$\kappa$ is a left inverse of $\iota$. Hence, $\kappa$ defines a splitting of
the extension (2.13) and thus it determines a right inverse of $\pi$; i. e. a
holomorphic gauge field $\psi$ on ${\mathscr{F}}$. We have proved the
following proposition.

######  Proposition 2.

If $a({\mathscr{F}})=0$, then the map $\nabla\mapsto\nabla+\eta$ defines a
bijective correspondence between ${\mathcal{C}}$ and the set of holomorphic
gauge fields on the coherent sheaf ${\mathscr{F}}$.

######  Proposition 3.

The set of holomorphic gauge fields on the coherent sheaf ${\mathscr{F}}$, if
it is nonempty, is an affine space associated to the finite dimensional vector
space
$\Gamma\big{(}Y,\,{\mathscr{H}om}({\mathscr{F}},\Omega^{1}({\mathscr{F}}))\big{)}.$

Proof. The exact sequence (2.13) gives rise to corresponding ${\rm
Ext}$-sequence

$0\to{\rm
Hom}({\mathscr{F}},\Omega^{1}({\mathscr{F}}))\overset{\lambda}{\to}{\rm
Hom}({\mathscr{F}},{\mathscr{J}}^{1}({\mathscr{F}}))\overset{\mu}{\to}{\rm
Hom}({\mathscr{F}},{\mathscr{F}})\overset{\nu}{\to}{\rm
Ext}^{1}({\mathscr{F}},\Omega^{1}({\mathscr{F}})).$

Since $\mu(\phi)=\pi\circ\phi$, the existence of a holomorphic gauge field
$\psi$ on ${\mathscr{F}}$, is equivalent to $1_{\mathscr{F}}\in{\rm
im}(\mu)={\ker}(\nu)$. In fact, the Atiyah class $a({\mathscr{F}})$ is the
image of $1_{\mathscr{F}}$ by $\nu$.

If $\psi$ and $\psi_{1}$ are gauge fields on the coherent sheaf
${\mathscr{F}}$, then $\mu(\psi_{1}-\psi)=0$; i. e. $\psi_{1}-\psi\in{\rm
im}(\lambda)$. Thus, the set of holomorphic gauge fields on ${\mathscr{F}}$,
if nonempty, is an affine space with vector space ${\rm
Hom}({\mathscr{F}},\Omega^{1}({\mathscr{F}})).$ This vector space is the space
of sections
$\Gamma(Y,\,{\mathscr{H}om}({\mathscr{F}},\Omega^{1}({\mathscr{F}}))$. As
${\mathscr{H}om}({\mathscr{F}},\Omega^{1}({\mathscr{F}}))$ is a coherent
sheaf, the space of holomorphic gauge fields on ${\mathscr{F}}$, is a finite
dimensional affine space [13, page 700 ]. ∎

######  Remark 1.

Taking into account the isomorphism [4, Prop. 2]

${\rm Ext}^{1}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))\simeq
H^{1}(Y,\,{\mathscr{H}om}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))),$

$a({\mathscr{F}})$ can be considered as a cohomology class. In particular,
when ${\mathscr{F}}$ is a locally free sheaf,
${\mathscr{H}om}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))\simeq\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{E}nd}({\mathscr{F}})$,
and $a({\mathscr{F}})$ is defined by a $\check{\rm C}$ech cocycle with
coefficients in
$\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{E}nd}({\mathscr{F}})$ (see [18, page
180]).

###  2.2. Holomorphic Yang-Mills fields

####  2.2.1. Hermitian sheaves

The fiber at $x\in Y$ of a coherent ${\mathscr{O}}$-module ${\mathscr{G}}$
will be denoted by
${\mathscr{G}}_{(x)}:={\mathscr{G}}_{x}/{\mathfrak{m}}_{x}{\mathscr{G}}_{x}$,
where ${\mathfrak{m}}_{x}$ is the maximal ideal of ${\mathscr{O}}_{x}$. If
${\mathcal{Z}}$ is a section of ${\mathscr{G}}$ the corresponding vector in
${\mathscr{G}}_{(x)}$ is denoted by $Z(x)$. On the other hand, the singular
set ${\mathcal{S}}$ of ${\mathscr{G}}$ is a closed analytic subset of $Y$
whose codimension is greater than or equal to $1$ [25, Chap. V, Thm. 5.8].
Moreover, ${\mathscr{G}}$ is locally free on $Y\setminus{\mathcal{S}}$. We set
${G}$ for the vector bundle over $Y\setminus{\mathcal{S}}$, with fibers
$G(x):={\mathscr{G}}_{(x)}$, determined by the locally free sheaf
${\mathscr{G}}|_{Y\setminus{\mathcal{S}}}$.

The following definition generalizes the given in [32, Chap. III, Sect 1] for
locally free sheaves.

######  Definition 2.

A Hermitian metric on the coherent sheaf ${\mathscr{G}}$ is a set
$\\{\langle\,,\,\rangle_{x}\\}_{x\in Y}$ of Hermitian metrics on the fibers of
${\mathscr{G}}$, satisfying the following condition: Given
${\mathcal{Z}}_{1},{\mathcal{Z}}_{2}$ two sections of ${\mathscr{G}}$ on an
open $U$ of $Y,$ the map

${\mathcal{Z}}_{1}\cdot{\mathcal{Z}}_{2}:x\in U\mapsto\langle
Z_{1}(x),\,Z_{2}(x)\rangle_{x}\in{\mathbb{C}}$

is bounded, and its restriction to $U\setminus{\mathcal{S}}$ is $C^{\infty}.$
A sheaf endowed with a Hermitian metric is called a Hermitian sheaf.

Given ${\mathcal{Z}}_{1},{\mathcal{Z}}_{2}\in\Gamma(Y,\,{\mathscr{G}})$,
${\mathcal{Z}}_{1}\cdot{\mathcal{Z}}_{2}$ is a bounded map with possible
discontinuities in the singular locus. If $Y$ is a Kähler manifold, the
function ${\mathcal{Z}}_{1}\cdot{\mathcal{Z}}_{2}$ may be non-integrable on
$Y$ with respect to the volume form defined by the Kähler metric.

If ${\mathscr{G}}$ is a torsion-free coherent sheaf on the Kähler manifold
$Y,$ then ${\rm codim}\,{\mathcal{S}}\geq 2$ [25, Cap. V, Cor. 5.15]. Hence,
there exists an analytic closed subset ${\mathcal{W}}$ of $Y,$ containing
${\mathcal{S}},$ and such that $1\leq{\rm codim}\,{\mathcal{W}}<{\rm
codim}\,{\mathcal{S}}.$ Let ${\mathcal{N}}$ be a tubular neighborhood of
${\mathcal{S}}$ in ${\mathcal{W}}.$ Then

  1. (1)

${\mathcal{Z}}_{1}\cdot{\mathcal{Z}}_{2}$ is a continuous map on the closed
space $Y\setminus{\mathcal{N}},$ which is disjoint with the singularity set
${\mathcal{S}}.$

  2. (2)

${\mathcal{W}},$ as a subset of $Y$ with codimension $\geq 1,$ has measure
zero with respect to the differential form ${\rm d}{\rm vol}.$

Thus, the following integral is well-defined

(2.15)
$({\mathcal{Z}}_{1},\,{\mathcal{Z}}_{2}):=\int_{Y\setminus{\mathcal{N}}}{\mathcal{Z}}_{1}\cdot{\mathcal{Z}}_{2}\,{\rm
d}{\rm vol}.$

From the boundedness of ${\mathcal{Z}}_{1}\cdot{\mathcal{Z}}_{2}$ together
with $(2),$ it follows that this definition does not depend on the choices of
the analytic closed set ${\mathcal{W}}$ and the neighborhood ${\mathcal{N}}.$

######  Remark 2.

A Hermitian structure $\langle\,,\,\rangle$ on a coherent sheaf
${\mathscr{F}}$ determines a Hermitian metric on the coherent sheaf
${\mathscr{E}nd}({\mathscr{F}})$ in a natural way. On the other hand, if $Y$
is a Kähler manifold, the metric defines a Hermitian structure on the locally
free sheaf $\Omega^{k}$ of holomorphic $k$-forms. Thus, if ${\mathscr{F}}$ is
a Hermitian sheaf over a Kähler manifold, we have a Hermitian metric on the
sheaf $\Omega^{k}\otimes_{\mathscr{O}}{\mathscr{E}nd}({\mathscr{F}}),$ which
will also denoted $\langle\,,\,\rangle$.

####  2.2.2. Yang-Mills functional

Given $\nabla$ a holomorphic connection on the coherent sheaf $\mathscr{F},$
it defines a morphism of ${\mathbb{C}}_{Y}$-modules
$\nabla^{(k)}:\Omega^{k}({\mathscr{F}}){\to}\Omega^{k+1}({\mathscr{F}})$ in
the usual way. The composition
${\mathcal{K}}_{\nabla}:=\nabla^{(1)}\circ\nabla:{\mathscr{F}}\to\Omega^{2}({\mathscr{F}})$
is the curvature of $\nabla$; and as it is well-known

(2.16) ${\mathcal{K}}_{\nabla}\in{\rm
Hom}({\mathscr{F}},\,\Omega^{2}({\mathscr{F}}))=\Gamma(Y,\,{\mathscr{H}om}({\mathscr{F}},\,\Omega^{2}({\mathscr{F}}))).$

The connection is said to be flat if ${\mathcal{K}}_{\nabla}=0$.

For each $x\in Y,$ we denote by $\alpha_{x}$ and $\lambda_{x}$ the natural
morphisms

$\big{(}{\mathscr{H}om}_{\mathscr{O}}({\mathscr{F}},\,\Omega^{k}({\mathscr{F}}))\big{)}_{x}\overset{\alpha_{x}}{\to}{\rm
Hom}_{{\mathscr{O}}_{x}}\big{(}{\mathscr{F}}_{x},\,\Omega^{k}_{x}\otimes_{{\mathscr{O}}_{x}}{\mathscr{F}}_{x}\big{)}\overset{\lambda_{x}}{\leftarrow}\Omega^{k}_{x}\otimes_{{\mathscr{O}}_{x}}{\rm
End}_{{\mathscr{O}}_{x}}({\mathscr{F}}_{x}).$

As ${\mathscr{F}}$ is coherent, $\alpha_{x}$ is isomorphism [12, page 239].
Furthermore, if ${\mathscr{F}}_{x}$ is free, then $\lambda_{x}$ is bijective.

If ${\mathcal{S}}$ is the singularity set of ${\mathscr{F}},$ it is also the
singular locus of
${\mathscr{H}om}({\mathscr{F}},\,\Omega^{2}({\mathscr{F}})).$ Hence, for each
point $x$ outside of ${\mathcal{S}},$ the fiber of
${\mathscr{H}om}({\mathscr{F}},\,\Omega^{k}({\mathscr{F}}))$ at $x$ can be
identified with the vector space $\Omega^{k}{(x)}\otimes{\rm End}({F}{(x)}).$
According to (2.16), the curvature ${\mathcal{K}}_{\nabla}$ of a holomorphic
connection $\nabla$ determines the vector

(2.17) $K_{\nabla}(x)\in\Omega^{2}(x)\otimes{\rm End}(F(x))$

for each $x\in Y\setminus{\mathcal{S}}$. That is, $K_{\nabla}$ is a $2$-form
${\rm End}(F)$-valued.

Let us assume that ${\mathscr{F}}$ is a Hermitian torsion-free sheaf on the
Kähler manifold $Y,$ according to (2.15), one defines

(2.18)
$\lVert{\mathcal{K}}_{\nabla}\rVert^{2}=({\mathcal{K}}_{\nabla},\,{\mathcal{K}}_{\nabla})=\int_{Y\setminus{\mathcal{N}}}{\mathcal{K}}_{\nabla}\cdot{\mathcal{K}}_{\nabla}\,{\rm
d}{\rm
vol}=\int_{Y\setminus{\mathcal{N}}}|K_{\nabla}\wedge\star\,K_{\nabla}|,$

where $|\,\cdot\,|$ is the corresponding norm on ${\rm End}(F)$ and $\star$ is
the Hodge star operator.

More concretely, if locally $K_{\nabla}$ can be expressed as $\alpha\otimes
A$, with $\alpha$ a $2$-form and $A$ a local section of ${\rm End}(F)$, then
the integrand in (2.18) is $(\alpha\wedge\star\alpha)\langle A\circ A\rangle$.
In a local unitary frame of ${\rm End}(F)$, if the connection is compatilble
with the metric, the matrix $\check{A}$ associated to $A$ is antihermitian and
$\langle A\circ A\rangle=-{\rm tr}(\check{A}\check{A})$. That is,

(2.19) $\langle K_{\nabla}\wedge\star\,K_{\nabla}\rangle=-{\rm
tr}\big{(}K_{\nabla}\wedge\star\,K_{\nabla}\big{)}.$

On the space of holomorphic gauge fields on the torsion-free sheaf
${\mathscr{F}}$ one can define the following map

$\mathcal{YM}:\nabla\in{\rm
Hom}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))\mapsto\lVert{\mathcal{K}}_{\nabla}\rVert^{2}=\int_{Y\setminus{\mathcal{N}}}{\mathcal{K}}_{\nabla}\cdot{\mathcal{K}}_{\nabla}\,{\rm
d}{\rm vol}.$

It is called the Yang-Mills’ functional. The $\nabla$ on which this functional
takes a stationary value are the holomorphic Yang-Mills fields.

######  Remark 3.

If ${\mathscr{F}}$ admits a flat connection, then ${\mathscr{F}}$ is a locally
free sheaf [10, Chap. 8, Prop. 2.3], [20, Thm. 1.4.10], [23, Prop. 8.8]. In
this case the minimum value taken by the corresponding Yang-Mills functional
is obviously zero. Conversely, if ${\mathscr{F}}$ is locally free, then the
minimum of the respective Yang-Mills functional is zero as well.

####  2.2.3. Yang-Mills fields on reflexive sheaves

If ${\mathscr{G}}$ is a reflexive sheaf on $Y,$ then it is torsion-free and
the codimension of the singular set is $\geq 3$ [25, Cap. V, Cor. 5.20], [16,
Cor. 1.4]. Furthermore, if $C$ is a closed subset of $Y$ with codimension
$\geq 2,$ then the restriction $\Gamma(Y,\,{\mathscr{G}})\to\Gamma(Y\setminus
C,\,{\mathscr{G}})$ is an isomorphism [25, Cap.V, Prop. 5.21] [17, Prop.
1.11].

Let us suppose that ${\mathscr{F}}$ is a reflexive sheaf on the Kähler
manifold $Y$ endowed with a Hermitian metric. Then
${\mathscr{E}nd}({\mathscr{F}})$ is also a reflexive sheaf [25, Chap. V, Prop.
4.15]. If ${\mathcal{S}}$ is the singular locus of ${\mathscr{F}},$ let
${\mathcal{W}}$ be a closed subspace of $Y$ containing ${\mathcal{S}}$ and
such that ${\rm codim}\,{\mathcal{W}}=2.$ Let ${\mathcal{N}}$ denote a tubular
neighborhood of ${\mathcal{S}}$ in ${\mathcal{W}}.$ Therefore, the vector
space
$\Gamma(Y\setminus{\mathcal{N}},\,\Omega^{k}({\mathscr{E}nd}({\mathscr{F}})))\simeq\Gamma(Y,\,\Omega^{k}({\mathscr{E}nd}({\mathscr{F}}))),$
and the latter is finite dimensional. Furthermore, as
$(Y\setminus{\mathcal{N}})\cap{\mathcal{S}}=\emptyset,$
${\mathscr{H}om}({\mathscr{F}},\,\Omega^{k}({\mathscr{F}}))|_{Y\setminus{\mathcal{N}}}$
is a locally free sheaf.

Let $\nabla$ be a holomorphic gauge field on ${\mathscr{F}}$, then

${\mathcal{K}}_{\nabla}\in\Gamma\big{(}Y,\,{\mathscr{H}om}({\mathscr{F}},\,\Omega^{2}({\mathscr{F}}))\big{)}=\Gamma(Y\setminus{\mathcal{N}},\,{\mathscr{H}om}({\mathscr{F}},\,\Omega^{2}({\mathscr{F}}))\big{)}$

can considered as a global section of the locally free sheaf on
$Y\setminus{\mathcal{S}}\,$ $\Omega^{2}\otimes{\rm End}(F).$ A similar
observation is also valid for $\nabla.$

Let us assume that the sheaf ${\mathscr{F}}$ supports a holomorphic gauge
field $\nabla_{0}$. By Proposition 3, given
${\mathcal{E}}_{1},\dots\mathcal{E}_{m}$, a basis of

${\rm
Hom}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))=\Gamma\big{(}Y,\,{\mathscr{H}om}({\mathscr{F}},\Omega^{1}({\mathscr{F}}))\big{)}=\Gamma(Y\setminus{\mathcal{S}},\,\Omega^{1}\otimes{\rm
End}(F)),$

any holomorphic gauge field can be written
$\nabla=\nabla_{0}+\sum\lambda_{i}{\mathcal{E}}_{i},$ with
$\lambda_{i}\in{\mathbb{C}}$. The curvature

(2.20)
${\mathcal{K}}_{\nabla}=\nabla\circ\nabla={\mathcal{K}}_{\nabla_{0}}+\sum_{i}\lambda_{i}{\mathcal{B}}_{i}+\sum_{ij}\lambda_{i}\lambda_{j}{\mathcal{B}}_{ij},$

where ${\mathcal{B}}_{i}:=\nabla_{0}({\mathcal{E}}_{i})$ and
${\mathcal{B}}_{ij}:={\mathcal{E}}_{i}\wedge{\mathcal{E}}_{j}$. Thus,

$\lVert{\mathcal{K}}_{\nabla}\rVert^{2}=({\mathcal{K}}_{\nabla},\,{\mathcal{K}}_{\nabla})=P(\lambda_{1},\dots,\lambda_{m}),$

where $P$ is a polynomial of degree $\leq 4$ in the variables $\lambda_{i}$.

The Yang-Mills fields are those $\nabla$ defined by constants $\lambda_{i}$
which satisfy the algebraic equations of degree $\leq 3$

(2.21) $\frac{\partial\,P}{\partial\lambda_{i}}=0,\;\;\;i=1,\dots,m.$

Therefore,

######  Theorem 1.

If the reflexive sheaf ${\mathscr{F}}$ admits a holomorphic gauge field and

$m={\rm dim}\,{\rm Hom}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}})).$

Then the set ${\sf YM}({\mathscr{F}})$ of holomorphic Yang-Mills fields on
${\mathscr{F}}$ is in bijective correspondence with the points of the
algebraic set in ${\mathbb{C}}^{m}$ defined by $m$ algebraic equations.

In particular, if $m=2$ and the cardinal of ${\sf YM}({\mathscr{F}})$ is
finite, then $\\#{\sf YM}({\mathscr{F}})\leq 9$.

Proof. The case $m=2$ is a consequence of Bézout’s theorem. ∎

If $\nabla$ is a holomorphic Yang-Mills field, given ${\mathcal{E}}\in{\rm
Hom}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))$, then $\nabla+{\mathcal{E}}$
is a new connection. Conversely, the difference of two connections on
${\mathscr{F}}$ is a global section of the sheaf
${\mathscr{H}om}({\mathscr{F}},\,\Omega^{1}({\mathscr{F}}))$. Thus, one can
consider the following “variation” of $\nabla$,
$\nabla_{\epsilon}=\nabla+\epsilon\,{\mathcal{E}},$ with
$\epsilon\in{\mathbb{C}}$. Conversely, if $\nabla$ and $\hat{\nabla}$ are
holomorphic connections on the Hermitian sheaf ${\mathscr{F}}$, then
$\nabla-\hat{\nabla}$ is determined by a vector
$\Gamma(Y,\,\Omega^{1}\otimes{\rm End}({F})).$

Considering a variation $\nabla_{\epsilon}$ as above, then
$K_{\nabla_{\epsilon}}=K_{\nabla}+\epsilon\nabla E+O(\epsilon^{2})$, where $E$
is the vector of $\Gamma(Y,\,\Omega^{1}\otimes{\rm End}(F)),$ defined by
${\mathcal{E}}$, and $\nabla E$ is the covariant derivative of $E$.

(2.22)
$(1/2)\frac{d}{d\epsilon}\Big{|}_{\epsilon=0}||K_{\nabla_{\epsilon}}||^{2}=\int_{Y\setminus{\mathcal{N}}}\langle
K_{\nabla},\,\nabla E\rangle\,{\rm dvol}=:(K_{\nabla},\,\nabla{E}).$

Therefore, $\nabla$ is a Yang-Mills field if for any “variation”
${\mathcal{E}}$ of $\nabla$

(2.23) $(K_{\nabla},\,\nabla{E})=0.$

In particular, the flat holomorphic gauge fields are Yang-Mills.

We denote

${}^{(p)}\nabla:\Gamma(Y,\,\Omega^{p}({\mathscr{E}nd}({\mathscr{F}})))\to\Gamma(Y,\,\Omega^{p+1}({\mathscr{E}nd}({\mathscr{F}}))),$

the operator defined by the connection $\nabla$. In this notation Bianchi’s
identity is read as

${}^{(2)}\nabla K_{\nabla}=0.$

On the other hand, the orthogonality condition (2.23) which satisfy the Yang-
Mills fields gives rise to the following proposition.

######  Proposition 4.

The holomorphic gauge field $\nabla$ on the reflexive sheaf ${\mathscr{F}}$ is
a Yang-Mills field iff its curvature
$K_{\nabla}\in\Gamma(Y,\,\Omega^{2}\otimes_{\mathscr{O}}{\rm End}(F))$ is
orthogonal to the vector space ${\rm im}\,(^{(1)}\nabla)$.

####  2.2.4. The case ${\rm rank}\,{\mathscr{F}}=1$.

In this case the reflexive sheaf ${\mathscr{F}}$ is a locally free sheaf [16,
Prop. 1.9]. Then ${\mathscr{E}nd}({\mathscr{F}})$ is the sheaf associated to
the trivial line bundle ${\mathbb{C}}\times Y\to Y$. If $s$ is a local frame
of the corresponding line bundle $F,$ a holomorphic connection $\nabla$ on
${\mathscr{F}}$ on this frame is determined by a ${\mathbb{C}}$-valued
$1$-form $A$, $\nabla s=As$. In this frame
${}^{(p)}\nabla(\beta)=\partial\beta+A\wedge\beta-(-1)^{p}\beta\wedge
A=\partial\beta,$ for any ${\mathbb{C}}$-valued $p$-form $\beta$. That is,

(2.24) $^{(p)}\nabla=\partial.$

In this case (2.20) reduces to $K_{\nabla}=K_{\nabla_{0}}$, since
$\nabla_{0}(E_{i})=\partial E_{i}=0.$ In particular, the Yang-Mills functional
is constant.

On the other hand, the Bianchi’s identity reduces to $\partial K_{\nabla}=0$.
If furthermore, $\nabla$ is a Yang-Mills field, from (2.23) it follows
$\partial^{\dagger}(K_{\nabla})=0$, where $\partial^{\dagger}$ is the adjoint
of $\partial$. That is, $K_{\nabla}$ is $\partial$-harmonic. As $Y$ is a
Kähler manifold, $K_{\nabla}$ is also $d$-harmonic. Hence, the norm of
$K_{\nabla}$ minimizes the corresponding norm in its cohomology class. That
is, denoting by ${\bf c}$ the cohomology class defined by $K_{\nabla},$

(2.25) $\mathcal{YM}(\nabla)=\lVert
K_{\nabla}\rVert^{2}={\min}\\{\lVert\beta\rVert^{2}\,|\,\beta\in{\bf c}\\}.$

As we are assuming that ${\mathscr{F}}$ supports a holomorphic gauge field,
the first Chern class $c_{1}({\mathscr{F}})=[(2\pi)^{-1}K_{\nabla}]$ vanishes
[4]. Consequently, ${\bf c}=0$ and $\mathcal{YM}$ is the functional zero.

##  3\. Fields on a brane

###  3.1. Gauge fields on a $B$-brane.

Let $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ be a $B$-brane on the
complex manifold $Y$; that is, ${\mathscr{F}}^{\bullet}$ is an object of the
category $D^{b}(Y)$, the bounded derived category of coherent sheaves over
$Y$. According to the observation at the beginning of Subsection 2.1 (see
(2.1)), we define a gauge field on the brane ${\mathscr{F}}^{\bullet}$ as an
element of ${\rm
Hom}_{D^{b}(Y^{(1)})}(L\pi_{1}^{*}{\mathscr{F}}^{\bullet},\,L\pi_{2}^{*}{\mathscr{F}}^{\bullet})$
which lifts the identity on $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$. We
explain below this condition. By the adjunction relation

(3.1) ${\rm
Hom}_{D^{b}(Y^{(1)})}(L\pi_{1}^{*}{\mathscr{F}}^{\bullet},\,L\pi_{2}^{*}{\mathscr{F}}^{\bullet})\simeq{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})),$

where

${\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}):=R\pi_{1*}L\pi_{2}^{*}{\mathscr{F}}^{\bullet}\simeq{\mathscr{O}}_{Y^{(1)}}\otimes^{L}{\mathscr{F}}^{\bullet}.$

Therefore, the gauge fields on ${\mathscr{F}}^{\bullet}$ are elements of the
group

${\rm
Ext}^{0}({\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}));$

i.e., open strings between ${\mathscr{F}}^{\bullet}$ and
${\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})$ with ghost number $0$ [2, Sect.
5.2], [24].

As ${\mathscr{O}}_{Y^{(1)}}$ is the locally free module
${\mathscr{O}}\oplus\Omega^{1}$, then
${\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})$ is the complex of abelian sheaves
$\sf{Forg}\big{(}{\mathscr{F}}^{\bullet}\oplus\Omega^{1}({\mathscr{F}}^{\bullet})\big{)},$
with ${\mathscr{O}}$-structure given by

(3.2)
$f\ast(\sigma^{\bullet}\oplus\alpha^{\bullet})=f\sigma^{\bullet}\oplus(df\otimes\sigma^{\bullet}+f\alpha^{\bullet}).$

More precisely, one can consider the complex of abelian sheaves direct sum of
${\mathscr{F}}^{\bullet}$ and $\Omega^{1}({\mathscr{F}}^{\bullet}).$ According
to the notation introduced at the end of Subsection 2.1.2, we write

(3.3) $\Big{(}\boldsymbol{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}):={\sf
Forg}\big{(}{\mathscr{F}}^{\bullet}\oplus\Omega^{1}({\mathscr{F}}^{\bullet})\big{)},\,\,\delta_{J}^{\bullet}\,\Big{)},$

where $\delta_{J}^{\bullet}=\delta^{\bullet}\oplus(1\otimes\delta^{\bullet}).$
The complex $\boldsymbol{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})$ can be
equipped with an ${\mathscr{O}}$-module structure (3.2). As $\delta$ is the
coboundary operator of a complex ${\mathscr{O}}$-modules, then
$\delta_{J}(f\ast(-))=f\ast(\delta_{J}(-)).$ Then
${\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})$ the complex (3.3) endowed with
this ${\mathscr{O}}$-module structure.

The exact sequence of complexes of ${\mathscr{O}}$-modules

(3.4)
$0\to\Omega^{1}({\mathscr{F}}^{\bullet})\overset{\iota}{\rightarrow}{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})\overset{\pi}{\rightarrow}{\mathscr{F}}^{\bullet}\to
0$

determines a distinguished triangle

$\Omega^{1}({\mathscr{F}}^{\bullet})\overset{\iota}{\rightarrow}{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})\overset{\pi}{\rightarrow}{\mathscr{F}}^{\bullet}\overset{+1}{\to}$

in the category $D^{b}(Y)$ [22, page 46], [11, page 157]. As ${\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,.\,)$ is a cohomological functor, it
follows that

$\displaystyle{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}))\to{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}))\overset{\pi\circ}{\longrightarrow}{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{F}}^{\bullet})$
$\displaystyle\to{\rm
Ext}^{1}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}))\to$

is an exact sequence. The Atiyah class of ${\mathscr{F}}^{\bullet}$ is the
image of $1\in{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{F}}^{\bullet})$ in ${\rm
Ext}^{1}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}))$.
Thus, we give the following definition.

######  Definition 3.

A gauge field on ${\mathscr{F}}^{\bullet}$ is an element $\psi\in{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}))$,
such that ${\pi\circ}\psi=1\in{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{F}}^{\bullet})$

From the above exact sequence, it follows the following proposition.

######  Proposition 5.

The vanishing of the Atiyah class of ${\mathscr{F}}^{\bullet}$ is a necessary
and sufficient condition for the existence of gauge fields on this brane.
Furthermore, the set of gauge fields on ${\mathscr{F}}^{\bullet}$, if is
nonempty, is an affine space over the finite dimensional vector space ${\rm
Ext}^{0}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}))$.

######  Remark 4.

The complex ${\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})$ gives rise to the
corresponding cohomology ${\mathscr{O}}$-modules
${\mathscr{H}}^{*}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})).$

On the other hand, the $B$-brane $({\mathscr{F}},\,{\delta}^{\bullet})$
determines the “complex”
$\big{(}{\mathscr{H}}^{*}({\mathscr{F}}^{\bullet}),\,0\big{)}$ consisting of
the cohomology ${\mathscr{O}}$-modules. In turn, we can consider
${\mathscr{J}}^{1}({\mathscr{H}}^{*}({\mathscr{F}}^{\bullet})),$ the jet sheaf
complex corresponding to this “complex”. It is immediate to check that
${\mathscr{H}}^{*}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}))$ and
${\mathscr{J}}^{1}({\mathscr{H}}^{*}({\mathscr{F}}^{\bullet}))$ are
canonically isomorphic

(3.5)
${\mathscr{J}}^{1}({\mathscr{H}}^{*}({\mathscr{F}}^{\bullet}))\simeq{\mathscr{H}}^{*}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})).$

###  3.2. Gauge fields on $B$-branes over ${\mathbb{P}}^{n}$.

In this subsection we construct a family of generators for the derived
category $D^{b}({\mathbb{P}}^{n}),$ and prove that the set of holomorphic
gauge fields on any $B$-brane over ${\mathbb{P}}^{n}$ has at most $1$ element.
Some general properties that will be used later are compiled beforehand.

####  3.2.1. Generators of a derived category

Let ${\mathbf{C}}$ be an abelian category and let ${\sf E}$ be a finite set of
generators of the derived category $D^{b}({\mathbf{C}}).$ Given $A$ a complex
in ${\mathbf{C}}$, as usual, we denote with ${A}[l]$, with $l\in{\mathbb{Z}}$,
the complex ${A}$ shifted $l$ to the left.

Given $E,\tilde{E}$ elements of the generating set ${\sf E},$ let us consider
morphisms $h$ between ${E}^{\prime}:={E}[l]$ and
${\tilde{E}}^{\prime}:={\tilde{E}}[l^{\prime}]$. We denote by ${\rm
Cone}(h)={E}^{\prime}[1]\oplus{\tilde{E}}^{\prime}$ the mapping cone of $h$
[11, page 154]. We define ${\sf E}^{(1)}$ the set obtained adding to ${\sf E}$
the elements of the form ${\rm Cone}(h)$. Hence, an element of ${\sf E}^{(1)}$
is a complex whose term at a position $p$ is either $0$, or $E$, or a direct
sum of ${E}\oplus\tilde{E},$ with $E,\tilde{E}\in{\sf E}.$

Repeating the process with the elements of ${\sf E}^{(1)}$ one obtains ${\sf
E}^{(2)}$, etc. The objects of the triangulated subcategory generated by the
family ${\sf E}$ are elements which belong to some ${\sf E}^{(m)}$. Therefore,
an object of the triangulated subcategory of $D^{b}({\mathbf{C}})$ generated
by ${\sf E}$ is a complex $(G,\,d)$, where ${G}^{p}$ is of the form

(3.6) $G^{p}=\bigoplus_{i\in I_{p}}E_{pi},$

with $E_{pi}\in{\sf E}$ and $i$ varying in a finite set $I_{p}$. (When $i$
“runs over the empty set”, the direct sum is taken to be $0$). As ${\sf E}$ is
a set generating $D^{b}({\mathbf{C}}),$ each object of this category is
isomorphic to one of the form (3.6).

Given two complexes $(A,\,\partial_{A})$ and $(B,\,\partial_{B})$ in an
additive category ${\mathbf{C}}$, the complex ${\rm Hom}^{\bullet}(A,\,B)$ is
defined by (see [21, page 17])

(3.7) ${\rm Hom}^{m}(A,\,B)=\prod_{q\in{\mathbb{Z}}}{\rm
Hom}_{\mathbf{C}}\big{(}A^{q},\,B^{q+m}\big{)},$

with the differential $\delta_{H}$.

(3.8)
$(\delta_{H}^{m}g)^{q}=\partial_{B}^{m+q}g^{q}+(-1)^{m+1}g^{q+1}\partial_{A}^{q},$

with $g\in{\rm Hom}^{m}(A,\,B).$

Denoting by ${\rm Com}({\mathbf{C}})$ the abelian category of complexes on
${\mathbf{C}}$ and fixed the object $A,$ one has the functor $F_{A}:{\rm
Com}(\mathbf{C})\to{\rm Com}(\mathbf{Ab})$ defined by

$F_{A}(B)={\rm
Hom}^{\bullet}(A,\,B),\;\;F_{A}(f)^{m}(g)=\big{(}f^{m+q}g^{q}\big{)},$

$f$ a morphism from $(B,\,\partial_{B})\to(C,\,\partial_{C}).$ Assumed that
${\mathbf{C}}$ has enough injectives, as $F_{A}$ is a left exact functor, one
has the corresponding derived functor $RF_{A}:D^{+}{(\mathbf{C}})\to
D^{+}(\mathbf{Ab}).$

Since the $0$th derived functor of a left exact functor is isomorphic to the
functor itself, $F_{A}=R^{0}F_{A}={\rm Ext}^{0}(A,\,.\,).$ On the other hand,
${\rm Hom}_{D({\mathbf{C}})}\big{(}{A},\,B\big{)}={\rm Ext}^{0}(A,\,{B})$ [31,
Sect. 10.7], so

(3.9) ${\rm Hom}_{D({\mathbf{C}})}\big{(}{A},\,B\big{)}={\rm
Hom}^{\bullet}({A},\,{B}).$

One has the following lemma.

######  Lemma 1.

Let ${G}$ be the complex in the category ${\mathbf{C}}$ defined in (3.6) and
$\hat{G}$ the complex ${\hat{G}}^{q}=\bigoplus_{j\in J_{q}}E_{qj}.$ If ${\rm
Hom}_{\mathbf{C}}\big{(}E_{pi},\,E_{qj}\big{)}=0$ for all $p,q,i,j,$ then

${\rm Hom}_{D^{b}({\mathbf{C}})}\big{(}{G},\,\hat{G}\big{)}=0.$

Proof. From (3.7) and (3.9) together with the additivity of the functor ${\rm
Hom}_{\mathbf{C}}(\,.\,,\,.\,),$ it follows

${\rm Hom}^{m}(G,\,\hat{G})=\prod_{p}\bigoplus_{ij}{\rm
Hom}_{\mathbf{C}}(E_{pi},\,E_{(p+m)j})=0.$

Thus, ${\rm Hom}_{D^{b}({\mathbf{C}})}({G},\,\hat{G})=0.$ ∎

We shall deal with the particular case where the category ${\mathbf{C}}$ is a
category $\mathbf{Coh}(X)$ of coherent sheaves. More precisely, let $X$ be a
smooth complex $n$-dimensional variety, let $\mathbf{Qcoh}(X)$ denote the
category of quasi-coherent sheaves on $X.$ One has the derived functor

$RF_{A}:D^{+}(\mathbf{Qcoh}(X))\to D^{+}(\mathbf{Ab}).$

Since ${\rm Ext}^{i}(A,\,B)=0$ for $A,B$ complexes of coherent sheaves and
$i\notin[0,\,n],$ we have the functor

$RF_{A}:D^{b}(X):=D^{b}(\mathbf{Coh}(X))\to D^{b}(\mathbf{Ab}).$

From Lemma 1 applied to the category $D^{b}(X)$, it follows the following
corollary.

######  Corollary 1.

Let ${\mathscr{G}}^{p}=\bigoplus_{pi}{\mathscr{E}}_{pi}$ be an object of
$D^{b}(X)$ and $\hat{\mathscr{G}}$ is an object of this category isomorphic
the complex ${\mathscr{G}}^{p}\otimes_{{\mathscr{O}}_{X}}\Omega^{1}_{X},$ then

${\rm Hom}_{D^{b}(X)}\big{(}{{\mathscr{G}}},\,\hat{\mathscr{G}}\big{)}=0,$

if ${\rm Hom}({\mathscr{E}}_{ip},\,\Omega^{1}_{X}({\mathscr{E}}_{jq}))=0$ for
all $p,q,i,j.$

####  3.2.2. Generators of the derived category $D^{b}(X)$

Let $X$ be a locally Noetherian scheme, $i:Z\hookrightarrow X$ a closed
subscheme of $X$ and $j:U\hookrightarrow X$ an open part of $X.$ Since $j$ is
an open embedding, the functor proper inverse image $j^{!}$ is defined between
the categories of modules
$\mathbf{Mod}({\mathscr{O}}_{X})\to\mathbf{Mod}({\mathscr{O}}_{U}).$ One has
the pair of adjoint functors $(j_{!}\dashv j^{!}=j^{*})$ and the pair
$(i^{*}\dashv i_{*}=i_{!}).$

$j^{!}:\mathbf{Mod}({\mathscr{O}}_{X})\leftrightarrows\mathbf{Mod}({\mathscr{O}}_{U}):j_{!}\;\;\;\text{and}\;\;\;i^{*}:\mathbf{Mod}({\mathscr{O}}_{X})\leftrightarrows\mathbf{Mod}({\mathscr{O}}_{Z}):i_{*}.$

For every sheaf ${\mathscr{F}}$ on $X,$ the corresponding adjunction morphisms
give rise to the sequence of morphisms of ${\mathscr{O}}_{X}$-modules

(3.10) $j_{!}j^{!}{\mathscr{F}}\to{\mathscr{F}}\to i_{*}i^{*}{\mathscr{F}}.$

If ${\mathscr{F}}$ is a coherent ${\mathscr{O}}_{X}$-module, then sheaf
$i^{*}{\mathscr{F}}$ is coherent. Since $i$ is a proper map,
$i_{*}i^{*}{\mathscr{F}}$ is a coherent ${\mathscr{O}}_{X}$-module. As
$j_{!}j^{!}{\mathscr{F}}$ is a subsheaf of ${\mathscr{F}}$, it is also
coherent.

In particular, when $U=X\setminus Z$, for any
${\mathscr{F}}\in\mathbf{Coh}(X)$, one has the exact sequence of
${\mathscr{O}}_{X}$-modules

(3.11) $0\to j_{!}j^{!}{\mathscr{F}}\to{\mathscr{F}}\to
i_{*}i^{*}{\mathscr{F}}\to 0.$

If ${\mathscr{F}}^{\bullet}$ is an object of the derived category
$D^{b}(X)=D^{b}(\mathbf{Coh}(X))$, then we have the distinguished triangle

(3.12) $Rj_{!}j^{!}{\mathscr{F}}^{\bullet}\to{\mathscr{F}}^{\bullet}\to
Ri_{*}i^{*}{\mathscr{F}}^{\bullet}\overset{+1}{\longrightarrow},$

where $j^{!}{\mathscr{F}}^{\bullet}\in D^{b}(\mathbf{Coh}(U))$ and
$i^{*}{\mathscr{F}}^{\bullet}\in D^{b}(\mathbf{Coh}(Z)).$ Hence, as $j_{!}$
and $i_{*}$ are exact functors, one has the following proposition.

######  Proposition 6.

If $\\{g_{k}\\}$ (resp. $\\{g^{\prime}_{k^{\prime}}\\}$) is a set generators
of the derived category $D^{b}(\mathbf{Coh}(U))$ (resp.
$D^{b}(\mathbf{Coh}(Z))$) as a triangulated category, then
$\\{j_{!}g_{k},\,i_{*}g^{\prime}_{k^{\prime}}\\}$ is a set that generates
$D^{b}(X).$

If $U={\rm Spec}\,R$, where $R$ is is a Noetherian ring, the category
$\mathbf{Coh}(U)$ of coherent sheaves on $U$ is equivalent to the category of
finitely generated $R$-modules, with $R=\Gamma(U,\,{\mathscr{O}}_{U}).$ The
functor $\Gamma(U,\,\cdot\,)$ defines the equivalence.

Moreover, if $R$ is a regular local ring, then its global dimension is finite,
and any object $N$ of the category $\mathbf{Mod}_{f}(R)$ (of the $R$-modules
of finite type) admits a finite free resolution

(3.13) $0\to S_{m}\overset{f_{m}}{\longrightarrow}S_{m-1}\to\dots\to
S_{1}\overset{f_{1}}{\longrightarrow}S_{0}\to N\to 0,$

with $S_{a}$ a finite direct sum $\oplus_{I_{a}}R.$

Denoting by $\mathbf{T}$ the triangulated subcategory of
$D^{b}(\mathbf{Mod}_{f}(R))$ generated by $R,$ then the $S_{a}$ are objects of
$\mathbf{T}.$ On the other hand, if we have the exact sequence $0\to
S_{1}\overset{f_{1}}{\longrightarrow}S_{0}\to N\to 0$ in the category
$\mathbf{Mod}_{f}(R),$ then one has the distinguished triangle

$S_{1}\overset{f_{1}}{\longrightarrow}S_{0}\to{\rm
Cone}(f_{1})\overset{+1}{\longrightarrow}.$

Thus, $N$ belongs to the category $\mathbf{T}.$ In the case that $N$ has a
resolution as (3.13), then one can consider the exact sequences

$0\to S_{m}\overset{f_{m}}{\longrightarrow}S_{m-1}\to{\rm im}(f_{m-1})\to 0,$

and

$0\to{\rm im}(f_{m-1})\to S_{m-2}\to\dots\to N\to 0.$

By the previous result, ${\rm im}(f_{m-1})$ belongs to $\mathbf{T}.$ By
induction, it follows that $N$ is also an object of $\mathbf{T}.$ Hence
$\mathbf{Mod}_{f}(R)\subset{\mathbf{T}}.$ As $D^{b}(\mathbf{Mod}_{f}(R))$ is
generated by $\mathbf{Mod}_{f}(R),$ it follows that $R$ generates the derived
category $D^{b}(\mathbf{Mod}_{f}(R)).$

From the equivalence between $\mathbf{Coh}(U)$ and $\mathbf{Mod}_{f}(R),$ one
deduces the following proposition.

######  Proposition 7.

If $U$ is a Noetherian affine scheme, then ${\mathscr{O}}_{U}$ generates
$D^{b}(U)$, the bounded derived category of coherent sheaves on $U$.

From Proposition 6, it follows the following corollary (see [5, Lem. 2.3.2]).

######  Corollary 2.

If $U\overset{j}{\hookrightarrow}X$ is an open part of $X$ isomorphic to a
Noetherian affine scheme and $Z=X\setminus U,$ then the derived category
$D^{b}(X)$ is generated by $j_{!}{\mathscr{O}}_{U}$ together with the elements
of $i_{*}\\{{\rm generators\;of}\,D^{b}(Z)\\},$ where $i$ is the inclusion
$Z\hookrightarrow X.$

An Example. Let $X$ be ${\mathbb{P}}^{1},$
$U=\\{[1:x]\in{\mathbb{P}}^{1}\,|\,x\in{\mathbb{C}}\\}$ and $Z=\\{[0:1]\\}.$
In this case, ${\mathscr{O}}_{Z}=i_{*}i^{*}{\mathscr{O}}_{X}$ is the
skyscraper sheaf at the point $P=[0:1],$ and
$j_{!}j^{!}{\mathscr{O}}_{X}={\mathscr{O}}_{X}(-1).$ That is,
$j_{!}{\mathscr{O}}_{U}\simeq{\mathscr{O}}_{X}(-1).$

As $U={\rm Spec}\,R$ with $R$ the polynomial ring ${\mathbb{C}}[x]$, then
${\mathscr{O}}_{U}$ is the generator of the corresponding category $D^{b}(U).$
The category $D^{b}(Z)$ is generated by $i^{*}{\mathscr{O}}_{X}$, obviously.
By Corollary 2), $\\{{\mathscr{O}}_{X}(-1),\,{\mathscr{O}}_{Z}\\}$ is a set of
generators of $D^{b}(X).$

In this case, the exact sequence (3.11) is

$0\to{\mathscr{O}}_{X}(-1)\to{\mathscr{O}}_{X}\to{\mathscr{O}}_{Z}\to 0.$

Therefore, the set $\\{{\mathscr{O}}_{X}(-1),\,{\mathscr{O}}_{X}\\}$ is also a
set of generators of $D^{b}(X)$ (which is in agreement with Beilinson’s
theorem). Hence, any object of $D^{b}(X)$ is isomorphic to one complex
${\mathscr{G}}^{\bullet}$ of the form (3.6), where
${\mathscr{E}}_{pi}\in\\{{\mathscr{O}_{X}},\,{\mathscr{O}}_{X}(-1)\\}.$

On the other hand, for
${\mathscr{E}},\,{\mathscr{E}}^{\prime}\in\\{{\mathscr{O}_{X}},\,{\mathscr{O}}_{X}(-1)\\}$,
one has

${\rm
Hom}({\mathscr{E}},\,{\mathscr{E}}^{\prime}\otimes_{\mathscr{O}}\Omega^{1}_{X})={\rm
Hom}({\mathscr{O}}_{X},\,({{\mathscr{E}}}^{\vee}\otimes{\mathscr{E}}^{\prime})\otimes_{\mathscr{O}}\Omega^{1}_{X})=H^{0}(X,\,\Omega^{1}(\epsilon)),$

where $\epsilon\in\\{0,+1,-1\\}$ and $\Omega^{1}(\epsilon)$ is the twisted
sheaf $\Omega^{1}_{X}\otimes_{{\mathscr{O}}_{X}}{\mathscr{O}}(\epsilon).$ But
$H^{0}(X,\,\Omega^{1}(\epsilon))=0$ [28, page 4]. Thus, by Corollary 1
together with Proposition 5, the set of of holomorphic gauge fields over any
brane on ${\mathbb{P}}^{1}$ has cardinal $\leq 1.$

######  Remark 5.

The family ${\sf
E}=\\{{\mathscr{O}}_{{\mathbb{P}}^{n}}(-n),\,{\mathscr{O}}_{{\mathbb{P}}^{n}}(-n+1),\dots,{\mathscr{O}}_{{\mathbb{P}}^{n}}\\}$
generates $D^{b}({\mathbb{P}}^{n}).$ If
${\mathscr{E}},\,{\mathscr{E}}^{\prime}\in{\sf E},$ then

${\rm
Hom}\big{(}{\mathscr{E}},\,\Omega^{1}_{{\mathbb{P}}^{n}}({\mathscr{E}}^{\prime})\big{)}=H^{0}\big{(}X,\,\Omega^{1}_{{\mathbb{P}}^{n}}(\epsilon)\big{)},$

with $\epsilon\in\\{-n,\,-n+1,\dots,n\\}.$

If $n>1,$ the cohomology group
$H^{0}(X,\,\Omega^{1}_{{\mathbb{P}}^{n}}(\epsilon))$ is not necessarily zero
(see [28, page 4]). Thus, Corollary 1 is not applicable in order to extend to
${\mathbb{P}}^{n}$ the above result obtained for the $B$-branes over
${\mathbb{P}}^{1}.$ To achieve such an extension it will be necessary to
construct another set of generators.

####  3.2.3. Generators of $D^{b}({\mathbb{P}}^{n})$

We will generalize the result of the above Example. Let $X$ be a smooth a
complex $n$-dimensional variety. Let $Z_{1}$ be an irreducible subvariety of
$X$ of codimension $1,$ and $i_{1}:Z_{1}\hookrightarrow X$ the corresponding
inclusion. We set $U_{1}:=X\setminus Z_{1}$ and $j_{1}:U_{1}\hookrightarrow
X.$ Let us assume that $U_{1}$ is isomorphic to an affine variety. The exact
sequence

$0\to j_{1!}j_{1}^{!}{\mathscr{O}}_{X}\to{\mathscr{O}}_{X}\to
i_{1*}i^{*}_{1}{\mathscr{O}}_{X}\to 0,$

can be written
$0\to{\mathscr{M}}_{1}\to{\mathscr{O}}_{X}\to{\mathscr{O}}_{Z_{1}}\to 0,$
where ${\mathscr{M}}_{1}$ is the invertible sheaf ${\mathscr{O}}_{X}(-Z_{1}),$
which is supported in $U_{1},$ i.e. the stalk $({\mathscr{M}}_{1})_{x}=0$ for
all $x\in Z_{1}.$

As $j_{1}^{!}{\mathscr{O}}_{X}={\mathscr{O}}_{U_{1}}$, then
${\mathscr{M}}_{1}=j_{1!}{\mathscr{O}}_{U_{1}}$. From Corollary 2, it follows
that the derived category $D^{b}(X)$ is generated by ${\mathscr{M}}_{1}$
together with the elements of $i_{1*}\\{{\rm
generators\;of}\,D^{b}(Z_{1})\\}.$

If $Z_{2}\overset{i_{2}}{\hookrightarrow}Z_{1}$ is an irreducible hypersurface
of $Z_{1}$ and $U_{2}:=Z_{1}\setminus Z_{2}$ is isomorphic to an affine
variety, then we have an exact sequence

$0\to{\mathscr{M}}_{2}\to{\mathscr{O}}_{Z_{1}}\to
i_{2*}i_{2}^{*}{\mathscr{O}}_{Z_{1}}\to 0,$

where ${\mathscr{M}}_{2}$ is the sheaf on $Z_{1},$
${\mathscr{O}}_{{Z}_{1}}(-Z_{2}).$ Hence, $D^{b}(Z_{1})$ is generated by
${\mathscr{M}}_{2}$ together with $i_{2*}\\{{\rm
generators\,of}\,D^{b}(Z_{2})\\}.$ Consequently, $D^{b}(X)$ is generated by

$\\{{\mathscr{M}}_{1},\,i_{1*}{\mathscr{M}}_{2}\\}\cup
i_{1*}i_{2*}\big{(}\\{{\rm generators\,of}\,D^{b}(Z_{2})\\}\big{)}.$

Let us assume that there exists a tower $X=Z_{0}\supset Z_{1}\supset
Z_{2}\supset\dots\supset Z_{n},$ where
$Z_{k}\overset{i_{k}}{\hookrightarrow}Z_{k-1}$ is an irreducible hypersurface
of $Z_{k-1}$ with $i=1,\dots,n.$ According to the above argument,
${\mathscr{M}}_{k}={\mathscr{O}}_{Z_{k-1}}(-Z_{k})$ is a sheaf supported on
$U_{k}:=Z_{k-1}\setminus Z_{k}$. We set $\iota_{k}=i_{1}\circ\dots\circ
i_{k}:Z_{k}\hookrightarrow X$ and $\iota_{0}:={\rm id}:Z_{0}\to X.$ For
$k=1,\dots,n$ we denote ${\mathscr{L}}_{k}:=\iota_{k-1*}({\mathscr{M}}_{k}).$

We set ${\mathscr{L}}_{n+1}$ for denoting the sheaf on $X$ direct sum of the
skyscraper sheaves on the points of the finite set $Z_{n}.$ Then

${\mathscr{L}}_{n+1}=\iota_{n*}\big{(}\\{{\rm
generators\,of}\,D^{b}(Z_{n})\\}).$

We have proved the following proposition.

######  Proposition 8.

With the above notations, the derived category $D^{b}(X)$ is generated by the
set of ${\mathscr{O}}_{X}$-modules

${\sf
E}=\\{{\mathscr{L}}_{1},\dots,{\mathscr{L}}_{n},\,{\mathscr{L}}_{n+1}\\}.$

  * •

When $j\neq k$ the $U_{j}\cap U_{k}=\emptyset.$ Hence, ${\mathscr{L}}_{j}$ and
$\Omega^{1}({\mathscr{L}}_{k})$ have disjoint supports.; thus,

(3.14) ${\rm
Hom}_{O_{X}}\big{(}{\mathscr{L}}_{j},\,\Omega^{1}_{X}\otimes_{{\mathscr{O}}_{X}}{\mathscr{L}}_{k}\big{)}=0.$

  * •

Since ${\mathscr{L}}_{1}$ is the invertible ${\mathscr{O}}_{X}(-Z_{1}),$
tensoring by the dual ${\mathscr{L}}_{1}^{\vee}$

${\rm
Hom}({\mathscr{L}}_{1},\,{\mathscr{L}}_{1}\otimes_{{\mathscr{O}}_{X}}\Omega^{1}_{X})={\rm
Hom}({\mathscr{O}}_{X},\,\Omega^{1}_{X})=H^{0}(X,\,\Omega^{1}_{X})=H^{1,0}(X).$

  * •

In this item, when $k>1,$ we set ${\mathscr{L}}:={\mathscr{L}}_{k},$
${\mathscr{M}}:={\mathscr{M}}_{k},$ and $\iota:=\iota_{k}$ for the inclusion
$Z:=Z_{k}\hookrightarrow X;$ thus ${\mathscr{L}}=\iota_{*}{\mathscr{M}}.$ We
will also delete the subscript $X$ in the sheaves
${\mathscr{O}}_{X}=:{\mathscr{O}}$ and $\Omega^{1}_{X}=:\Omega^{1}.$ By
${\mathscr{H}om}\,_{\mathscr{R}}(\,.\,,\,.\,)$ we denote the sheaf of linear
homomorphisms between the corresponding ${\mathscr{R}}$-modules.

By [22, Prop. 2.5.13] and [22, Cor. 2.3.4]

${\mathscr{H}om}\,_{\mathscr{O}}({\mathscr{L}},\,{\mathscr{L}}\otimes_{\mathscr{O}}\Omega^{1})\simeq\iota_{*}{\mathscr{H}om}\,_{{\mathscr{O}}_{Z}}({\mathscr{M}},\,{\mathscr{M}}\otimes_{{\mathscr{O}}_{Z}}\Omega^{1}_{Z}).$

Hence,

(3.15)
$\displaystyle{\mathscr{H}om}\,_{\mathscr{O}}({\mathscr{L}},\,{\mathscr{L}}\otimes_{\mathscr{O}}\Omega^{1})\simeq\Gamma\big{(}X,\,\iota_{*}{\mathscr{H}om}\,_{{\mathscr{O}}_{Z}}({\mathscr{M}},\,{\mathscr{M}}\otimes_{{\mathscr{O}}_{Z}}\Omega^{1}_{Z})\big{)}=$
$\displaystyle\Gamma\big{(}Z,\,{\mathscr{H}om}\,_{{\mathscr{O}}_{Z}}({\mathscr{M}},\,{\mathscr{M}}\otimes_{{\mathscr{O}}_{Z}}\Omega^{1}_{Z})\big{)}\simeq{\rm
Hom}_{{\mathscr{O}}_{Z}}({\mathscr{O}}_{Z},\Omega^{1}_{Z})=H^{1,0}(Z).$

From the above items, together with Corollary 1, Proposition 8 and Proposition
5, it follows the theorem.

######  Theorem 2.

Let $X$ be an $n$-dimensional smooth variety and suppose that

  1. (1)

There exists $X=Z_{0}\supset Z_{1}\supset\dots\supset Z_{n},$ where $Z_{i+1}$
is a smooth irreducible subvariety of $Z_{i},$ with ${\rm
codim}_{Z_{i}}Z_{i+1}=1$.

  2. (2)

$U_{j}=Z_{j-1}\setminus Z_{i}$ is isomorphic to an affine variety.

If $H^{1,0}(Z_{j})=0$ for all $j,$ then the set of holomorphic gauge fields on
any $B$-brane over $X$ has cardinal $\leq 1.$

####  3.2.4. Holomorphic gauge fields on $B$-branes over ${\mathbb{P}}^{n}$

Let $X={\mathbb{P}}^{n}$, we define

$Z_{1}:=\\{[0:x_{1}:\dots:x_{n}]\simeq{\mathbb{P}}^{n-1},\;U_{1}={\mathbb{P}}^{n}\setminus
Z_{1}\simeq{\mathbb{C}}^{n}.$

Analogously, one defines $Z_{2}=\\{[0:0:x_{2}:\dots:x_{n}]\\}$ and
$U_{2}=Z_{1}\setminus Z_{2},$ etc. Moreover, $H^{1,0}(Z_{j})=0.$ By Theorem 2,
one has the following theorem.

######  Theorem 3.

The set of holomorphic gauge fields on any $B$-brane on ${\mathbb{P}}^{n}$ has
cardinal $\leq 1.$

######  Remark 6.

For the $B$-brane on $X={\mathbb{P}}^{n}$ consisting only of the line bundle
${\mathscr{O}}(k),$

${\rm
Hom}({\mathscr{O}}(k),\,\Omega^{1}_{X}\otimes_{{\mathscr{O}}_{X}}{\mathscr{O}}(k))={\rm
Hom}({\mathscr{O}},\,\Omega^{1}_{X})=H^{0}(X,\,\Omega^{1}_{X})=0.$

This fact is in agreement with the previous general result.

On the other hand, if a line bundle admits a holomorphic gauge field then its
Chern class vanishes [4]. Thus, among the elements of the set
$\\{{\mathscr{O}}_{X}(k)\\}_{k\in{\mathbb{Z}}}$, only ${\mathscr{O}}_{X}$ can
support a holomorphic gauge field, and this field is precisely
$\partial:{\mathscr{O}}_{X}\to\Omega^{1}_{X}.$

###  3.3. Flag varieties

In this section, we will first recall some properties of the Schubert
varieties. We denote by $X:={\rm GL}(n,\,{\mathbb{C}})/B$, where $B$ is the
Borel subgroup of ${\rm GL}(n,\,{\mathbb{C}})$ consisting of the upper
triangular matrices. $X$ is the manifold of complete flags in
${\mathbb{C}}^{n}.$

Let $T$ denote the maximal torus of ${\rm GL}(n,\,{\mathbb{C}})$ formed by the
diagonal matrices. We set $W:=N(T)/T$, where $N(T)$ is the normalizer of $T$
in the group ${\rm GL}(n,\,{\mathbb{C}}).$ As it is well-known, the Weyl group
$W$ is isomorphic to the symmetric group $S_{n}$. The Bruhat partial order [9,
Sect 10.5] in $S_{n}$ will be denoted by $\leq$.

Given $w\in W,$ we set $C^{0}_{w}$ for the Schubert cell of $X$ associated to
$w$ [9]. The closure (in Zariski topology) of this cell is the corresponding
Schubert subvariety of $X$

(3.16) $C_{w}:=\bigcup_{v\leq w}C_{v}^{0}.$

The Poincaré polynomial of $C_{w}$ given in [1] shows that the Betti number
$b_{1}(C_{w})$ vanishes, if $C_{w}$ is any smooth Schubert variety.

Theorem 1 of [26] gives a criterion to determine whether a Schubert variety in
a flag variety is singular. Let $w$ be the permutation
$w=(t_{1},\dots,t_{n})$. The variety $C_{w}$ is singular iff there are
integers $i,j,k,l$ with $1\leq i<j<k<l\leq n$ such that, either

$t_{k}<t_{l}<t_{i}<t_{j},\;\;\text{or}\;\;t_{l}<t_{j}<t_{k}<t_{i}.$

In particular, all the Schubert varieties of $X={\rm GL}(3,\,{\mathbb{C}})/B$
are smooth. Thus, if $C_{w}$ is such a variety, as the $b_{1}(C_{w})=0$, the
Hodge numbers $h^{1,0}(C_{w})$ and $h^{0,1}(C_{w})$ vanish

(3.17) $h^{1,0}(C_{w})=h^{0,1}(C_{w})=0.$

From now on in this section, $X$ will denote the variety of complete flags in
${\mathbb{C}}^{3}.$ The Bruhat order in the group $S_{3}$ is showed in the
following diagram [6].

$\textstyle{w_{1}=(321)}$$\textstyle{w_{3}=(231)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w_{2}=(312)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w_{5}=(213)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w_{4}=(132)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w_{6}=(123)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Hence, there are only two pairs of permutations that are not comparable in the
Bruhat order, namely, $(w_{2},\,w_{3})$ and $(w_{4},\,w_{5}).$ For any other
pair $(w,\,w^{\prime})$ the intersection $C_{w}\cap C_{w^{\prime}}=C_{v}$,
where $v={\rm min}\\{w,\,w^{\prime}\\}.$

The length $l(w)$ of the permutation $w$ is defined by

$l(w)=\\#\\{a<b\,|\,w(a)>w(b)\\}.$

We will set $C^{0}_{k}$ for the cell $C^{0}_{w_{k}}$ and $C_{k}:=C_{w_{k}}.$
The cell $C^{0}_{w}$ is an affine variety isomorphic to
${\mathbb{C}}^{l(\omega)}$ [9, page 157]. Hence, ${\rm dim}\,C_{4}=1={\rm
dim}\,C_{5}$ and ${\rm dim}\,C_{2}=2={\rm dim}\,C_{3}.$ Furthermore, $X$ is
the disjoint union of the cells

$X=\bigsqcup_{k=1}^{6}C^{0}_{k}.$

On the other hand, the inclusion relations among the Schubert subvarieties of
$X$ can be obtained from (3.16) by means of the Bruhat order depicted above.

We consider the following tower of subspaces of $X$

$Z_{0}:=X\supset Z_{1}:=C_{2}\cup C_{3}\supset Z_{2}:=C_{4}\cup C_{5}\supset
Z_{3}:=C_{6}.$

That is, $Z_{i}$ is the union of the Schubert varieties of dimension $3-i.$ We
set $U_{i}:=Z_{i-1}\setminus Z_{i}$. From (3.16), it follows the expression of
each $U_{i}$ in terms of Schubert cells. The corresponding inclusion relations
are showed in the diagram.

$\textstyle{U_{1}:=C^{0}_{1}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{C_{1}=X=:}$$\textstyle{Z_{0}}$$\textstyle{U_{2}:=C^{0}_{2}\cup
C^{0}_{3}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{2}}$$\textstyle{C_{2}\cup
C_{3}=:}$$\textstyle{Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{U_{3}:=C^{0}_{4}\cup
C^{0}_{5}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{3}}$$\textstyle{C_{4}\cup
C_{5}=:}$$\textstyle{Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{C_{6}=:}$$\textstyle{Z_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{3}}$

A family of generators for $D^{b}(X)$ will then be constructed by successive
applications of Corollary 2.

$U_{1}$ open part of $Z_{0}=X.$ In the flag variety $X$, we have the divisor
$Z_{1}=X\setminus C^{0}_{1}=C_{2}\cup C_{3},$ and the partition of $X$ defined
by the affine variety $U_{1}=C^{0}_{1}$ and $Z_{1}.$ We denote by $j_{1}$ and
$i_{1}$ the respective inclusions

$U_{1}\overset{j_{1}}{\hookrightarrow}X\overset{i_{1}}{\hookleftarrow}Z_{1}.$

On the other hand, we have the exact sequence of ${\mathscr{O}}_{X}$-modules

$0\to{\mathscr{M}}_{1}:=j_{1!}j^{!}_{1}{\mathscr{O}}_{X}\to{\mathscr{O}}_{X}\to
i_{1*}i^{*}_{1}{\mathscr{O}}_{X}\to 0.$

The nontrivial term on the left satisfies

  1. (1)

${\mathscr{M}}_{1}$ is the invertible sheaf ${\mathscr{O}}_{X}(-C_{2}-C_{3})$
associated to the divisor $-C_{2}-C_{3}.$

  2. (2)

${\rm Supp}\,{\mathscr{M}}_{1}\subset U_{1}.$

  3. (3)

As $j^{!}_{1}{\mathscr{O}}_{X}={\mathscr{O}}_{U_{1}}$, from Corollary 2, it
follows that $D^{b}(X)$ is generated by ${\mathscr{M}}_{1}$ together with
$i_{1*}\\{{\rm generators\;of}D^{b}(Z_{1})\\}.$ We write

(3.18) $D^{b}(X)=\langle{\mathscr{M}}_{1},\,i_{1*}D^{b}(Z_{1})\rangle.$

$U_{2}$ open subset of $Z_{1}.$ Next we will consider a divisor
$Z_{2}=C_{4}+C_{5}$ of $Z_{1}$ and we apply to it the above analysis. We will
first state the following simple lemma.

######  Lemma 2.

Let $Y$ be a complex variety, and $Y_{1},\,Y_{2}$ affine subvarieties of $Y$,
such that $Y=Y_{1}\cup Y_{2}.$ If $Y_{k}$, for $k=1,2,$ is isomorphic to an
affine variety, then the derived category $D^{b}(Y)$ is generated by

$\\{\alpha_{1!}{\mathscr{O}}_{Y_{1}},\,\alpha_{2!}{\mathscr{O}}_{Y_{2}}\\}\cup\alpha_{!}\\{{\rm
generators\,of}\,D^{b}(Y_{1}\cap Y_{2})\\},$

where $\alpha$ is the inclusion $\alpha:Y_{1}\cap Y_{2}\hookrightarrow Y,$ and
$\alpha_{k}:Y_{k}\hookrightarrow Y$ denotes the corresponding embedding.

Proof. Given ${\mathscr{F}}\in\mathbf{Coh}(Y),$ one has the coherent sheaves
${\mathscr{F}}_{k}:=\alpha_{k!}\alpha_{k}^{!}{\mathscr{F}}$ and
${\mathscr{F}}_{Y_{1}\cap Y_{2}}=\alpha_{!}\alpha^{!}{\mathscr{F}}$ and the
exact sequence of ${\mathscr{O}}_{Y}$-modules [22, Prop. 2.3.6 (vii)]

$0\to{\mathscr{F}}_{Y_{1}\cap
Y_{2}}\to{\mathscr{F}}_{1}\oplus{\mathscr{F}}_{2}\to{\mathscr{F}}\to 0.$

By Proposition 7, $D(Y_{k})$ is generated by ${\mathscr{O}}_{Y_{k}}.$ From the
above exact sequence, it follows that the sheaf ${\mathscr{F}}$ belongs to the
subcategory of $D^{b}(Y)$ generated by $\alpha_{1!}{\mathscr{O}}_{Y_{1}},\,$
$\alpha_{2!}{\mathscr{O}}_{Y_{2}}$ together with the elements of form
$\alpha_{!}(g),$ where $g$ belongs to a set of generators of $D^{b}(Y_{1}\cap
Y_{2}).$ ∎

Going back to the variety $X$. Next we will consider a divisor
$Z_{2}=C_{4}+C_{5}$ of $Z_{1}.$ Then $U_{2}:=Z_{1}\setminus
Z_{2}=C_{2}^{0}\cup C_{3}^{0}.$ For $k=2,3$ we have the inclusions

$C^{0}_{k}\overset{\alpha_{k}}{\hookrightarrow}U_{2}\overset{j_{2}}{\hookrightarrow}Z_{1}\overset{i_{2}}{\hookleftarrow}Z_{2}.$

As $C^{0}_{2}\cap C^{0}_{3}=\emptyset,$ from Lemma 2 together with Corollary
2, it follows that the set
$\\{\alpha_{k!}{\mathscr{O}}_{C^{0}_{k}}\\}_{k=2,3}$ is a set of generators of
$D^{b}(U_{2}).$

For $k=2,3$ the ${\mathscr{O}}_{Z_{1}}$-module
$(j_{2}\alpha_{k})_{!}{\mathscr{O}}_{C^{0}_{k}},$ will be denoted
${\mathscr{M}}_{k}.$

  1. (1)

Since $Z_{1}\setminus C^{0}_{k}=C_{k^{\prime}},$ with
$k^{\prime}\in\\{2,\,3\\}\setminus\\{k\\},$ then ${\mathscr{M}}_{k}$ is the
invertible sheaf in $Z_{1}$ associated to the divisor $-C_{k^{\prime}}.$ That
is, ${\mathscr{M}}_{k}={\mathscr{O}}_{Z_{1}}(-C_{k^{\prime}}).$

  2. (2)

${\rm Supp}\,{\mathscr{M}}_{k}\subset C^{0}_{k}.$

  3. (3)

From Corollary 2, we conclude

$D^{b}(Z_{1})=\langle{\mathscr{M}}_{2},\,{\mathscr{M}}_{3},\,i_{2*}D^{b}(Z_{2})\rangle.$

From (3.18), it follows

(3.19)
$D^{b}(X)=\langle{\mathscr{M}}_{1},\,i_{1*}{\mathscr{M}}_{2},\,i_{1*}{\mathscr{M}}_{3},\,i_{1*}i_{2*}D^{b}(Z_{2})\rangle.$

$U_{3}$ open subset of $Z_{2}.$ In a similar way, we define
$U_{2}=C^{0}_{4}\cup C^{0}_{5}$ and $Z_{3}=Z_{2}\setminus U_{2}=C_{6}.$ For
$k=4,5$ one has the embeddings

$C^{0}_{k}\overset{\alpha_{k}}{\hookrightarrow}U_{3}\overset{j_{3}}{\hookrightarrow}Z_{2}\overset{i_{3}}{\hookleftarrow}Z_{3}.$

Since $C^{0}_{4}\cap C^{0}_{5}=\emptyset,$ from Lemma 2 together with
Corollary 2, it follows that the set
$\\{\alpha_{k!}{\mathscr{O}}_{C^{0}_{k}}\\}_{k=4,5}$ is a set of generators of
$D^{b}(U_{3}).$

For $k=4,5$ the ${\mathscr{O}}_{Z_{2}}$-module
$(j_{3}\alpha_{k})_{!}{\mathscr{O}}_{C^{0}_{k}},$ will be denoted
${\mathscr{M}}_{k}.$

  1. (1)

Since $Z_{2}\setminus C^{0}_{k}=C_{k^{\prime}},$ with
$k^{\prime}\in\\{4,\,5\\}\setminus\\{k\\},$ then ${\mathscr{M}}_{k}$ is the
invertible sheaf in $Z_{2}$ associated to the divisor $-C_{k^{\prime}}.$ That
is, ${\mathscr{M}_{k}}={\mathscr{O}}_{Z_{2}}(-C_{k^{\prime}}).$

  2. (2)

${\rm Supp}\,{\mathscr{M}}_{k}\subset C^{0}_{k}.$

  3. (3)

By Corollary 2,
$D^{b}(Z_{2})=\langle{\mathscr{M}}_{3},\,{\mathscr{M}}_{4},\,i_{3*}D^{b}(Z_{3})\rangle.$

From (3.19), it follows

(3.20)
$D^{b}(X)=\langle{\mathscr{M}}_{1},\,i_{1*}{\mathscr{M}}_{2},\,i_{1*}{\mathscr{M}}_{3},\,(i_{1}i_{2})_{*}{\mathscr{M}}_{4},\,(i_{1}i_{2})_{*}{\mathscr{M}}_{5},\,(i_{1}i_{2}i_{3})_{*}D^{b}(Z_{3})\rangle.$

For the sake of simplicity in the notation we set

${\mathscr{L}}_{1}:={\mathscr{M}}_{1},\,{\mathscr{L}}_{2}:=i_{1*}{\mathscr{M}}_{2},\,{\mathscr{L}}_{3}:=i_{1*}{\mathscr{M}}_{3},\,{\mathscr{L}}_{4}:=(i_{1}i_{2})_{*}{\mathscr{M}}_{4},\,{\mathscr{L}}_{5}:=(i_{1}i_{2})_{*}{\mathscr{M}}_{5}.$

The point $Z_{3}.$ ${\mathscr{L}}_{6}$ will denote the skyscraper sheaf on $X$
at the point $C_{6}.$ Thus,

######  Proposition 9.

The derived category $D^{b}(X)$ is generated by the set
$\\{{\mathscr{L}}_{1},\dots,{\mathscr{L}}_{6}\\}.$

Since ${\rm Supp}({\mathscr{L}}_{k})\subset C^{0}_{k}$ and $C^{0}_{r}\cap
C^{0}_{s}=\emptyset.$ for $r\neq s.$ Then

(3.21) ${\rm
Hom}_{{\mathscr{O}}_{X}}\big{(}{\mathscr{L}}_{r},\,{\mathscr{L}}_{s}\otimes_{{\mathscr{O}}_{X}}\Omega^{1}_{X}\big{)}=0.$

######  Proposition 10.

For $r=1,\dots,6$

${\rm
Hom}_{{\mathscr{O}}_{X}}({\mathscr{L}}_{r},\,{\mathscr{L}}_{r}\otimes_{{\mathscr{O}}_{X}}\Omega^{1}_{X})=0.$

Proof. As ${\mathscr{L}}_{1}={\mathscr{M}}_{1}$ is the invertible sheaf
${\mathscr{O}}_{X}(-Z_{1}),$

${\rm
Hom}_{{\mathscr{O}}_{X}}({\mathscr{L}}_{1},\,{\mathscr{L}}_{1}\otimes_{{\mathscr{O}}_{X}}\Omega^{1}_{X})={\rm
Hom}_{{\mathscr{O}}_{X}}({\mathscr{O}}_{X},\,\Omega^{1}_{X})=H^{1,0}(X).$

By (3.17), $H^{1,0}(X)=0.$

The remaining cases can be treated similarly. To simplify the notation we will
write ${\mathscr{X}}$ for ${\mathscr{O}}_{X}$ and ${\mathscr{Z}}_{1}$ for
${\mathscr{O}}_{Z_{1}};$ we also write $i:=i_{1}.$

As $\Omega^{1}_{X}$ is a locally free ${\mathscr{X}}$-module, by [22, Prop.
2.5.13]

${\mathscr{L}}_{2}\otimes_{\mathscr{X}}\Omega^{1}_{X}\simeq
i_{!}\big{(}{\mathscr{M}}_{2}\otimes_{{\mathscr{Z}}_{1}}\Omega^{1}_{Z_{1}}\big{)}.$

Hence,

$\displaystyle{\mathscr{H}om}_{\mathscr{X}}\big{(}{\mathscr{L}}_{2},\,{\mathscr{L}}_{2}\otimes_{\mathscr{X}}\Omega^{1}_{X}\big{)}\simeq{\mathscr{H}om}_{\mathscr{X}}\big{(}i_{*}{\mathscr{M}}_{2},\,i_{!}({\mathscr{M}}_{2}\otimes_{{\mathscr{Z}}_{1}}\Omega^{1}_{Z_{1}})\big{)}$
$\displaystyle\simeq
i_{*}{\mathscr{H}om}_{{\mathscr{Z}}_{1}}\big{(}{\mathscr{M}}_{2},\,{\mathscr{M}}_{2}\otimes_{{\mathscr{Z}}_{1}}\Omega^{1}_{Z_{1}}\big{)}.$

Since ${\mathscr{M}}_{2}$ is an invertible ${\mathscr{Z}}_{1}$-module

$\displaystyle{\rm
Hom}_{\mathscr{X}}\big{(}{\mathscr{L}}_{2},\,{\mathscr{L}}_{2}\otimes_{\mathscr{X}}\Omega^{1}_{X}\big{)}\simeq\Gamma\big{(}Z_{1},\,{\mathscr{H}om}_{{\mathscr{Z}}_{1}}\big{(}{\mathscr{M}}_{2},\,{\mathscr{M}}_{2}\otimes_{{\mathscr{Z}}_{1}}\Omega^{1}_{Z_{1}}\big{)}\big{)}$
$\displaystyle={\rm
Hom}_{{\mathscr{Z}}_{1}}\big{(}{\mathscr{M}}_{2},\,{\mathscr{M}}_{2}\otimes_{{\mathscr{Z}}_{1}}\Omega^{1}_{Z_{1}}\big{)}=H^{0}(Z_{1},\,\Omega^{1}_{Z_{1}})$

On the other hand, $Z_{1}=C_{2}\cup C_{3}$, and
$H^{1,0}(C_{2})=H^{1,0}(C_{3})=0.$ Thus,
$H^{0}(Z_{1},\,\Omega^{1}_{Z_{1}})=0.$ That is, the proposition is proved for
the case $r=2.$ The proof of the remaining cases is analogous. ∎

######  Theorem 4.

The set of holomorphic gauge fields on any $B$-brane over the variety of
complete flags in ${\mathbb{C}}^{3}$ is $\leq 1$.

Proof. An object of $D^{b}(X)$ is isomorphic to one of the triangulated
category generated by the family ${\mathscr{L}}_{1},\dots,{\mathscr{L}}_{6}$
in $D^{b}(X).$ Let $({G}^{\bullet},\,d^{\bullet})$ be an object of this
triangulated category. Then ${G}^{p}$ is a sheaf of the form

(3.22) ${G}^{p}=\bigoplus_{i\in S_{p}}{\mathscr{L}}_{pi},$

$S_{p}$ being a finite set. From Corollary 1 together with (3.21) and
Proposition 10, it follows

${\rm Hom}_{D^{b}(X)}\big{(}{G}^{\bullet},\,\Omega^{1}(G^{\bullet})\big{)}=0.$

The theorem is a consequence of Proposition 5. ∎

###  3.4. The homotopy category

Let $X$ be a smooth projective variety. The category $\mathbf{Coh}(X)$ of
coherent sheaves over $X$ has not enough injectives, for this reason it is
convenient to regard $D^{b}(X)$ as a subcategory of
$D^{b}(\mathbf{Mod}({\mathscr{O}_{X}}))$, the bounded derived category of the
${\mathscr{O}}_{X}$-modules. In fact, $D^{b}(X)$ is equivalent to
$D^{b}(\mathbf{Mod}({\mathscr{O}_{X}}))_{\rm coh}$, the full subcategory of
$D^{b}(\mathbf{Mod}({\mathscr{O}_{X}}))$ consisting of the complexes with
coherent cohomology [29, Exp II] [15, Chapter II] (for a brief description
[19, page 63]). Thus, $D^{b}(X)$ can be identified with the homotopy category
whose objects are the complexes ${\mathscr{G}}^{\bullet}$ of injective
${\mathscr{O}}_{X}$-modules, such that its cohomogy is bounded and coherent;
i.e. ${\mathscr{H}}^{j}({\mathscr{G}}^{\bullet})$ is coherent and vanishes for
$|j|>>0$. We will denote this homotopy category by $K^{b}(X)_{\rm coh}$.

In the identification of $D^{b}(X)$ with the homotopy category $K^{b}(X)_{\rm
coh},$ the quasi-isomorphisms correspond to homotopy equivalences. Thus, the
independence of quasi-isomorphisms in the frame of $K(X)_{\rm coh}$ means
independence with respect to homotopy equivalences. In particular, if
${\mathscr{F}}^{\bullet}$ and ${\mathscr{G}}^{\bullet}$ are objects of
$K^{b}(X)_{\rm coh}$ homotopically equivalent, the respective cohomology
sheaves ${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$ and
${\mathscr{H}}^{i}({\mathscr{G}}^{\bullet})$ are isomorphic.

If $Y$ is a Hodge manifold, then it is a smooth projective variety, according
to a well-known Kodaira’s theorem. By the GAGA correspondence, the coherent
analytic sheaves on $Y$ can be identified with the algebraic ones. Hence, the
$B$-branes on $Y$ can be considered as objects of $K^{b}(Y)_{\rm coh}$.

From now on, we assume that $Y$ is a Hodge manifold. In accordance with the
preceding paragraph, one can assume that the $B$-brane
$({\mathscr{F}}^{\bullet},\delta^{\bullet})$ over $Y$ is a complex of
injective ${\mathscr{O}}$-modules with coherent cohomology modules satisfying
${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})=0$ for $|i|>>0$. Moreover,

${\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}))\simeq{\rm
Hom}_{K^{b}(Y)_{\rm
coh}}({\mathscr{F}}^{\bullet},\,\widehat{{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}})),$

where $\widehat{{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}})$ is an object of
$K^{b}(Y)_{\rm coh}$ $q$-isomorphic to
${\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})$. Thus, the elements of the space
(3.1) can be identified with morphisms in the homotopyy category
$K^{b}(Y)_{\rm coh}$. Given a gauge $\psi\in{\rm
Hom}_{D^{b}(Y)}({\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})),$
we denote by $\hat{\psi}$ the corresponding morphism in the category
$K^{b}(Y)_{\rm coh}.$ The exact sequence (3.4) gives rise to the following
sequence of arrows in the additive category $K^{b}(Y)_{\rm coh}$

$\widehat{\Omega^{1}({\mathscr{F}}^{\bullet})}\overset{\hat{\iota}}{\rightarrow}\widehat{{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})}\overset{\hat{\pi}}{\rightarrow}{\mathscr{F}}^{\bullet}$

where $\widehat{\Omega^{1}({\mathscr{F}}^{\bullet})}$ is quasi-isomorphic to
$\Omega^{1}({\mathscr{F}}^{\bullet})$ and $\hat{\pi}\circ\hat{\psi}=1.$ We
will take to the cohomology of those complexes taking into account that:

  * •

The cohomology of an object with a hat is isomorphic to the cohomology of the
corresponding complex without hat.

  * •

A morphism of the homotopy category, determines a well-defined morphism
between the cohomologies; i. e. a morphism of complexes of
${\mathscr{O}}$-modules.

  * •

${\mathscr{H}}^{*}(\Omega^{1}({\mathscr{F}}^{\bullet}))\simeq\Omega^{1}({\mathscr{H}}^{*}({\mathscr{F}}^{\bullet})).$

Then, one has

(3.23)
$0\to\Omega^{1}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet}))\overset{\boldsymbol{\iota}^{j}}{\longrightarrow}{\mathscr{H}}^{j}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}))\overset{\boldsymbol{\pi}^{j}}{\longrightarrow}{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\to
0,$

where the maps induced between the cohomologies are denoted in bold. In turn,
$\hat{\psi}$ determines a well-defined morphism of ${\mathscr{O}}$-modules
between the cohomologies

$\boldsymbol{\psi}^{j}:{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\to{\mathscr{H}}^{j}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})),$

that obviously satisfies $\boldsymbol{\pi}^{j}\circ\boldsymbol{\psi}^{j}=1.$

Hence, the sequence (3.23) of ${\mathscr{O}}$-modules splits. Furthermore, by
(3.5),
${\mathscr{J}}^{1}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet}))\simeq{\mathscr{H}}^{j}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})).$
Thus, we have a splitting, in the category of ${\mathscr{O}}$-modules, of
${\mathscr{J}}^{1}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet}))$ as direct sum
${\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})$ and
$\Omega^{1}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})).$

According to Definition 1, $\boldsymbol{\psi}^{j}$ defines a holomorphic
connection on the ${\mathscr{O}}$-module
${\mathscr{H}}^{j}({\mathscr{F}}^{\bullet}).$

We set $\boldsymbol{\eta}^{j}$ for the morphism of abelian sheaves defined by
the inclusion in the direct sum

$\boldsymbol{\eta}^{j}:\sf{Forg}\big{(}{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\big{)}\rightarrow\sf{Fog}\big{(}{\mathscr{H}}^{j}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet}))\big{)}=\sf{Forg}\big{(}{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\oplus\Omega^{1}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet}))\big{)},$

Hence, $\boldsymbol{\pi}^{j}(\boldsymbol{\psi}^{j}-\boldsymbol{\eta}^{j})=0$,
and thus $\boldsymbol{\psi}^{j}-\boldsymbol{\eta}^{j}$ defines a morphism of
abelian sheaves

(3.24)
$\vartheta^{j}:{\mathscr{H}}^{j}({\mathscr{F}})\to{\Omega^{1}}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})),$

which, by (3.2), satisfies the Leibniz’s rule (2.14). That is,

######  Proposition 11.

The gauge field $\psi$ on the brane ${\mathscr{F}}^{\bullet}$ determines on
each ${\mathscr{O}}$-module ${\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})$ a
holomorphic connection $\vartheta^{j}$.

######  Definition 4.

The gauge field $\psi$ is called flat, if the curvature of the connection
$\vartheta^{j}$ vanishes, for all $j$.

######  Remark 7.

Let $\psi,\phi$ be two gauge fields on ${\mathscr{F}}$, we set

$\xi:=\phi-\psi\in{\rm
Hom}_{D^{b}(Y)}\big{(}{\mathscr{F}}^{\bullet},\,{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})\big{)},$

and denote by $\hat{\xi}$ the corresponding morphism
${\mathscr{F}}^{\bullet}\to\widehat{{\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})}$
in the category $K^{b}(Y)_{\rm coh}.$ Thus, $\hat{\xi}$ determines a well
defined morphism of ${\mathscr{O}}$-modules between the cohomologies,
$\boldsymbol{\xi}^{j}:{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\to{\mathscr{H}}^{j}({\mathscr{J}}^{1}({\mathscr{F}}^{\bullet})).$

On the other hand, as $\phi$ and $\psi$ are gauge fields,
$\hat{\pi}\hat{\xi}=0$. Hence, $\boldsymbol{\xi}^{j}$ defines morphisms of
${\mathscr{O}}$-modules
$\zeta^{j}({\xi}):{\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})\to\Omega^{1}({\mathscr{H}}^{j}({\mathscr{F}}^{\bullet})).$
We denote by $\vartheta^{j}$ and $\chi^{j}$ the connections on
${\mathscr{H}}^{j}({\mathscr{F}})$ determined by $\hat{\psi}$ and
$\hat{\phi}$, respectively. Since
$\boldsymbol{\xi}^{j}=(\boldsymbol{\phi}^{j}-\boldsymbol{\eta}^{j})-(\boldsymbol{\psi}^{j}-\boldsymbol{\eta}^{j})$,
it follows that $\chi^{j}=\vartheta^{j}+{\zeta}^{j}(\xi)$. In short,
$\zeta^{j}(\xi)$ is the “variation” on the connection $\vartheta^{j}$ induced
by the “variation” $\xi$ of the gauge field $\psi$.

###  3.5. Yang-Mills fields on a brane.

The result deduced in the following paragraph gives us a suggestion for the
definition of the Yang-Mills functional over the gauge fields on a brane over
a Hodge manifold.

####  3.5.1. An Euler-Poincaré mapping.

Let ${\mathscr{A}}$ be a coherent sheaf on the Hodge manifold $Y,$ and
$\alpha:{\mathscr{A}}\to\Omega^{1}({\mathscr{A}})$ a holomorphic connection on
${\mathscr{A}}$. Denoting by ${\mathcal{S}}_{\mathscr{A}}$ the singular set of
${\mathscr{A}}$, on $Y\setminus{\mathcal{S}}_{\mathscr{A}}$ we define
differential form

$\Phi({\mathscr{A}},\alpha):={\rm tr}\big{(}K_{\alpha}\wedge\star
K_{\alpha}\big{)}\in\Gamma(Y\setminus{\mathcal{S}}_{\mathscr{A}},\,\Omega^{\rm
top}\big{)},$

$K_{\alpha}$ being the curvature of $\alpha$ considered as an ${\rm
End}(A)$-valued $2$-form.

By $\mathfrak{C}$, we denote the category whose objects are pairs
$({\mathscr{A}},\,\alpha)$. A morphism
$f:({\mathscr{A}},\,\alpha)\to({\mathscr{B}},\,\beta)$ is a morphism of
coherent sheaves compatible with the connections; i.e. such that $(1\otimes
f)\circ\alpha=\beta\circ f$.

######  Proposition 12.

If
$0\to({\mathscr{A}},\,\alpha)\overset{f}{\to}({\mathscr{B}},\,\beta)\overset{g}{\to}({\mathscr{C}},\,\gamma)\to
0,$ is an exact sequence in $\mathfrak{C}$, then on $Y\setminus{\mathcal{S}}$

$\Phi({\mathscr{B}},\beta)=\Phi({\mathscr{A}},\alpha)+\Phi({\mathscr{C}},\gamma),$

where ${\mathcal{S}}$ is the union of the singular sets of ${\mathscr{A}},$
${\mathscr{B}},$ and ${\mathscr{C}}$.

Proof. Let $y_{0}\in Y\setminus{\mathcal{S}}$. As the exact sequence splits
locally on $Y\setminus{\mathcal{S}}$, there exists an open neighborhood $U$ of
$y_{0}$ such that $g|_{U}$, in the sequence of locally free modules
$0\to{\mathscr{A}}|_{U}\overset{f|_{U}}{\to}{\mathscr{B}}|_{U}\overset{g|_{U}}{\to}{\mathscr{C}}|_{U}\to
0,$ has a right inverse $h$.

Let $a$ be a frame for ${\mathscr{A}}|_{U}$, then $\alpha(a)={\sf A}\cdot a$,
where ${\sf A}$ is a matrix of $1$-forms on $U$. Furthermore, $a$ can be
chosen so that ${\sf A}(y_{0})=0$. Similarly, let $c$ be a frame for
${\mathscr{C}}|_{U}$, then $\gamma(c)={\sf C}\cdot c$ and we choose $c$ so
that ${\sf C}(y_{0})=0$. From the splitting, it follows that
$\\{f(a),\,h(c)\\}$ is a frame for ${\mathscr{B}}|_{U}$. By the compatibility
of the connections with $f$ and $g$,

$\beta(f(a))=(1\otimes f)(\alpha(a))=(1\otimes f)({\sf A}\cdot a)={\sf A}\cdot
f(a).$

On the other hand, $\beta(h(c))={\sf R}\cdot f(a)+{\sf S}\cdot h(c)$, with
${\sf R}$ and ${\sf S}$ matrices of $1$-forms. But,

${\sf C}\cdot c=\gamma(c)=\gamma(gh(c))=(1\otimes g)(\beta(h(c))=(1\otimes
g)\big{(}{\sf R}\cdot f(a)+{\sf S}\cdot h(c)\big{)}.$

As $g\circ f=0$ and $g\circ h=1$, it follows that ${\sf C}={\sf S}.$ That is,
the matrix of the connection $\beta$ in the frame $\\{f(a),\,h(c)\\}$ is

(3.25) ${\sf M}:=\begin{pmatrix}{\sf A}&{\sf R}\\\ 0&{\sf C}\end{pmatrix}$

Since ${\sf A}(y_{0})=0$ and ${\sf C}(y_{0})=0$, the matrix of
$K_{\alpha}(y_{0})$, of the curvature of $\alpha$ at the point $y_{0}$, is
$d{\sf A}$. Analogously, the matrix of $K_{\gamma}(y_{0})$ is $d{\sf C}$. The
one of $K_{\beta}(y_{0})$ is the exterior derivative of (3.25), since ${\sf
M}\wedge{\sf M}=0$ at $y_{0}$. Then

$\displaystyle{\rm tr}\big{(}K_{\beta}(y_{0})\wedge\star
K_{\beta}(y_{0})\big{)}$ $\displaystyle={\rm tr}(d{\sf A}\wedge\star d{\sf
A})+{\rm tr}(d{\sf C}\wedge\star d{\sf C})$ $\displaystyle={\rm
tr}\big{(}K_{\alpha}(y_{0})\wedge\star K_{\alpha}(y_{0})\big{)}+{\rm
tr}\big{(}K_{\gamma}(y_{0})\wedge\star K_{\gamma}(y_{0})\big{)}.$

As $y_{0}$ is an arbitrary point of $Y\setminus{\mathcal{S}}$, it follows the
proposition. ∎

Let $({\mathscr{G}}^{\bullet},\,{\delta}^{\bullet})$ be a bounded complex of
coherent sheaves on the manifold $Y$. Let $\nabla^{\bullet}$ be a family of
holomorphic connections, compatible with the operators $\delta^{\bullet}$.
That is,
$\nabla^{i}:{\mathscr{G}}^{i}\rightarrow\Omega^{1}({\mathscr{G}}^{i})$ is a
holomorphic gauge field on the coherent sheaf ${\mathscr{G}}^{i}$ such that

$(1\otimes\delta^{i})\nabla^{i}=\nabla^{i+1}\delta^{i}.$

Hence,
$\nabla^{i}({\mathscr{K}er}({\delta}^{i}))\subset{\mathscr{K}er}(1\otimes{\delta}^{i})$
and a similar relation for the image ${\mathscr{I}m}(\delta^{i-1})$. It
follows that $\nabla^{i}$ induces a connection $\theta^{i}$ on the cohomology

$\theta^{i}:{\mathscr{H}}^{i}({\mathscr{G}}^{\bullet})\to\Omega^{1}({\mathscr{H}}^{i}({\mathscr{G}}^{\bullet})).$

Obviously, the restrictions of $\nabla^{i}$ determine connections on
${\mathscr{K}er}(\delta^{i})$ and ${\mathscr{I}m}(\delta^{i+1})$,
respectively. One has the exact sequence

(3.26)
$0\to\big{(}{\mathscr{K}er}(\delta^{i}),\,\nabla^{i}\big{)}\to\big{(}{\mathscr{G}}^{i},\,\nabla^{i}\big{)}\to\big{(}{\mathscr{I}m}(\delta^{i}),\,\nabla^{i+1}\big{)}\to
0$

in the category $\mathfrak{C}$. Similarly, we have the exact sequence

(3.27)
$0\to\big{(}{\mathscr{I}m}(\delta^{i-1}),\,\nabla^{i}\big{)}\to\big{(}{\mathscr{K}er}(\delta^{i}),\,\nabla^{i}\big{)}\to\big{(}{\mathscr{H}}^{i},\,\theta^{i}\big{)}\to
0.$

######  Corollary 3.

Denoting with ${\mathcal{S}}$ the union of the singular sets of the sheaves
${\mathscr{G}}^{i}$, then on $Y\setminus{\mathcal{S}}$

(3.28) $\sum_{i}(-1)^{i}{\rm tr}\big{(}K_{\nabla^{i}}\wedge\star
K_{\nabla^{i}}\big{)}=\sum_{i}(-1)^{i}{\rm tr}\big{(}K_{\theta^{i}}\wedge\star
K_{\theta^{i}}\big{)}.$

Proof. From Proposition 12 together with (3.26), it follows

$\Phi({\mathscr{G}}^{i},\,\nabla^{i})=\Phi({\mathscr{K}er}(\delta^{i}),\,\nabla^{i})+\Phi({\mathscr{I}m}(\delta^{i}),\,\nabla^{i+1}).$

From (3.27), one obtains an analogous relation. Taking the alternate sums

$\sum_{i}(-1)^{i}\Phi({\mathscr{G}}^{i},\,\nabla^{i})=\sum_{i}(-1)^{i}\Phi({\mathscr{H}}^{i},\,\theta^{i}).$

∎

####  3.5.2. The Yang-Mills functional.

We propose a definition for the Yang-Mills functional over gauge fields on a
brane. This proposal is based on the following considerations:

  1. (1)

It is reasonable to require that this definition generalizes the one for
coherent sheaves.

  2. (2)

As a gauge field is a homotopy class of a morphism of complexes, it seems
convenient to move on the cohomology of these complexes.

  3. (3)

Let $E^{\bullet}$ be a bounded complex of Hermitian vector bundles over the
Kähler manifold $Y,$ and $\nabla^{\bullet}$ a family of connections compatible
with the Hermitian metrics and the coboundary operators. Denoting by
$H^{i}(E^{\bullet})$ the cohomology bundles, there exist connections
$\theta^{i}$ on those bundles, induced by the family $\nabla^{\bullet}$. By
Corollary 3 together with (2.19), one has the following equality of Euler-
Poincaré type.

$\sum_{i}(-1)^{i}\lVert
K_{\nabla^{i}}\rVert^{2}=\sum_{i}(-1)^{i}\lVert(K_{\theta^{i}}\rVert^{2}.$

On the basis of the above considerations, it seems appropriate to define the
value of the Yang-Mills functional on the gauge $\psi$ on the brane
${\mathscr{F}}^{\bullet}$ as $\sum_{i}(-1)^{i}\lVert
K_{\vartheta_{i}}\rVert^{2}.$

More precisely, taking into account Proposition 11, we adopt the following
definitions.

######  Definition 5.

The brane $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ is called a Hermitian
brane, if the cohomology sheaves ${\mathscr{H}}^{j}$ are Hermitian.

Let $({\mathscr{F}}^{\bullet},\,{\delta}^{\bullet})$ be a Hermitian brane on
the Hodge manifold $Y$. Given a gauge field $\psi$ on the brane
$({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$, by Proposition 11, one has the
family of curvatures ${\mathcal{K}}_{\vartheta^{i}}$ of the connections
induced on the cohomologies. Whose norms
$\lVert{\mathcal{K}}_{\vartheta^{i}}\rVert$ are defined in accordance with
(2.18).

######  Definition 6.

Given a gauge field $\psi$ on the Hermitian $B$-brane
$({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$, if the sheaves
${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$, are reflexive, we define the
value of the Yang-Mills functional at $\psi$ by

(3.29)
$\mathcal{YM}(\psi)=\sum_{i}(-1)^{i}\lVert{\mathcal{K}}_{\vartheta_{i}}\rVert^{2}.$

Thus, $\mathcal{YM}(\psi)$ is a kind of Euler characteristic of the gauge
field. The Yang-Mills fields on the brane
$({\mathscr{F}}^{\bullet},\,{\delta}^{\bullet})$ are the critical points of
the functional $\psi\mapsto\mathcal{YM}(\psi).$

Note that, if $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ is an acyclic
complex, then the Yang-Mills functional for this complex is identically zero.

Let
${\mathscr{A}}^{\bullet}:=\big{(}{\mathscr{A}}^{\bullet},\,d_{A}^{\bullet},\,\alpha^{\bullet}\big{)}$
be a complex in the category ${\mathfrak{C}}$; i.e, a complex of coherent
sheaves with a family of holomorphic connections compatible with the
coboundary operator $d_{A}$. Let $f:=(f^{\bullet})$ a morphism
$f^{\bullet}:{\mathscr{A}}^{\bullet}\to{\mathscr{B}}^{\bullet}$ in
${\mathfrak{C}}$; that is, $f$ is a morphism of complexes compatible with the
connections. Let ${\mathscr{C}}^{\bullet}$ denote the mapping cone of $f$.
Thus,
${\mathscr{C}}^{\bullet}=\big{(}{\mathscr{A}}^{\bullet}[1]\oplus{\mathscr{B}}^{\bullet},d_{C}^{\bullet},\nabla^{\bullet},\big{)}$,
with $d_{C}(a,\,b)=\big{(}d(a),\,(-1)^{{\rm degree}\,a}f(a)+db\big{)}$ and
$\nabla(a,\,b)=(\alpha(a),\,\beta(b)).$ In fact, $(1\otimes
d_{C})\circ\nabla=\nabla\circ d_{C}$ and thus ${\mathscr{C}}^{\bullet}$ is a
complex of the category ${\mathfrak{C}}$

For each $i$ one has the following exact sequence in the category
${\mathfrak{C}}$

$0\to{\mathscr{B}}^{i}\to{\mathscr{C}}^{i}\to{\mathscr{A}}^{i}[1]\to 0.$

From Proposition 12,
$\Phi({\mathscr{B}}^{i})+\Phi({\mathscr{A}}^{i+1})=\Phi({\mathscr{C}}^{i})$.
Multiplying by $(-1)^{i}$ and summing

(3.30) $\displaystyle\sum_{i}(-1)^{i}{\rm tr}(K_{\beta^{i}}\wedge\star
K_{\beta^{i}})+$ $\displaystyle\sum_{i}(-1)^{i}{\rm
tr}(K_{\alpha^{i+1}}\wedge\star K_{\alpha^{i+1}})$
$\displaystyle=\sum_{i}(-1)^{i}{\rm tr}(K_{\nabla^{i}}\wedge\star
K_{\nabla^{i}}).$

Let us assume that

  * •

${\mathscr{A}}^{i}$ and ${\mathscr{B}}^{i}$ Hermitian sheaves for all $i$.

  * •

$\alpha^{i}$ and $\beta^{i}$ are Hermitian gauge fields (i.e., compatible with
the metric) on ${\mathscr{A}}^{i}$ and ${\mathscr{B}}^{i}$, respectively.

Then one defines on ${\mathscr{C}}^{i}$ the metric
$\langle(a,b),\,(a^{\prime},b^{\prime})\rangle:=\langle
a,\,a^{\prime}\rangle+\langle b,\,b^{\prime}\rangle$. The connection
$\nabla^{i}$ is compatible with this metric. From the equality (3.30) together
with (2.19), one deduces the following proposition.

######  Proposition 13.

With the above notations and under the above hypotheses, $\alpha$ and $\beta$
determine in a natural way a gauge field $\nabla$ on the mapping cone of
$f^{\bullet}$ satisfying

(3.31) $\mathcal{YM}(\beta)-\mathcal{YM}(\alpha)=\mathcal{YM}(\nabla).$

On the other hand, in the context of the branes theory, the fact that the
branes ${\mathscr{A}}^{\bullet},\,{\mathscr{B}}^{\bullet}$ and
${\mathscr{C}}^{\bullet}$ are the members of the distinguished triangle
${\mathscr{A}}^{\bullet}\to{\mathscr{B}}^{\bullet}\to{\mathscr{C}}^{\bullet}\overset{+1}{\to}$
means that ${\mathscr{A}}^{\bullet}$ and ${\mathscr{C}}^{\bullet}$ can
potentially bind together to form the membrane ${\mathscr{B}}^{\bullet}$ [2,
Section 6.2.1]. Thus, the additive nature of equation (3.31) is consistent
with this interpretation.

From now on, we assume that ${\mathscr{F}}^{\bullet}$ is a Hermitian $B$-brane
such that the cohomology sheaves ${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$
are reflexive.

Let us suppose that the set of gauge fields on the brane
${\mathscr{F}}^{\bullet}$ is nonempty. Let $m$ be the dimension of the vector
space ${\rm
Ext}^{0}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet})).$ We
denote by $\xi_{1},\dots,\xi_{m}$ a basis of this vector space. According to
Proposition 5, any gauge field $\psi$ on the brane can be expressed

$\psi=\tilde{\psi}+\sum_{a}\lambda_{a}\xi_{a}$

$\tilde{\psi}$ being a fixed gauge field and $\lambda_{a}\in{\mathbb{C}}.$
Hence, the connections on the cohomology sheaves ${\mathscr{H}}^{i}$ can be
written in the form (see Remark 7)

$\vartheta^{i}=\tilde{\vartheta}^{i}+\sum_{a}\lambda_{a}\zeta^{i}_{a},$

with $\zeta^{i}_{a}\in{\rm
Hom}({\mathscr{H}}^{i},\,\Omega^{1}({\mathscr{H}}^{i}))$. The corresponding
curvatures satisfy

$K_{\vartheta^{i}}=K_{\tilde{\vartheta}^{i}}+\sum_{a}\lambda_{a}\tilde{\vartheta}^{i}(\zeta^{i}_{a})+\sum_{a,b}\lambda_{a}\lambda_{b}\zeta^{i}_{a}\wedge\zeta^{i}_{b}.$

Therefore, $\lVert K_{\vartheta^{i}}\rVert^{2}$ is a polynomial
$P^{i}(\lambda_{1},\dots,\lambda_{m})$ of degree $\leq 4.$ Thus, the critical
points of the Yang-Mills functional correspond to the points
$(\lambda_{1},\dots,\lambda_{m})\in{\mathbb{C}}^{m}$ which satisfy the
equations $\frac{\partial P}{\partial\lambda_{a}}=0,$ where $P$ is the
polynomial $\sum_{i}(-1)^{i}P^{i}(\lambda_{1},\dots,\lambda_{m})$. We have the
following result, which generalizes Theorem 1.

######  Theorem 5.

Assumed the set of gauge fields on the brane ${\mathscr{F}}^{\bullet}$ is
nonempty and $m={\rm dim}\,{\rm
Ext}^{0}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}))$. Then
the set of Yang-Mills fields on ${\mathscr{F}}^{\bullet}$ are in bijective
correspondence with the points of a subvariety of ${\mathbb{C}}^{m}$ defined
by $m$ polynomials of degree $\leq 3$.

Let $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ be a $B$-brane, such that
the ${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$ are reflexive sheaves. If
$\phi$ and $\psi$ are gauge fields on ${\mathscr{F}}^{\bullet}$ and
$\xi=\phi-\psi$, using the notations introduced in Remark 7, the connections
on the cohomologies induced by $\phi$ and $\psi$ satisfy
$\chi^{j}(\xi)=\vartheta^{j}+\zeta^{j}(\xi)$, with

$\zeta^{j}(\xi)\in\Gamma(Y\setminus{\mathcal{S}},\,\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{E}nd}(H^{j})),$

${\mathcal{S}}$ being the union of the singularity sets of the reflexive
sheaves ${\mathscr{H}}^{j}.$

With the mentioned notation, an infinitesimal variation $\psi_{\epsilon}$ of
$\psi$ is an element of the form $\epsilon\xi,$ where
$\epsilon\in{\mathbb{C}}$ and $\xi\in{\rm
Ext}^{0}({\mathscr{F}}^{\bullet},\,\Omega^{1}({\mathscr{F}}^{\bullet}).$ The
relation $\psi_{\epsilon}=\psi+\epsilon\xi$ gives rise to the following
equality between the connections on the cohomologies

$\vartheta^{j}_{\epsilon}=\vartheta^{j}+\epsilon\zeta^{j}.$

Furthermore, on $Y\setminus{\mathcal{S}}$ the curvatures satisfy

$K_{\vartheta^{j}_{\epsilon}}=K_{\vartheta^{j}}+\epsilon\vartheta^{j}({\zeta^{j}})+O((\epsilon)^{2}),$

$\vartheta^{j}({\zeta^{j}})$ being the covariant derivative of ${\zeta^{j}}$
considered as a section of
$\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{E}nd}(H^{j})$.

From (2.22) together with Definition 6, the functional $\mathcal{YM}$ takes at
the gauge field ${\psi}$ a stationary value if

(3.32)
$(1/2)\frac{d}{d\epsilon}\mathcal{YM}(\psi_{\epsilon})\Big{|}_{\epsilon=0}=\sum_{j}(-1)^{j}\langle
K_{\vartheta^{j}},\,\vartheta^{j}(\zeta^{j})\rangle=0,$

for all any variation of $\psi$.

In particular, if $\vartheta^{i}$ es a Yang-Mills field for all $i,$ then by
(2.23)

(3.33) $\langle K_{\vartheta^{i}},\,\vartheta^{i}(\zeta^{i})\rangle=0,$

and we have the following proposition.

######  Proposition 14.

Let $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ be a Hermitian brane, such
that the sheaves ${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$ are reflexive,
and let $\psi$ gauge field on ${\mathscr{F}}^{\bullet}$. If $\vartheta^{i}$ is
a Yang-Mills field on ${\mathscr{H}}^{i}$ for all $i$, then $\psi$ is a
stationary point of the Yang-Mills functional; i.e. $\psi$ is a Yang-Mills
field on the brane.

The following theorem is a converse to Proposition 14.

######  Theorem 6.

Let ${\mathscr{F}}^{\bullet}$ be a Hermitian $B$-brane as in Proposition 14.
If $\nabla^{\bullet}$ is a Yang-Mills field on ${\mathscr{F}}^{\bullet},$ then
the connection $\vartheta^{j}$ induced on ${\mathscr{H}}^{j}$ is a Yang-Mills
field on this sheaf.

Proof. We will consider $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ as an
object of the category $K^{b}_{\rm coh}(Y)$; that is, we assume that
$({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ is a complex of injective
${\mathscr{O}}$-modules such that its cohomology is bounded and coherent.

As ${\mathscr{K}er}(\delta^{i})$ is a submodule of the injective
${\mathscr{O}}$-module ${\mathscr{F}}^{i}$, then ${\mathscr{K}er}(\delta^{i})$
is a retract of ${\mathscr{F}}^{i}$ and thus it is also an injective
${\mathscr{O}}$-module. Therefore, the following short exact sequence of
coherent ${\mathscr{O}}$-modules

$0\to{\mathscr{K}er}(\delta^{i})\to{\mathscr{F}}^{i}\to{\mathscr{C}oim}(\delta^{i})\to
0$

splits. That is,
${\mathscr{F}}^{i}\simeq{\mathscr{K}er}(\delta^{i})\oplus{\mathscr{C}oim}(\delta^{i}).$

Since ${\mathscr{I}m}({\delta}^{i-1})$ is a retract of the injective
${\mathscr{O}}$-module ${\mathscr{K}er}(\delta^{i})$, the following short
exact sequence also splits

$0\to{\mathscr{I}m}({\delta}^{i-1})\to{\mathscr{K}er}(\delta^{i})\to{\mathscr{H}}^{i}\to
0.$

Thus,

(3.34) ${\mathscr{F}}^{i}\simeq{\mathscr{H}}^{i}\oplus{\mathscr{G}}^{i},$

where ${\mathscr{G}}^{i}$ is isomorphic to the direct sum of
${\mathscr{C}oim}(\delta^{i})$ and ${\mathscr{I}m}(\delta^{i-1})$. As
${\mathscr{H}}^{i}$ and ${\mathscr{G}}^{i}$ are summands in a direct sum
decomposition of an injective ${\mathscr{O}}$-module, they are also injective.

On the other hand, the coboundary operator
$\delta^{i}:{\mathscr{F}}^{i}\to{\mathscr{F}}^{i+1}$ induces via the
isomorphisms (3.34) to the morphism

(3.35)
$\delta^{i}:{\mathscr{H}}^{i}\oplus{\mathscr{G}}^{i}\to{\mathscr{H}}^{i+1}\oplus{\mathscr{G}}^{i+1},\;\;\;(a,\,b)\mapsto(0,\,\delta^{i}b).$

Given $\xi\in{\rm Hom}_{K^{b}_{\rm
coh}(Y)}\big{(}{\mathscr{F}}^{\bullet},\,\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{F}}^{\bullet}\big{)}$,
according to Remark 7, it determines $\zeta^{i}\in{\rm
Hom}\big{(}{\mathscr{H}}^{i},\,\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{H}}^{i}\big{)}.$
As $\nabla^{\bullet}$ is, by hypothesis, a Yang-Mills field then (3.32) is
satisfied.

Given $j$, a general “variation” of $\vartheta^{j}$ is defined by an element
$\tau\in{\rm
Hom}\big{(}{\mathscr{H}}^{j},\,\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{H}}^{j}\big{)}.$
Under the hypotheses of the proposition, we need to prove that

$\langle K_{\vartheta^{j}},\,\vartheta^{j}(\tau)\rangle=0,$

for any variation $\tau$.

The morphism $\tau$ can be extended

$C^{i}:{\mathscr{H}}^{i}\oplus{\mathscr{G}}^{i}\to\Omega^{1}\otimes_{\mathscr{O}}\big{(}{\mathscr{H}}^{i}\oplus{\mathscr{G}}^{i}\big{)},$

where

$C^{i}(a,\,b)=\begin{cases}(\tau(a),\,0),\;\;\text{if}\;\,i=j\\\
(0,\,0),\;\;\text{if}\;\,i\neq j\end{cases}$

Moreover, the $C^{i}$ are compatible with the coboundaries. For example for
$i=j$, by (3.35),
$((1\otimes\delta^{j})\circ{C}^{j})(a,\,b)=(1\otimes\delta^{j})(\tau(a),\,0)=0$;
and ${C}^{j+1}\circ\delta^{j}(a,\,b)=0$. Thus, by the isomorphism (3.34) the
$C^{i}$ determine a morphism
$\xi:{\mathscr{F}}^{\bullet}\to\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{F}}^{\bullet}$
in the category $K^{b}_{\rm coh}(Y)$, and the corresponding $\zeta^{i}$
induced in the cohomologies are all $0$ except when $i=j$, in which case
$\zeta^{j}=\tau$. Hence, by (3.32)

$0=\sum_{i}(-1)^{i}\langle
K_{\vartheta^{i}},\,\vartheta^{i}(\zeta^{i})\rangle=(-1)^{j}\langle
K_{{\vartheta}^{j}},\,\vartheta^{j}(\zeta^{j})\rangle=(-1)^{j}\langle
K_{{\vartheta}^{j}},\,\vartheta^{j}(\tau)\rangle.$

This holds for any “variation” $\tau$ of $\vartheta^{j}$. That is, by (2.23),
$\vartheta^{j}$ is a Yang-Mills field on ${\mathscr{H}}^{j}$. ∎

From Theorem 6 together with Proposition 14, it follows the corollary.

######  Corollary 4.

Let $({\mathscr{F}}^{\bullet},\,\delta^{\bullet})$ be a Hermitian $B$-brane on
the Hodge manifold $Y,$ such that sheaves
${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$ are reflexive. A gauge field
$\psi$ on the brane is a Yang-Mills field iff $\vartheta^{i}$ is a Yang-Mills
field on ${\mathscr{H}}^{i}({\mathscr{F}}^{\bullet})$ for all $i$.

## References

  * [1] Akyildiz, E. and Carrell, J. B.: Betti Numbers of smooth Schubert varieties and the remarkable formula of Kostant, Macdonald, Shapiro, and Steinberg. Michigan Math. J. 61, 543-553 (2012)
  * [2] Aspinwall, P. S.: $D$-branes on Calabi-Yau manifolds. In J.M. Maldacena (ed) Progress in String Theory, World Sci. Publ., 2005, 1-152.
  * [3] Aspinwall, P. S. et al.: Dirichlet branes and mirror symmetry. Clay mathematics monographs vol 4. Amer. Math. Soc., Providence 2009.
  * [4] Atiyah, M.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85, 181-207 (1957).
  * [5] Böhning, Ch.: Derived categories of coherent sheaves on rational homogeneous manifolds. Documenta Math. 11, 261-331 (2006)
  * [6] Brion, M.: Lectures on the geometry of flag varieties. In Pragacz P. (ed.) Topics on Cohomological Studies of Algebraic Varieties. pp 87-103 Birkhäuser 2005.
  * [7] Deligne, P.: Equations différentialles à points singuliers réguliers. Lecture Notes 163. Springer-Verlag, Berlin 1970.
  * [8] Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Springer-Verlag 1995.
  * [9] Fulton, W.: Young tableaux with applications to representation theory and geometry. London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge 1997.
  * [10] Gelfand, S. I. and Manin, Y. I.: Homological algebra. Springer, Berlin 1999.
  * [11] Gelfand, S. I. and Manin, Y. I.: Methods of homological algebra. Springer, Berlin 2002.
  * [12] Grauert, H. and Remmert, R.: Coherent analytic sheaves. Springer-Verlag. Berlin 1984.
  * [13] Griffiths, P. and Harris, J.: Principles of algebraic geometry. John Wiley, New York 1994.
  * [14] Hamilton, M. J. D.: Mathematical gauge theory. Springer 2017.
  * [15] Hartshorne, R.: Residues and Duality. Lecture Notes in Math., vol. 20, Springer-Verlag, Berlin 1966.
  * [16] Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254, 121-176 (1980).
  * [17] Hartshorne, R.: Generalized divisors on Gorenstein schemes. Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin. Part III, vol 8 1994 287-339.
  * [18] Huybrechts, D.: Complex geometry. Springer, Berlin 2005.
  * [19] Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Oxford U. P., Oxford 2006.
  * [20] Hotta, R., Takeuchi, K. and Tanisaki, T.: D-Modules, perverse sheaves, and representation theory. Bikhäuser, Basel 2008.
  * [21] Iversen, B.: Cohomology of sheaves. Springer-Verlag, Berlin 1986.
  * [22] Kashiwara, M. and Schapira, P.: Sheaves on manifolds. Springer-Verlag 2002.
  * [23] Katz, N.: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin. Publ. Math. I.H.E.S. 39 175-232 (1970).
  * [24] Katz, S. and Sharpe, E.: D-branes, open string vertex operators, and Ext groups. Adv. Theor. Math. Phys. 6 979-1030 (2003).
  * [25] Kobayashi, S.: Differential geometry of complex vector bundles. Iwanami Shoten Publishers and Princeton University Press, Princeton 1987.
  * [26] Lakshmibai, V. and Sandya, B.: Criterion for smoothness of Schubert varieties in ${\rm Sl}(n)/B.$ Proc. Indian Acad. Sci. Math. Sci. 100, 45-52 (1990)
  * [27] Moore, J. D.: Lectures on Seiberg-Witten Invariants. Springer, Berlin 2001.
  * [28] Okonek Ch., Schneider, M. and Spindler, H.: Vector bundles on complex projective spaces. Birkhäuser, Basel 1988.
  * [29] Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), dirigé par P. Berthelot, A. Grothendieck, L. Illusie, avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. Lecture Notes in Math., vol. 225, Springer-Verlag, Berlin 1971.
  * [30] Viña, A.: Gauge fields on coherent sheaves. arXiv:2109.11841 [math. AG].
  * [31] Weibel, Ch. A.: Homological algebra. Cambridge U.P., Cambridge 1997.
  * [32] Wells, R. O.: Differential Analysis on Complex Manifolds. Springer, New York 2008.