# Degenerate Hermitian geometry and
curvatures of holomorphic fibrations

Gunnar Þór Magnússon  Hafnarfjörður, Iceland<EMAIL_ADDRESS>
###### Abstract.

We calculate curvature tensors of metrics on the total spaces of holomorphic
fibrations. Our main tool is a theory of Chern connections and curvature forms
for possibly degenerate Hermitian forms on holomorphic vector bundles. We
prove a version of the Codazzi–Griffiths equations for curvatures of sub- and
quotient bundles in that setting and apply them to the study of honest
Hermitian metrics on fibrations. We apply this to calculate the curvature of a
metric on Grassmannian bundles and prove they have positive holomorphic
sectional curvature if the base does.

## Introduction

Let $\pi:X\to S$ be a family of complex manifolds, and consider the associated
short exact sequence

${0}$${T_{X/S}}$${T_{X}}$${\pi^{*}T_{S}}$${0.}$

We often find ourselves in the situation of having a metric $h_{S}$ on the
base $S$ and a Hermitian form $h_{X/S}$ on the $X$ that is positive-definite
on the relative tangent bundle $T_{X/S}$, and then construct a metric on $X$
as the sum $h_{X}:=h_{X/S}+\pi^{*}h_{S}$. This happens when we consider a
projective or Grassmannian bundle associated to a vector bundle [1, 2, 4]; the
blow-up of a point like in the original proof of the Kodaira embedding theorem
[11]; metrics on holomorphic fibrations [5]; or metrics on versal families of
deformations [12].

In these situations, the metric on the base has a curvature tensor, and the
relative form often has something that resembles such a tensor. As neither
form is positive-definite on $X$, we can’t use our usual tools to calculate a
curvature tensor for $h_{X}$ from this information. Instead, we are often
reduced to committing difficult calculations in local coordinates; see for
example Chaturvedi and Heier [6, Theorem 1.1].

If we had the equivalent of Chern connections and curvature forms for
arbitrary Hermitian forms, we could try to prove a version of the
Codazzi–Griffiths equations [10] that relate the curvature form of sub- and
quotient bundles to that of the ambient bundle. In the positive-definite case,
those equations can be used to calculate the curvature tensor of the sum
$h_{1}+h_{2}$ of two metrics on a given bundle $E$ by considering the short
exact sequence

${0}$${E}$${E\oplus E}$${E}$${0,}$

where the metrics on the bundles are (in order) $h_{1}+h_{2}$, $h_{1}\oplus
h_{2}$, and the induced metric on the “quotient”; see [15, Chapter 7, Exercise
6].

In this note, we fill in the details of this sketch. In section 2 we show that
a holomorphic vector bundle $E\to X$ with a smooth Hermitian form $b$ admits a
Chern connection. Such a connection is not unique when the form is degenerate,
but we can always get rid of the ambiguity between different choices of
connections by applying the form $b$, as the difference of two such
connections must take values in its kernel. This yields a unique curvature
_tensor_ for such a form. We then prove the Codazzi–Griffiths equations hold
in this setting and calculate what they imply for sums of Hermitian forms.

As an application, in section 3 we first recover a result of Chaturvedi and
Heier [6], and then dot some i’s and cross some t’s from works of Álvarez [1]
and Álvarez, Heier and Zheng [2] on the curvature of metrics on projectivized
vector bundles (see also Yang and Zheng [4]). They prove that such bundles
admit metrics of positive holomorphic sectional curvature, and finish by
saying the same arguments can prove that so do Grassmannian bundles. We treat
the slightly more general case of Grassmannian bundles without mention of an
underlying vector bundle, and as a corollary conclude that bundles of flag
manifolds admit metrics of positive holomorphic sectional curvature once the
base manifold does. Along with our insistence on coordinate-invariant
calculations, these are the new results of our note.

To do this, we require a degenerate analogue of many statements about
adjoints, orthogonal subspaces and induced inner products that are classical
in the positive-definite case. We do not know a reference for these results,
which would make for fun exercises in a linear algebra course, and thus
provide statements and proofs. We do this in section 1 as we need to refer to
those results for our work on connections and curvature.

A seeming novelty of our approach is that we prove all our results using the
connection formulation of complex differential geometry (versus the local
coordinate or moving frame formulations). While this connection approach is
common in modern Riemannian geometry, complex geometers seem to prefer the
older, index-heavy, ways. We hope this note also serves as an advertisement
for the merits of the connection approach.

There is some prior art on degenerate metrics in Riemannian geometry; see for
example [3, 13, 14]. The very quick version seems to be that one can define a
Levi-Civita connection for such an object and thus a curvature tensor, but
neither will generally be unique. This nicely mirrors what we find here.

We finish this introduction by noting that we think the technical results
developed here on general Hermitian forms are only interesting when applied to
the study of honest Hermitian metrics, and do not have merit on their own.
Some examples may explain this opinion:

######  Example 0.1.

More degenerate the form is, the less information the connection gives: Let
$E\to X$ be a holomorphic vector bundle, let $b=0$, and let $D$ be any
connection whose $(0,1)$-part is $\bar{\partial}$. Then $D$ is a Chern
connection of $b$.

######  Example 0.2.

Let $X$ be a compact complex manifold of dimension $n$ that admits a nowhere-
zero holomorphic one-form $\alpha$. It gives us a smooth Hermitian form
$b=i\alpha\wedge\overline{\alpha}$ whose kernel forms a holomorphic subbundle
of rank $n-1$ and whose quotient is the trivial line bundle. Any of our Chern
connections for this form will yield a curvature form whose top exterior power
vanishes; so if we were to develop Chern–Weil theory in this setting we should
expect to conclude the top Chern class of the manifold is zero. However, a
much easier way to conclude the same is to use the additivity property of
total Chern classes on the short exact sequence we began with.

######  Example 0.3.

Even when a form is non-degenerate, having a Chern connection and curvature
form just doesn’t let us say very much: Let $\pi:X\to S$ be a family of
compact complex Kähler manifolds of dimension $2n$ over a smooth base $S$. Let
$E\to S$ be the holomorphic vector bundle whose sheaf of sections is
$\mathcal{R}^{n}\pi_{*}\mathbb{C}\otimes\mathcal{O}_{S}$ and whose fiber over
a point $s$ is $E_{s}=H^{n}(X_{s},\mathbb{C})$. This vector bundle admits a
non-degenerate Hermitian form given by the cup product, and the flat
Gauss–Manin connection is the Chern connection of this Hermitian form.

Also note that in the non-degenerate case the most useful results are found
when the form is positive-definite. For example the Laplacian of a non-
degenerate form of mixed signature will not be strongly elliptic,111For
example, on the torus $\mathbb{C}^{2}/\mathbb{Z}^{4}$ with the form
$b=\smash{\frac{i}{2}}dz_{1}\wedge
d\bar{z}_{1}-\smash{\frac{i}{2}}dz_{2}\wedge d\bar{z}_{2}$ the kernel of the
Laplacian is infinite-dimensional. so the Hodge isomorphism theorem fails, and
with it go most of the tools and techniques developed over the last hundred
years or so.

We’ll finish with one final example we don’t have any applications in mind
for, but find amusing in its inaccessibility.

######  Example 0.4.

Let $(E,h)\to X$ be a holomorphic vector bundle with a Hermitian metric $h$.
The curvature tensor of the metric,

$R(\alpha,\overline{\beta},s,\overline{t})=h(\tfrac{i}{2}\Theta_{\alpha,\overline{\beta}}s,\overline{t}),$

defines a Hermitian form on $E\otimes T_{X}$ that is usually not positive-
definite. Good luck saying anything about its curvature tensor.

##  1\. Linear algebra

All vector spaces are finite-dimensional and over the complex numbers. Let $V$
be a vector space and let $b$ be a Hermitian form on $V$. We want to study the
geometry of the pair $(V,b)$.222Everything we say also applies to symmetric
bilinear forms on finite-dimensional vector spaces over arbitrary fields. The
reader who is interested in those can mentally erase all the conjugations of
vectors or spaces. Our objective is to establish a version of the
Codazzi–Griffiths equations for sub- and quotient bundles of a holomorphic
vector bundle, so the main questions we want to answer here are:

  * •

When does an adjoint of a linear morphism exist?

  * •

Does a Hermitian form on $V$ induce Hermitian forms on all spaces derived from
its tensor algebra?

  * •

Given a short exact sequence $0\to S\to V\to Q\to 0$ and a Hermitian form on
$V$, do we get Hermitian forms on $S$ and $Q$?

######  Definition 1.1.

A _Hermitian space_ is a finite-dimensional vector space $V$ with a Hermitian
form $b$. A _morphism_ of Hermitian spaces is a linear morphism $f:V\to W$
such that

$b_{W}(fx,\overline{fy})=b_{V}(x,\overline{y})$

for all $x,y\in V$.

The category this forms is the one that has Hermitian spaces for objects and
the above morphisms between them. Given two objects in this category, there is
not necessarily any morphism between them; if $V$ has a non-degenerate form
while $W$ has the zero form, there is no morphism $V\to W$.

A Hermitian form can be seen as a linear morphism $b:V\to\overline{V}^{*}$. As
such, it has a kernel. We clearly have

$\operatorname{Ker}b=\\{x\in V\mid b(x,\overline{y})=0\text{ for all $y\in
V$}\\}.$

The results we want to prove are all known for spaces with a non-degenerate
form. Our main technical tool will be to reduce to that case by quotienting
out the kernel of the form we have.

######  Proposition 1.2.

Let $V$ be a vector space. For every Hermitian form $b$ on $V$, there exists a
unique non-degenerate Hermitian form $\hat{b}$ on $V/\operatorname{Ker}V$ such
that $q:V\to V/\operatorname{Ker}b$ is a Hermitian morphism.

###### Proof.

We define

$\hat{b}(qx,\overline{qy})=b(x,\overline{y}).$

This is well-defined, as any other element in $qx$ differs from the one we
chose by an element of $\operatorname{Ker}q=\operatorname{Ker}b$. The form
$\hat{b}$ is also non-degenerate, as if $qx\in\operatorname{Ker}\hat{b}$ then
$x\in\operatorname{Ker}b$, so $qx=0$. The form is also defined in such a way
that $q$ is a morphism of Hermitian spaces.

If $b^{\prime}$ is another non-degenerate Hermitian form on
$V/\operatorname{Ker}b$ that makes $q$ into a Hermitian morphism, then

$b^{\prime}(qx,\overline{qy})=b(x,\overline{y})=b(qx,\overline{qy})$

for all $qy\in V/\operatorname{Ker}b$. Thus $b(qx)=b^{\prime}(qx)$ as
$(0,1)$-forms for all $qx\in V/\operatorname{Ker}b$, so $b=b^{\prime}$. ∎

If this operation should be called anything, it is clearly a _purge_ of the
original space, as it removes all the degeneracy from the Hermitian form.

### Existence of adjoints

######  Definition 1.3.

Let $(V,b_{V})$ and $(W,b_{W})$ be Hermitian spaces, and let $f:V\to W$ be a
linear morphism. An _adjoint_ of $f$ is a linear morphism $f^{\dagger}:W\to V$
such that

$b_{V}(f^{\dagger}x,\overline{y})=b_{W}(x,\overline{fy})$

for all $x\in W$ and $y\in V$.

If $b_{V}$ is non-degenerate, every morphism $f:V\to W$ admits a unique
adjoint. It is defined by

$f^{\dagger}=b_{V}^{-1}\circ\overline{f}^{*}\circ b_{W},$

where $\overline{f}^{*}$ is the conjugate dual morphism
$f^{*}:\overline{W}^{*}\to\overline{V}^{*}$ that $f$ induces. In general
adjoints may not exist.

######  Theorem 1.4.

Let $f:V\to W$ be a linear morphism. The morphism $f$ admits an adjoint if and
only if $f(\operatorname{Ker}b_{V})\subset\operatorname{Ker}b_{W}$. If the set
of adjoints of $f$ is not empty, it is a
$\operatorname{Hom}(W,\operatorname{Ker}b_{V})$ torsor.

###### Proof.

If $f$ admits an adjoint $f^{\dagger}$, then

$b_{V}(f^{\dagger}x,\overline{y})=b_{W}(x,\overline{fy})$

for all $x\in W$ and $y\in V$. Taking $y\in\operatorname{Ker}b$, we see that
$b_{W}(x,\overline{fy})=0$ for all $x\in W$, so
$f(y)\in\operatorname{Ker}b_{W}$.

Suppose then that $f(\operatorname{Ker}b_{V})\subset\operatorname{Ker}b_{W}$.
Then $f$ defines a linear morphism $\hat{f}:V/\operatorname{Ker}b_{V}\to
W/\operatorname{Ker}b_{W}$. As the induced forms on those spaces are non-
degenerate, we get a unique adjoint
$\hat{f}^{\dagger}:W/\operatorname{Ker}b_{W}\to V/\operatorname{Ker}b_{V}$. It
induces a morphism $W\to V/\operatorname{Ker}b_{V}$. Picking a lift
$V/\operatorname{Ker}b_{V}\to V$ we then get an adjoint $f^{\dagger}$ of $f$.

If $g$ and $h$ are adjoints of $f^{\dagger}$, then

$b_{V}((g-h)x,\overline{y})=b_{V}(gx,\overline{y})-b_{V}(hx,\overline{y})=b_{W}(x,\overline{fy})-b_{W}(x,\overline{fy})=0$

for all $x\in W$ and $y\in V$, so the image of $g-h$ is contained in
$\operatorname{Ker}b_{V}$. ∎

######  Corollary 1.5.

The subset of $\operatorname{Hom}(V,W)$ of morphisms that have adjoints is a
linear subspace of codimension $\dim\operatorname{Ker}b_{V}(\dim
W-\dim\operatorname{Ker}b_{W})$.

###### Proof.

The subset is not empty because the zero morphism always has an adjoint, and
it is clearly a subspace.

Let’s pick splittings $V=\operatorname{Ker}b_{V}\oplus
V/\operatorname{Ker}b_{V}$ and $W=\operatorname{Ker}b_{W}\oplus
W/\operatorname{Ker}b_{W}$. Then we get a splitting

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\operatorname{Hom}(V,W)=\operatorname{Hom}(\operatorname{Ker}b_{V},\operatorname{Ker}b_{W})\oplus\operatorname{Hom}(\operatorname{Ker}b_{V},W/\operatorname{Ker}b_{W})\hfill\cr
0.0pt{\hfil$\displaystyle\hfill{}\oplus\operatorname{Hom}(V/\operatorname{Ker}b_{v},\operatorname{Ker}b_{W})\oplus\operatorname{Hom}(V/\operatorname{Ker}b_{v},W/\operatorname{Ker}b_{W}).\crcr}}}$

A morphism $f:V\to W$ satisfies $f(\operatorname{Ker}b_{V})\subset
f(\operatorname{Ker}b_{W})$ if and only if its projection onto
$\operatorname{Hom}(\operatorname{Ker}b_{V},W/\operatorname{Ker}b_{W})$ in
this splitting is zero. ∎

######  Corollary 1.6.

If $f:V\to W$ has an adjoint $f^{\dagger}$, then $f^{\dagger}$ also has an
adjoint.

###### Proof.

If $x\in\operatorname{Ker}W$, then

$b_{V}(f^{\dagger}x,\overline{y})=b_{W}(x,\overline{fy})=0$

for all $y\in V$, so $f^{\dagger}x\in\operatorname{Ker}b_{V}$. ∎

The difference between a morphism $f$ and an adjoint of its adjoint is an
element of $\operatorname{Hom}(V,\operatorname{Ker}b_{V})$, so a double
adjoint is unique and equal to $f$ if and only if $b_{V}$ is non-degenerate.

######  Proposition 1.7.

Let $j:S\to V$ be a linear morphism, let $b_{V}$ be a Hermitian form on $V$,
and set $b_{S}=j^{*}b_{V}$. Then $j$ admits an adjoint.

###### Proof.

We have

$\operatorname{Ker}b_{S}=j^{-1}(\operatorname{Ker}b_{V})+\operatorname{Ker}j$

so
$j(\operatorname{Ker}b_{S})=j(j^{-1}(\operatorname{Ker}b_{V}))\subset\operatorname{Ker}b_{V}$.
∎

### Orthogonal complements

The objective of this section is to show how to obtain a Hermitian form on a
quotient space given one on the ambient space. It is well-known how to do this
when the forms are non-degenerate (for example, take duals to embed the
quotient dual in the ambient dual, restrict the dual metric, and invert) but
less clear how this should work in the general case.

######  Definition 1.8.

Let $(V,b)$ be a Hermitian space, and let $S\subset V$ be a subspace. The
_orthogonal complement_ of $S$ is

$S^{\perp}=\\{v\in V\mid b(v,\overline{w})=0\text{ for all $w\in S$}\\}.$

The orthogonal complement of a subspace is clearly again a subspace.

######  Proposition 1.9.

$\operatorname{Ker}b\subset S^{\perp}.$

###### Proof.

Let $v\in\operatorname{Ker}b$. Then $b(v,\overline{w})=0$ for all $w\in S$. ∎

As before we have the quotient space $q:V\to V/\operatorname{Ker}b$, and the
induced non-degenerate form $\hat{b}$ on $V/\operatorname{Ker}b$.

######  Proposition 1.10.

Let $S\subset V$. Then

$q(S^{\perp})=q(S)^{\perp},$

where the orthogonal complement on the right-hand side is with respect to
$\hat{b}$.

###### Proof.

Let $v\in S^{\perp}$, so $b(v,\overline{w})=0$ for all $w\in S$. Then
$\hat{b}(qv,\overline{qw})=0$ for all $w\in S$, which means that
$\hat{b}(qv,\overline{w})=0$ for all $w\in q(S)$, so $qv\in q(S)^{\perp}$.

Suppose then that $v\in q(S)^{\perp}$, so $\hat{b}(v,\overline{w})=0$ for all
$w\in q(S)$. If $x\in V$ is such that $qx=v$, then this means that
$b(x,\overline{w})=0$ for all $w\in S$. Then $x\in S^{\perp}$, so $qx\in
q(S^{\perp})$. ∎

######  Proposition 1.11.

$S+S^{\perp}=V.$

###### Proof.

We have

$q(S+S^{\perp})=q(S)+q(S^{\perp})=q(S)+q(S)^{\perp}=V/\operatorname{Ker}b$

as $\hat{q}$ is non-degenerate. As $q$ is surjective, this implies the result.
∎

######  Proposition 1.12.

$S\cap S^{\perp}=S\cap\operatorname{Ker}b.$

###### Proof.

We have

$q(S\cap S^{\perp})=q(S)\cap q(S^{\perp})=q(S)\cap q(S)^{\perp}=0$

as $\hat{q}$ is non-degenerate, so $S\cap
S^{\perp}\subset\operatorname{Ker}q=\operatorname{Ker}b$. Conversely,
$\operatorname{Ker}b\subset S^{\perp}$, so $S\cap\operatorname{Ker}b\subset
S\cap S^{\perp}$. ∎

If $f:S\to V$ is a linear morphism and $b_{V}$ is a Hermitian form on $V$,
then we get an induced Hermitian form $b_{S}:=f^{*}b_{V}$ on $S$. We can ask
whether the same happens when we have a morphism $f:V\to Q$ and a form
$b_{V}$. Clearly we can only expect something on the image of $f$, so we
should take it to be surjective. When the form on $b_{V}$ is positive-
definite, we can define a form on $Q$ by using the injection $0\to Q^{*}\to
V^{*}$ and the dual Hermitian forms. In general, we can still get a Hermitian
form on the quotient space by using orthogonal complements.

######  Theorem 1.13.

Let $q:V\to Q$ be a surjective morphism. A Hermitian form $b_{V}$ on $V$
induces a Hermitian form $b_{Q}$ on $Q$.

###### Proof.

Let $S=\operatorname{Ker}q$, and consider $S^{\perp}\subset V$. We have a
short exact sequence

${0}$${S\cap\operatorname{Ker}b_{V}}$${S^{\perp}}$${Q}$${0}$$\scriptstyle{q}$

because $S^{\perp}\cap\operatorname{Ker}q=S^{\perp}\cap
S=S\cap\operatorname{Ker}b_{V}$, and $q$ restricted to $S^{\perp}$ is still
surjective. We define

$b_{Q}(qx,\overline{qy})=b_{V}(x,\overline{y})$

for $x,y\in S^{\perp}$. This is well-defined by the above, as any other
element $x^{\prime}$ that maps to $qx$ satisfies $x^{\prime}-x\in
S\cap\operatorname{Ker}b_{V}\subset\operatorname{Ker}b_{V}$. ∎

######  Proposition 1.14.

Let $q:V\to Q$ be surjective, and let $b_{V}$ and $b_{Q}$ be as above. Then
$q$ admits an adjoint $q^{\dagger}:Q\to V$.

###### Proof.

We need to show that
$q(\operatorname{Ker}b_{V})\subset\operatorname{Ker}b_{Q}$. If $x,y$ are
elements of $V$, we have by definition

$b_{Q}(qx,\overline{qy})=b_{V}(x,\overline{y}).$

If $x\in\operatorname{Ker}b_{V}$, this implies that
$b_{Q}(qx,\overline{qy})=b_{V}(x,\overline{y})=0$ for all $y\in V$, so
$qx\in\operatorname{Ker}b_{Q}$. ∎

### Induced forms on tensor algebras

If $V$ is a vector space with a Hermitian form $b_{V}$, we get induced
Hermitian forms on any vector space derived from the tensor algebra of $V$ if
we restrict ourselves to spaces that only involve tensor powers of $V$ and not
$V^{*}$. Handling $V^{*}$, or more generally $\operatorname{Hom}(V,W)$, is
more delicate when the forms are possibly degenerate.

If $V$ and $W$ are vector spaces with Hermitian forms $b_{V}$ and $b_{W}$,
we’d like to use those forms to define a Hermitian form on
$\operatorname{Hom}(V,W)$. When the form $b_{V}$ is non-degenerate, every
linear morphism $f:V\to W$ admits a unique adjoint $f^{\dagger}:W\to V$, and
the Fröbenius form on $\operatorname{Hom}(V,W)$ is

$b(f,\overline{g})=\operatorname{tr}(g^{\dagger}f).$

In general, not every morphism $f:V\to W$ admits an adjoint, and the ones that
do may not admit unique adjoints. We could try to define a Hermitian form on
the subspace of morphisms that admit adjoints by
$\operatorname{tr}(g^{\dagger}f)$, but as adjoints may not be unique this is
not well-defined.

The way around this difficulty is again to pass to the quotient spaces. A
morphism $f:V\to W$ admits an adjoint if and only if
$f(\operatorname{Ker}b_{V})\subset\operatorname{Ker}b_{W}$. In the latter
case, there exists a unique morphism $\hat{f}:V/\operatorname{Ker}b_{V}\to
W/\operatorname{Ker}b_{W}$ such that the diagram

${V}$${W}$${V/\operatorname{Ker}b_{V}}$${W/\operatorname{Ker}b_{W}}$$\scriptstyle{\pi_{V}}$$\scriptstyle{f}$$\scriptstyle{\pi_{W}}$$\scriptstyle{\hat{f}}$

commutes; it is defined by $\hat{f}([v])=\pi_{W}(f(v))$.

######  Definition 1.15.

Let $(V,b_{V})$ and $(W,b_{W})$ be Hermitian spaces. We define

$b_{\operatorname{Hom}(V,W)}(f,\overline{g}):=\operatorname{tr}((\hat{g})^{\dagger}\hat{f})$

on the subspace of $\operatorname{Hom}(V,W)$ of morphisms that admit adjoints.

If we denote by $\operatorname{Hom}_{b}(V,W)\subset\operatorname{Hom}(V,W)$
the subspace of morphisms that admit adjoints, the definition says that we
have a surjective morphism

$\operatorname{Hom}_{b}(V,W)\to\operatorname{Hom}(V/\operatorname{Ker}b_{V},W/\operatorname{Ker}b_{W})$

and that we define a Hermitian form upstairs by pullback of the one
downstairs, which exists because the Hermitian forms there are non-degenerate.

The special case $V^{*}=\operatorname{Hom}(V,\mathbb{C})$, where $\mathbb{C}$
is equipped with the standard inner product, is interesting. Here an element
$\lambda\in V^{*}$ admits an adjoint if and only if
$\lambda(\operatorname{Ker}b_{V})=0$, that is, if
$\operatorname{Ker}b_{V}\subset\operatorname{Ker}\lambda$. Note however that
the dimension of the space of dual vectors that admit adjoints is $\dim
V-\dim\operatorname{Ker}b_{V}$. Meanwhile, the dimension of the space of
morphisms $V\to W$ that admit adjoints is $\dim V\dim
W-\dim\operatorname{Ker}b_{V}(\dim W-\dim\operatorname{Ker}b_{W})$. We then
have

$\dim\operatorname{Hom}_{b}(V,W)-\dim V^{*}_{b}\dim
W=\dim\operatorname{Ker}b_{v}\dim\operatorname{Ker}b_{W},$

which is generally positive, so the canonical map $V_{b}^{*}\otimes
W\to\operatorname{Hom}_{b}(V,W)$ is generally only injective and not bijective
like in the non-degenerate case. The problem is that the image of any element
on the left-hand side will be a morphism $f$ that satisfies
$f(\operatorname{Ker}b_{V})=0$, while there are more elements on the right-
hand side.

##  2\. Chern connections

Let $E\to X$ be a holomorphic vector bundle with a smooth Hermitian form $b$.
We say that a connection $D$ on $E$ is _compatible with $b$_ if

$db(s,\overline{t})=b(Ds,\overline{t})+b(s,\overline{Dt})$

for all sections $s,t$ of $E$. By composing a connection with the projection
onto $(1,0)$\- or $(0,1)$-forms, we get the $(1,0)$\- and $(0,1)$-parts of the
connection.

######  Proposition 2.1.

Let $E\to X$ be a holomorphic vector bundle and let $b$ be a smooth Hermitian
form on $E$. Then there exists a connection $D$ on $E$ that’s compatible with
$b$ and has $D^{0,1}=\bar{\partial}$.

###### Proof.

The plan is to construct a connection locally on trivializing neighborhoods of
$E$ and to glue the pieces together with a partition of unity.

Let $s$ be a smooth section of $E$ over a neighborhood $U$. We want to solve
the equation

(1) $\partial_{\alpha}b(s,\overline{t})=b(A(\alpha),\overline{t}),$

where $\alpha\in\mathcal{C}^{\infty}(U,T_{X})$,
$t\in\mathcal{C}^{\infty}(U,E)$ and $A\in\mathcal{A}^{1,0}(U,E)$, where the
last space is the one of smooth $(1,0)$-forms with values in $E$. Consider the
short exact sequence

${0}$${\operatorname{Ker}b}$${E}$${\operatorname{Im}b}$${0,}$$\scriptstyle{b}$

and tensor it by the vector bundle $\Omega^{1}_{X}$ to obtain

${0}$${\Omega^{1}_{X}\otimes\operatorname{Ker}b}$${\Omega^{1}_{X}\otimes
E}$${\Omega^{1}_{X}\otimes\operatorname{Im}b}$${0.}$$\scriptstyle{\operatorname{id}\otimes
b}$

We claim that $\alpha\otimes t\mapsto\partial_{\alpha}b(s,\overline{t})$ is a
smooth section of $\Omega^{1}_{X}\otimes\operatorname{Im}b$. By the above,
this is true if it is zero on the sheaf generated by sections of the form
$\alpha\otimes t$, where $t\in\operatorname{Ker}b$. But for such a section we
have $b(s,\overline{t})=0$ on $U$, so $\partial_{\alpha}b(s,\overline{t})=0$.
By taking a small enough neighborhood so the sheaf morphism above is
surjective, or noting that the sheaves are ones of smooth modules which are
fine, we obtain a section
$A_{s}\in\mathcal{C}^{\infty}(U,\Omega_{X}^{1}\otimes E)$ such that (1) holds
for any $\alpha$ and $t$.

Suppose now that there exists a holomorphic frame $(e_{1},\ldots,e_{r})$ of
$E$ over $U$. Such a frame exists over a small enough neighborhood around any
point. Let $A_{1},\ldots,A_{r}$ be the sections we just found for
$e_{1},\ldots,e_{r}$. Any section $s\in\mathcal{C}^{\infty}(U,E)$ can be
written as $s=\sum_{j}f_{j}e_{j}$ for smooth functions $f_{j}$ on $U$. We
define

$Ds:=\sum_{j=1}^{r}df_{j}\otimes e_{j}+f_{j}A_{j}$

on $U$. We claim that $D$ is a connection on $E$ over $U$ that is compatible
with $b$ and whose $(0,1)$-part is $\bar{\partial}$.

This object $D$ is a $1$-form with values in $E$ by construction of the
$A_{j}$. If $s$ is a section of $E$ and $f$ a smooth function, we write
$s=\sum_{j}f_{j}e_{j}$. Then

$\displaystyle D(fs)$ $\displaystyle=\sum d(ff_{j})\otimes e_{j}+ff_{j}A_{j}$
$\displaystyle=\sum df\otimes f_{j}e_{j}+fdf\otimes
e_{j}+ff_{j}A_{j}=df\otimes s+fDs,$

so $D$ is a connection on $E$. The forms $A_{j}$ are $(1,0)$-forms by
construction, so the $(0,1)$-part of $Ds$ is
$\sum_{j}\bar{\partial}f_{j}e_{j}=\bar{\partial}s$.

Finally, let $f$ and $g$ be smooth functions. We have

$\displaystyle db(fe_{j},\overline{ge_{k}})$
$\displaystyle=d\bigl{(}f\overline{g}b(e_{j},\overline{e}_{k})\bigr{)}$
$\displaystyle=df\cdot\overline{g}b(e_{j},\overline{e}_{k})+fdg\cdot
b(e_{j},\overline{e}_{k})+\partial
b(e_{j},\overline{e}_{k})+\overline{\partial b(e_{k},\overline{e}_{j})}$
$\displaystyle=b(df\otimes
e_{j},\overline{ge_{k}})+b(fe_{j},\overline{dg\otimes
e_{k}})+b(A_{j},\overline{e}_{k})+b(e_{j},\overline{A_{k}})$
$\displaystyle=b(D(fe_{j}),\overline{ge_{k}})+b(fe_{j},\overline{D(ge_{k})}).$

For smooth sections $s=\sum f_{j}e_{j}$ and $t=\sum g_{k}e_{k}$ we deduce from
the above and linearity that $D$ is compatible with $b$ over the neighborhood
$U$.

We now cover $X$ by neighborhoods $U_{\mu}$ as above and construct connections
$D_{\mu}$ on each $U_{\mu}$. We then take a partition of unity $(\phi_{\mu})$
relative to this covering, and define a connection $D$ on $E$ by setting

$Ds=\sum_{\mu}D_{\mu}(\phi_{\mu}s).$

This defines a connection on $E$ that’s compatible with $b$ and whose
$(0,1)$-part is $\bar{\partial}$. ∎

We will call a connection $D$ that’s compatible with $b$ and has
$D^{0,1}=\bar{\partial}$ a _Chern connection_ of $b$. If $b$ has a nontrivial
kernel, such a connection is not unique.

######  Proposition 2.2.

Let $D_{1}$ and $D_{2}$ be Chern connections of $b$. Then $D_{1}-D_{2}$ is an
element of $\mathcal{A}^{1,0}(\operatorname{Hom}(E,\operatorname{Ker}b))$.

###### Proof.

It is well-known that the difference of two connections is an element of
$\mathcal{A}^{1}(\operatorname{End}E)$. Both connections here have the same
$(0,1)$-part, so their difference is a $(1,0)$-form. If $s$ and $t$ are
holomorphic sections, we have

$b((D^{\prime}_{1}-D^{\prime}_{2})s,\overline{t})=\partial
b(s,\overline{t})-\partial b(s,\overline{t})=0,$

so $D_{1}-D_{2}=D_{1}^{\prime}-D_{2}^{\prime}$ takes values in
$\operatorname{Ker}b$. ∎

######  Proposition 2.3.

If $s$ is a section of $\operatorname{Ker}b$ and $D$ is a Chern connection of
$E$, then $Ds$ takes values in $\operatorname{Ker}b$.

###### Proof.

If $s$ is a section of $\operatorname{Ker}b$, then

$0=\partial b(s,\overline{t})=b(Ds,\overline{t})$

for all holomorphic sections $t$ of $E$, so
$Ds\in\mathcal{A}^{1}(\operatorname{Ker}b)$. ∎

######  Theorem 2.4.

Let $(E,b)\to X$ be a holomorphic vector bundle with a Hermitian form, and let
$D$ be a Chern connection of $E$. The curvature tensor

$R(\alpha,\overline{\beta},s,\overline{t})=b\bigl{(}\tfrac{i}{2}D^{2}_{\alpha\overline{\beta}}s,\overline{t}\bigr{)}$

is independent of the choice of $D$.

###### Proof.

Let $D_{1}$ and $D_{2}$ be Chern connections of $b$. Then $D_{1}=D_{2}+A$,
where $A\in\operatorname{Hom}(E,\operatorname{Ker}b)$, and

$D_{1}^{2}=(D_{2}+A)^{2}=D_{2}^{2}+AD_{2}+D_{2}A+A^{2}.$

The difference $D_{1}^{2}-D_{2}^{2}$ thus takes values in
$\operatorname{Ker}b$. ∎

######  Proposition 2.5.

Let $(E,b_{E})\to X$ and $(F,b_{F})\to X$ be holomorphic vector bundles with
Hermitian forms and let $D_{E}$ and $D_{F}$ be Chern connections of $E$ and
$F$. Let $f:E\to F$ be a smooth bundle morphism that admits an adjoint
$f^{\dagger}$. Then $D_{\operatorname{Hom}(E,F)}f$ admits an adjoint and
$(D_{\operatorname{Hom}(E,F)}f)^{\dagger}$ and
$D_{\operatorname{Hom}(F,E)}f^{\dagger}$ are both its adjoints.

###### Proof.

The morphism $f$ admits an adjoint if and only if
$f(\operatorname{Ker}b_{E})\subset\operatorname{Ker}b_{F}$. Let $s$ be a
section of $\operatorname{Ker}b_{E}$, so $f(s)$ is a section of
$\operatorname{Ker}b_{F}$. Then

$\displaystyle 0=db_{F}(f(s),\overline{t})$
$\displaystyle=b_{F}(D_{F}(f(s)),\overline{t})+b_{F}(f(s),\overline{Dt})$
$\displaystyle=b_{F}(D_{F}(f(s)),\overline{t})$
$\displaystyle=b_{F}(D_{\operatorname{Hom}(E,F)}f(s),\overline{t})+b_{F}(f(D_{E}s),\overline{t})$
$\displaystyle=b_{F}(D_{\operatorname{Hom}(E,F)}f(s),\overline{t})$

for all sections $t$, as $D_{E}s$ is a one-form with values in
$\operatorname{Ker}b_{E}$. Then
$D_{\operatorname{Hom}(E,F)}f(s)\in\mathcal{A}^{1}(\operatorname{Ker}b_{F})$
for all sections $s$ of $\operatorname{Ker}b_{E}$, so
$D_{\operatorname{Hom}(E,F)}f$ admits an adjoint.

We have $b_{F}(f(s),\overline{t})=b_{E}(s,\overline{f^{\dagger}t})$ for all
sections $s,t$. Taking the exterior derivative of both sides we get

$b_{F}(D_{F}(f(s)),\overline{t})+b_{F}(f(s),\overline{D_{F}t})=b_{E}(D_{E}s,\overline{f^{\dagger}t})+b_{E}(s,\overline{D_{E}(f^{\dagger}t)}).$

The left-hand side equals

$b_{F}(D_{\operatorname{Hom}(E,F)}f(s),\overline{t})+b_{F}(f(D_{E}s),\overline{t})+b_{E}(s,\overline{f^{\dagger}(D_{F}t)})$

while the right-hand side equals

$b_{E}(f(D_{E}s),\overline{t})+b_{E}(s,\overline{D_{\operatorname{Hom}(F,E)}f^{\dagger}(t)})+b_{E}(s,\overline{f^{\dagger}(D_{F}t)})$

so after cancelling terms we have

$b_{E}(s,\overline{D_{\operatorname{Hom}(F,E)}f^{\dagger}(t)})=b_{F}(D_{\operatorname{Hom}(E,F)}f(s),\overline{t})=b_{E}(s,\overline{(D_{\operatorname{Hom}(E,F)}f)^{\dagger}(t)})$

for all sections $s,t$. ∎

When $b_{E}$ is non-degenerate morphisms from $E$ to $F$ admit a unique
adjoint, so in that case
$D_{\operatorname{Hom}(F,E)}f^{\dagger}=(D_{\operatorname{Hom}(E,F)}f)^{\dagger}$.
Note that taking the adjoints of forms with values in a bundle conjugates the
form part.

######  Proposition 2.6.

Let $f:X\to Y$ be a holomorphic morphism and $E\to Y$ be a holomorphic vector
bundle. Let $b$ be a Hermitian form on $E$ with a Chern connection $D$. Then
$f^{*}D$ is a Chern connection of $f^{*}b$ on $f^{*}E\to X$.

###### Proof.

If $s$ and $t$ are holomorphic sections of $E$ and $\xi$ a tangent vector at a
point on $X$ we have

$\partial_{\xi}f^{*}b(f^{*}s,\overline{f^{*}t})=f^{*}(\partial_{f_{*}\xi}b(s,\overline{t}))=f^{*}(b(D_{f_{*}\xi}s,\overline{t}))=f^{*}b(f^{*}D_{\xi}f^{*}s,\overline{f^{*}t}).$

As $f^{*}E$ is generated by sections of the form $f^{*}s$, this completes the
proof. ∎

######  Proposition 2.7.

Let $X$ be a complex manifold and let $b$ be a Hermitian form on $T_{X}$. Let
$\omega=-\operatorname{Im}b$ be the associated Kähler form. Then $d\omega=0$
if and only if the torsion tensor $\tau(\alpha,\beta)=D_{\alpha}\beta-
D_{\beta}\alpha-[\alpha,\beta]$ of $b$ takes values in $\operatorname{Ker}b$.

###### Proof.

For holomorphic tangent fields we have

$\partial
b(\alpha,\beta,\overline{\gamma})=\partial_{\alpha}b(\beta,\overline{\gamma})-\partial_{\beta}b(\alpha,\overline{\gamma})-b([\alpha,\beta],\overline{\gamma})=b(\tau(\alpha,\beta),\overline{\gamma})$

and $\partial b=0$ if and only if $\partial\omega=0$. ∎

### A word on notation

In a lot of situations we care about, we now only have equality up to things
that take values in the kernel of a Hermitian form. For example, if $f:E\to F$
is a Hermitian morphism, it admits an adjoint $f^{\dagger}$ and we have

$b_{E}(f^{\dagger}fs,\overline{t})=b_{F}(fs,\overline{ft})=b_{E}(s,\overline{t})$

for all sections $s$ and $t$, so
$f^{\dagger}f-\operatorname{id}_{E}\in\operatorname{Hom}(E,\operatorname{Ker}b_{E})$.
When working on vector spaces we could deal with this by quotienting out the
kernel. If we try the same trick here, we run into the fact that the rank of
the sheaf $\operatorname{Ker}b\to X$ may not be constant. We want to do
differential geometry to the sections of this sheaf, so this is a problem.

This difficulty is a mirage as it can be taken care of by notation. We define
an equivalence relation on sections by saying that two sections are equivalent
if their difference is in the kernel of a Hermitian form of interest:

######  Definition 2.8.

If $(E,b)\to X$ is a holomorphic vector bundle with a Hermitian form $b$, and
$s,t\in\mathcal{A}^{k}(E)$, we write $s\sim t$ if
$s-t\in\mathcal{A}^{k}(\operatorname{Ker}b)$.

In general it’s problematic to define differential forms with values in a
sheaf, but here it’s hopefully clear what
$\mathcal{A}^{k}(\operatorname{Ker}b)$ means as $\operatorname{Ker}b$ is a
subsheaf of the sheaf of sections of a vector bundle.

Using this notation we can rephrase some of our earlier results on degenerate
forms:

  * •

If $f:E\to F$ is a Hermitian morphism and $f^{\dagger}$ an adjoint, then
$f^{\dagger}f\sim\operatorname{id}_{E}$.

  * •

If $f:E\to F$ is a morphism and $f_{1}^{\dagger}$ and $f_{2}^{\dagger}$ are
adjoints of $f$, then $f_{1}^{\dagger}\sim f_{2}^{\dagger}$.

  * •

If $f:E\to F$ is a morphism that admits an adjoint, then
$D_{\operatorname{Hom}(E,F)}$ has adjoints
$(D_{\operatorname{Hom}(E,F)}f)^{\dagger}\sim
D_{\operatorname{Hom}(F,E)}f^{\dagger}$.

  * •

If $D$ is a Chern connection of $(E,b)$ and $s\sim t$ then $Ds\sim Dt$.

  * •

If $D_{1}$ and $D_{2}$ are Chern connections of $(E,b)$, then $D_{1}s\sim
D_{2}s$ for all sections $s$.

As adjoints and connections behave well with respect to this relation, we can
prove the result we want by essentially the same arguments as in the case
where the Hermitian forms are positive-definite; we simply replace “$=$” in
those proofs by “$\sim$” and note that everything else works as before.

An example of this notation’s usefulness is the following result, which is
annoying to state and use without it. For Hermitian metrics, the $(2,0)$-part
of the curvature form is zero. For our Chern connections, the same is not
necessarily true; we can’t say what happens in the kernel of the form.

######  Proposition 2.9.

Let $(E,b)\to X$ be a Hermitian bundle, and let $D$ be a Chern connection for
$b$. Then $(D^{\prime})^{2}\sim 0$.

###### Proof.

We have

$\partial
b(s,\overline{t})=b(D^{\prime}s,\overline{t})+b(s,\overline{\bar{\partial}t})$

so

$0=\partial^{2}b(s,\overline{t})=b((D^{\prime})^{2}s,\overline{t})-b(D^{\prime}s,\overline{\bar{\partial}t})+b(D^{\prime}s,\overline{\bar{\partial}t})+b(s,\overline{\bar{\partial}^{2}t})=b((D^{\prime})^{2}s,\overline{t})$

for all sections $s$ and $t$. ∎

### Sub- and quotient bundles

Let

${0}$${S}$${E}$${Q}$${0}$$\scriptstyle{j}$$\scriptstyle{q}$

be a short exact sequence of holomorphic bundles over $X$, and let $b_{E}$ be
a smooth Hermitian form on $E$. It induces Hermitian forms $b_{S}$ and $b_{Q}$
on $S$ and $Q$ as we have seen. Both morphisms $j$ and $q$ also admit adjoints
$j^{\dagger}:E\to S$ and $q^{\dagger}:Q\to E$. We denote Chern connections of
$S$, $E$ and $Q$ by $D_{S}$, $D_{E}$ and $D_{Q}$.

######  Definition 2.10.

The _second fundamental form_ of $S$ in $E$ is

$\sigma(s):=q(D_{E}(js)-jD_{S}(s)).$

######  Proposition 2.11.

The second fundamental form is an element of
$\mathcal{A}^{1,0}(\operatorname{Hom}(S,Q))$. We have

$\sigma(s)=q(D_{\operatorname{Hom}(S,E)}j(s))=q(D_{E}(js))$

for sections $s$ of $S$.

###### Proof.

As $qj=0$ it is clear that $b(s)=q(D_{E}(js))$. We also have

$D_{E}(js)=D_{\operatorname{Hom}(S,E)}(j)(s)+j(D_{S}s),$

so $b(s)=q(D_{\operatorname{Hom}(S,E)}(j)(s))$.

As $j$ is holomorphic, we have
$D_{\operatorname{Hom}(S,E)}j=D^{\prime}_{\operatorname{Hom}(S,E)}j$ so
$\sigma$ has no $(0,1)$-part. It is also clearly $\mathcal{C}^{\infty}$-linear
in its tensor field variable. If $f$ is a smooth function, we have

$D_{E}(j(fs))=df\otimes js+fD_{E}(js)$

so $\sigma(fs)=q(D_{E}(j(fs)))=q(fD_{E}(js))=f\sigma(s)$. ∎

######  Proposition 2.12.

The second fundamental form admits an adjoint.

###### Proof.

We have to show that $\sigma(s)\in\operatorname{Ker}b_{Q}$ when $s$ is a
section of $\operatorname{Ker}b_{S}$. If $s$ is such a section, then $js$ is a
section of $\operatorname{Ker}b_{E}$, so $D_{E}(js)$ is a section of
$\operatorname{Ker}b_{E}$. Then $q(D_{E}(js))$ is a section of
$\operatorname{Ker}b_{Q}$ by the construction of $b_{Q}$. ∎

Our goal is to calculate the curvature forms of the sub- and quotient bundles.
The first stop on the way is a wildly useful list of formulas from Demailly
[8, Theorem 14.3].

######  Proposition 2.13.

$\displaystyle D^{\prime}_{\operatorname{Hom}(S,E)}j$ $\displaystyle\sim
q^{\dagger}\circ\sigma,$ $\displaystyle\qquad\bar{\partial}j$
$\displaystyle=0,$ $\displaystyle D^{\prime}_{\operatorname{Hom}(E,Q)}q$
$\displaystyle\sim-\sigma\circ j^{\dagger},$ $\displaystyle\bar{\partial}q$
$\displaystyle=0,$ $\displaystyle
D^{\prime}_{\operatorname{Hom}(E,S)}j^{\dagger}$ $\displaystyle\sim 0,$
$\displaystyle\bar{\partial}j^{\dagger}$ $\displaystyle\sim b^{\dagger}\circ
q,$ $\displaystyle D^{\prime}_{\operatorname{Hom}(Q,E)}q^{\dagger}$
$\displaystyle\sim 0,$ $\displaystyle\bar{\partial}q^{\dagger}$
$\displaystyle\sim-j\circ\sigma^{\dagger},$ $\displaystyle
D^{\prime}_{\operatorname{Hom}(S,Q)}\sigma$ $\displaystyle\sim 0,$
$\displaystyle\bar{\partial}\sigma^{\dagger}$ $\displaystyle\sim 0.$

###### Proof.

Let $s$ be a section of $S$. Then

$j(D_{S}s)+q^{\dagger}\sigma(s)\sim
D_{E}(js)=D_{\operatorname{Hom}(S,E)}j(s)+j(D_{S}s),$

where the first relation is by definition of $D_{S}$ and $\sigma$, so

$D_{\operatorname{Hom}(S,E)}j\sim q^{\dagger}\circ\sigma.$

The morphism $j$ is holomorphic, so $\bar{\partial}j=0$. This proves the first
line. The third line follows from the first by taking adjoints.

For the second line, the morphism $q$ is also holomorphic, so
$\bar{\partial}q=0$. Taking adjoints, we get that
$D^{\prime}_{\operatorname{Hom}(Q,E)}q^{\dagger}\sim 0$. We have
$\operatorname{id}_{E}\sim j\circ j^{\dagger}+q^{\dagger}\circ q$, so

$\displaystyle 0$
$\displaystyle=D^{\prime}_{\operatorname{End}E}\operatorname{id}_{E}$
$\displaystyle\sim D^{\prime}_{\operatorname{Hom}(S,E)}j\circ
j^{\dagger}+j\circ
D^{\prime}_{\operatorname{Hom}(E,S)}j^{\dagger}+D^{\prime}_{\operatorname{Hom}(Q,E)}q^{\dagger}\circ
q+q^{\dagger}D^{\prime}_{\operatorname{Hom}(E,Q)}q$ $\displaystyle\sim
q^{\dagger}\circ\sigma\circ j^{\dagger}+q^{\dagger}\circ
D^{\prime}_{\operatorname{Hom}(E,Q)}q.$

Applying $q$, we conclude that
$D^{\prime}_{\operatorname{Hom}(E,Q)}q\sim-\sigma\circ j^{\dagger}$. This
proves the second line. Taking adjoints we get the last piece of the fourth
line.

Finally we note that

$\displaystyle D^{\prime}_{\operatorname{Hom}(S,Q)}\sigma$ $\displaystyle\sim
D^{\prime}_{\operatorname{Hom}(S,Q)}(qD^{\prime}_{\operatorname{Hom}(S,E)}j)$
$\displaystyle\sim-\sigma\circ j^{\dagger}\circ
q^{\dagger}\circ\sigma+q^{\dagger}(D^{\prime}_{\operatorname{Hom}(S,Q)})^{2}j\sim
0$

as we’re dealing with curvature tensors of Hermitian forms which are of type
$(1,1)$ up to things that are $\sim 0$. The statement about
$\bar{\partial}\sigma^{\dagger}$ follows by taking adjoints. ∎

######  Proposition 2.14.

Under the smooth splitting $E\to S\oplus Q$ defined by $s\mapsto
j^{\dagger}s\oplus qs$ the Chern connection and curvature form of $E$ are

$\displaystyle D_{E}s$
$\displaystyle\sim\begin{pmatrix}j^{\dagger}&q\end{pmatrix}^{\dagger}\begin{pmatrix}D_{S}&-\sigma^{\dagger}\\\
\sigma&D_{Q}\end{pmatrix}\begin{pmatrix}j^{\dagger}\\\ q\end{pmatrix}(s),$
$\displaystyle D_{E}^{2}s$
$\displaystyle\sim\begin{pmatrix}j^{\dagger}&q\end{pmatrix}^{\dagger}\begin{pmatrix}D^{2}_{S}-\sigma^{\dagger}\wedge\sigma&-D^{\prime}_{\operatorname{Hom}(Q,S)}\sigma^{\dagger}\\\
\bar{\partial}\sigma&D^{2}_{Q}-\sigma\wedge\sigma^{\dagger}\end{pmatrix}\begin{pmatrix}j^{\dagger}\\\
q\end{pmatrix}(s).$

###### Proof.

Let $s$ be a section of $E$. We write

$s\sim j(j^{\dagger}s)+q^{\dagger}(qs).$

Then

$D_{E}s\sim q^{\dagger}\circ\sigma(j^{\dagger}s)+j\circ
D_{S}(j^{\dagger}s)-j\circ\sigma^{\dagger}(qs)+q^{\dagger}\circ D_{Q}(qs).$

This proves the first statement. For each of the terms here, we have

$\displaystyle D_{E}(q^{\dagger}\circ\sigma(j^{\dagger}s))$
$\displaystyle\sim-j\circ\sigma^{\dagger}\circ\sigma(j^{\dagger}s)+q^{\dagger}\circ\bar{\partial}\sigma(j^{\dagger}s)-q^{\dagger}\circ\sigma\circ
D_{S}(j^{\dagger}s),$ $\displaystyle D_{E}(j\circ D_{S}(j^{\dagger}s))$
$\displaystyle\sim q^{\dagger}\circ\sigma\circ D_{S}(j^{\dagger}s)+j\circ
D_{S}^{2}(j^{\dagger}s),$ $\displaystyle D_{E}(-j\circ\sigma^{\dagger}(qs))$
$\displaystyle\sim-q^{\dagger}\circ\sigma\circ\sigma^{\dagger}(qs)-j\circ
D^{\prime}_{\operatorname{Hom}(Q,S)}\sigma^{\dagger}(qs)+j\circ\sigma^{\dagger}\circ
D_{Q}(qs),$ $\displaystyle D_{E}(q^{\dagger}\circ D_{Q}(qs))$
$\displaystyle\sim-j\circ\sigma^{\dagger}\circ D_{Q}(qs)+q^{\dagger}\circ
D_{Q}^{2}(qs).$

Grouping these together by pre- and postfix morphisms, we get

$\displaystyle D_{E}^{2}s$ $\displaystyle\sim
j\bigl{(}D_{S}^{2}-\sigma^{\dagger}\circ\sigma\bigr{)}j^{\dagger}s-j\bigl{(}D^{\prime}_{\operatorname{Hom}(Q,S)}\sigma^{\dagger}\bigr{)}qs$
$\displaystyle\qquad+q^{\dagger}\bigl{(}\bar{\partial}\sigma\bigr{)}j^{\dagger}s+q^{\dagger}\bigl{(}D^{2}_{Q}-\sigma\circ\sigma^{\dagger}\bigr{)}qs$
$\displaystyle=\begin{pmatrix}j^{\dagger}&q\end{pmatrix}^{\dagger}\begin{pmatrix}D^{2}_{S}-\sigma^{\dagger}\wedge\sigma&-D^{\prime}_{\operatorname{Hom}(Q,S)}\sigma^{\dagger}\\\
\bar{\partial}\sigma&D^{2}_{Q}-\sigma\wedge\sigma^{\dagger}\end{pmatrix}\begin{pmatrix}j^{\dagger}\\\
q\end{pmatrix}(s),$

which completes the proof. ∎

######  Corollary 2.15 (Codazzi–Griffiths equations).

The curvature tensors of $S$ and $Q$ satisfy

$\displaystyle R_{S}(\alpha,\overline{\beta},s,\overline{t})$
$\displaystyle=R_{E}(\alpha,\overline{\beta},js,\overline{jt})-b_{Q}(\sigma(\alpha,s),\overline{\sigma(\beta,t)}),$
$\displaystyle R_{Q}(\alpha,\overline{\beta},s,\overline{t})$
$\displaystyle=R_{E}(\alpha,\overline{\beta},q^{\dagger}s,\overline{q^{\dagger}t})+b_{S}(\sigma^{\dagger}(\overline{\beta},s),\overline{\sigma^{\dagger}(\overline{\alpha},t)}).$

######  Corollary 2.16.

Let $E\to X$ be a holomorphic vector bundle with two Hermitian forms $b_{1}$
and $b_{2}$. Assume that $h:=b_{1}+b_{2}$ is positive-definite. The curvature
tensor of $h$ is

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
R_{h}(\alpha,\overline{\beta},s,\overline{t})=R_{b_{1}}(\alpha,\overline{\beta},s,\overline{t})+R_{b_{2}}(\alpha,\overline{\beta},s,\overline{t})-q(\sigma(\alpha)s,\overline{\sigma(\beta)t}),\crcr}}$

where the second fundamental form is

$\sigma(\alpha)s=(D_{b_{1},\alpha}-D_{b_{2},\alpha})s$

and the Hermitian form on the “quotient” bundle $E$ is

$q(s,\overline{t})=b_{1}(h^{-1}b_{2}s,\overline{h^{-1}b_{2}t})+b_{2}(h^{-1}b_{1}s,\overline{h^{-1}b_{1}t}).$

###### Proof.

We have a short exact sequence

$0\longrightarrow E\longrightarrow E\oplus E\longrightarrow E\longrightarrow
0,$

where the inclusion is $j(s)=s\oplus s$, the quotient map is $\pi(s\oplus
t)=s-t$, and the Hermitian metrics on the bundles are, in order,
$h_{1}+h_{2}$, $h_{1}\oplus h_{2}$, and the “quotient” metric. The Chern
connection of the direct sum is $D_{1}\oplus D_{2}$ and its curvature tensor
is $R_{1}\oplus R_{2}$. The second fundamental form is

$\sigma(\alpha)s=\pi(D_{1,\alpha}s\oplus
D_{2,\alpha}s)=D_{1,\alpha}s-D_{2,\alpha}s.$

To compute the quotient form we first note that the adjoint of $j:E\to E\oplus
E$ is defined by

$h(j^{\dagger}(s\oplus t),\overline{r})=h(s\oplus t,\overline{r\oplus
r})=b_{1}(s,\overline{r})+b_{2}(t,\overline{r})$

so

$j^{\dagger}(s\oplus t)=h^{-1}(b_{1}(s)+b_{2}(t)).$

Then $s\oplus t=jj^{\dagger}(s\oplus t)+\pi^{\dagger}\pi(s\oplus t)$, so we
find that

$\pi^{\dagger}\pi(s\oplus
t)=\bigl{(}s-h^{-1}(b_{1}(s)+b_{2}(t))\bigr{)}\oplus\bigl{(}t-h^{-1}(b_{1}(s)+b_{2}(t))\bigr{)}.$

We write $s=h^{-1}(b_{1}+b_{2})(s)$ and get

$s-h^{-1}(b_{1}(s)+b_{2}(t))=h^{-1}b_{2}(s-t).$

Similarly we get

$t-h^{-1}(b_{1}(s)+b_{2}(t))=-h^{-1}b_{1}(s-t).$

Then

$\pi^{\dagger}\pi(s\oplus
t)=\bigl{(}h^{-1}b_{2}(s-t)\bigr{)}\oplus\bigl{(}-h^{-1}b_{1}(s-t)\bigr{)}.$

In particular, the adjoint of $\pi$ is

$\pi^{\dagger}(s)=\bigl{(}h^{-1}b_{2}(s)\bigr{)}\oplus\bigl{(}-h^{-1}b_{1}(s)\bigr{)}.$

The inner product on the quotient is the pullback of $h$ by $\pi^{\dagger}$,
which completes the calculation. ∎

##  3\. Positive holomorphic sectional curvature

Our objective in this section is to prove the following technical theorem and
apply it to things relevant to our interests. This theorem was first proved by
Chaturvedi and Heier [6, Theorem 1.1] by classical calculations in local
coordinates. We hope our (mostly) coordinate-free proof is also of interest.

######  Theorem 3.1.

Let $\pi:X\to(B,h_{B})$ be a holomorphic fibration over a compact Hermitian
manifold. Suppose $h_{B}$ has positive holomorphic sectional curvature.
Suppose there exists a Hermitian form $h_{X/B}$ on $X$ whose restriction to
each fiber is a Hermitian metric that has positive holomorphic sectional
curvature. Then there exists a Hermitian metric on $X$ that has positive
holomorphic sectional curvature, and if $h_{B}$ is Kähler and the Kähler form
of $h_{X/B}$ is closed then this metric is also Kähler.

###### Proof.

We consider the Hermitian forms $h_{\lambda}=h_{X/B}+e^{\lambda}\pi^{*}h_{B}$
for real $\lambda$. First off, it’s clear that if $h_{B}$ is Kähler and the
Kähler form of $h_{X/B}$ is closed, then $h_{\lambda}$ is Kähler.

As the base $B$ is compact, $h_{X/B}$ is positive on $T_{X/B}$, and
$\pi^{*}h_{B}$ is positive on $\pi^{*}T_{B}$, we see that $h_{\lambda}$ is an
honest Hermitian metric for all $\lambda\gg 0$. Its curvature tensor is then

$R_{\lambda}=R_{X/B}+e^{\lambda}\pi^{*}R_{B}-\sigma^{*}q_{\lambda}.$

We need to investigate what happens first in $T_{X/B}$ and then the rest of
$T_{X}$. The calculations can get a little messy, so we split them into a few
technical lemmas. The first one will help us say what happens on $T_{X/B}$, as
the form $\pi^{*}h_{B}$ is zero there.

######  Lemma 3.2.

Let $b_{1}$ and $b_{2}$ be Hermitian forms on a vector space $V$ such that
$h:=b_{1}+b_{2}$ is positive-definite. Let $q$ be the associated “quotient”
form. Then $\operatorname{Ker}b_{j}\subset\operatorname{Ker}q$ for $j=1,2$.

###### Proof.

By symmetry it’s enough to prove the statement for $j=1$. Recall that

$q=(h^{-1}b_{2})^{*}b_{1}+(h^{-1}b_{1})^{*}b_{2}.$

We have $x=h^{-1}h(x)=h^{-1}b_{1}(x)+h^{-1}b_{2}(x)$, so if
$x\in\operatorname{Ker}b_{1}$ then $x=h^{-1}b_{2}(x)$. But then

$q(x)=(h^{-1}b_{2})^{*}b_{1}(x)=b_{1}(x)=0.\qed$

We can now deal with $T_{X/B}$, where $\pi^{*}R_{B}=0$. Let $(x,b)$ be a point
in $X$, so that $x\in X_{b}$. There exists a local normal frame of $T_{X_{b}}$
for $h_{X/B,b}$ centered at $x$. In this frame, we have
$D_{X/B}-\pi^{*}D_{B}=0$ at $x$. The only part of our curvature tensor that
survives at $x$ is thus $R_{\lambda}=R_{X/B}$, which is positive, so
$R_{\lambda}$ is positive on $T_{X/B}$.

Next we’ll consider what happens on the rest of $T_{X}$. As we don’t know what
$R_{X/B}$ does there, we have to increase the contribution of $\pi^{*}R_{B}$
and try to control the “quotient” form. For that we need to know what happens
to it as $\lambda$ grows.

######  Lemma 3.3.

Let $0\to S\to V\stackrel{{\scriptstyle\pi}}{{\to}}Q\to 0$ be a short exact
sequence of vector spaces. Let $b_{1}$ be a Hermitian form on $V$ that is
positive-definite on $S$ and $b_{2}$ be a Hermitian inner product on $Q$ such
that $h_{\lambda}:=b_{1}+e^{\lambda}\pi^{*}b_{2}$ is positive-definite for all
$\lambda\geq 0$. Let $q$ be the associated “quotient” form. Then there exists
a semipositive Hermitian form $q_{\infty}$ on $V$ that is positive-definite on
$S$ such that $h_{\lambda}\to h_{\infty}$ as $\lambda\to\infty$.

###### Proof.

As $h_{0}$ is positive-definite we can simultaneously diagonalize it and
$b_{1}$. Then $\pi^{*}b_{2}=h_{0}-b_{1}$ is also diagonal. Denote the
coefficients of $b_{1}$ by $x_{j}$ and the ones of $\pi^{*}b_{2}$ by $y_{j}$,
then the coefficients of $h_{\lambda}$ are

$\frac{e^{\lambda}x_{j}y_{j}}{x_{j}+e^{\lambda}y_{j}}=\frac{x_{j}y_{j}}{x_{j}e^{-\lambda}+y_{j}}\to\begin{cases}x_{j}&y_{j}\not=0\\\
0&y_{j}=0\end{cases}$

as $\lambda\to\infty$. The form this defines is semipositive on $V$ and
positive-definite on $S$.∎

The limit form is $(jj^{\dagger})^{*}b_{1}$, where $j^{\dagger}$ is the
adjoint of $j$ with respect to $h$. There’s probably an invariant proof of
this somewhere, but we don’t need it.

Now pick an auxiliary Hermitian metric on $X$ and define a compact sphere
bundle $S(T_{X})\subset T_{X}$ to test positivity on. Let

$\displaystyle m_{\lambda}$
$\displaystyle=\inf_{S(T_{X})}R_{X/B}-(D_{X/B}-\pi^{*}D_{B})^{*}q_{\lambda},$
$\displaystyle m_{\infty}$
$\displaystyle=\inf_{S(T_{X})}R_{X/B}-(D_{X/B}-\pi^{*}D_{B})^{*}q_{\infty}$

As $S(T_{X})$ is compact, the infimum $m_{\infty}$ is attained at a point $v$.
If $v\in\operatorname{Ker}\pi^{*}$, then we’re done as we know that $R_{X/B}$
is positive there. Otherwise $\pi^{*}(v)\not=0$, so
$m_{2}:=\pi^{*}R_{B}(v,\bar{v},v,\bar{v})>0$. We then pick $\lambda_{0}$ big
enough so that $m_{\infty}+e^{\lambda_{0}}m_{2}>0$. Now note that
$m_{\lambda}\to m_{\infty}$ as $\lambda\to\infty$ because $q_{\lambda}\to
q_{\infty}$. For $\lambda>\lambda_{0}$ we then have

$m_{\lambda}+e^{\lambda}m_{2}\geq m_{\lambda}+e^{\lambda_{0}}m_{2}\to
m_{\infty}+e^{\lambda_{0}}m_{2}>0.$

For all large enough $\lambda$, it follows that $R_{\lambda}$ will be
positive. ∎

As a corollaries, we get a family of theorems where a holomorphic family whose
fibers have positive holomorphic sectional curvature also has positive
holomorphic sectional curvature as soon as we can pick the metrics on the
fibers to vary smoothly between them. The trick is to find conditions that
guarantee we can extend the fiberwise metrics to a single smooth Hermitian
form. It is tempting to guess that if the metrics on the fibers are Kähler
then we also get a Kähler metric on the total space, but this is delicate.
Given a family of smoothly varying Kähler metrics on each fiber, there does
not need to exist a closed Hermitian form on the total space that restricts to
each of them. Remember for example the Hopf surface, which is the total space
of a fibration of projective lines over an elliptic curve; picking any
Hermitian metric on the total space we get a family of Kähler metrics on the
fibers, but if they came from a closed form on the total space the Hopf
surface would be Kähler.

### Grassmannian bundles

Recall that a Grassmannian manifold of $k$-planes in $n$-dimensional space
admits a unique Kähler–Einstein metric $\omega_{\operatorname{Gr}}$ such that
$\operatorname{Ric}\omega_{\operatorname{Gr}}=n\omega_{\operatorname{Gr}}$.
When we speak of “the” Kähler–Einstein metric on a Grassmannian, this is the
one we mean. This metric can be obtained in two ways:

  * •

It is the curvature form of the Hermitian metric on $\det
K_{\operatorname{Gr}}$ induced by the choice of any Hermitian inner product on
the underlying vector space $V$. This agrees with the metric obtained by
pullback of the Fubini–Study metric by the Plücker embedding.

  * •

It is the pullback of the Hermitian metric on
$\operatorname{Hom}(\mathcal{S},\mathcal{Q})$ induced by a Hermitian inner
product on the underlying vector space by the (holomorphic) second fundamental
form, where $0\to\mathcal{S}\to\underline{V}\to\mathcal{Q}\to 0$ is the short
exact sequence of tautological sub- and quotient bundles of the trivial bundle
with fiber $V$.

Either way, the holomorphic sectional curvature $H$ of this metric satisfies

$\frac{2}{k^{2}}\leq H\leq 2$

and is thus positive. It’s easiest to calculate its holomorphic sectional
curvature by going the $\operatorname{Hom}(\mathcal{S},\mathcal{Q})$ route, in
case the reader wants to try it out.

We can now slightly extend the results of Álvarez and her collaborators on
projectivized vector bundles by not requiring our Grassmannian bundle to come
from a vector bundle.

######  Proposition 3.4 (relative Plücker embedding).

Let $\pi:X\to B$ be a holomorphic fibration over a connected base whose fibers
are Grassmannian manifolds. Then there exists a holomorphic embedding $j:X\to
Y$ of holomorphic fibrations

${X}$${Y}$${B}$${B,}$$\scriptstyle{\pi}$$\scriptstyle{j}$$\scriptstyle{\hat{\pi}}$$\scriptstyle{\operatorname{id}}$

where the fibers of $Y\to B$ are projective spaces, and the fiberwise maps
$j_{b}:X_{b}\to Y_{b}$ are Plücker embeddings.

###### Proof.

Let $(U_{\nu})$ be a covering of $B$ by connected and simply connected charts.
Then there are local holomorphic isomorphisms $\theta_{\nu}:X_{|U_{\nu}}\to
U_{\nu}\times\operatorname{Gr}(k)$, where $\operatorname{Gr}(k)$ is the
Grassmannian of $k$-planes in $n$-dimensional space [9]. We define

$j_{\nu}:U_{\nu}\times\operatorname{Gr}(k)\to
U_{\nu}\times\mathbb{P}^{N},\quad(x,\operatorname{span}(z_{1},\ldots,z_{k}))\mapsto(x,[z_{1}\wedge\cdots\wedge
z_{k}]),$

where $N=\binom{n}{k}$, to be the usual Plücker embedding.

The coordinate change maps $\theta_{\mu\nu}=\theta_{\mu}\theta_{\nu}^{-1}$ can
be written as

$\theta_{\mu\nu}=\operatorname{id}\times f_{\mu\nu},$

where $f_{\mu\nu}$ is an automorphism of $\operatorname{Gr}(k)$. But an
automorphism of $\operatorname{Gr}(k)$ is exactly an automorphism of
$\mathbb{P}^{N}$ that preserves decomposable $k$-vectors [7], so these induce
automorphisms $\hat{f}_{\mu\nu}$ of $\mathbb{P}^{N}$. We define the
holomorphic family $Y\to B$ by the products $U_{\nu}\times\mathbb{P}^{N}$ and
the gluing maps $\operatorname{id}\times\hat{f}_{\mu\nu}$.

By construction and the shape of automorphisms of the Grassmannian, it follows
that the maps $j_{\nu}$ glue together to define an embedding $j:X\to Y$. ∎

Above we only really used that the fibers of $X$ embed in a manifold $V$ and
the automorphism group of the fibers is a subgroup of the automorphism group
of $V$, in case the reader has a favorite set of fibrations they’d like to try
this out on. We’ll call the embedding $j:X\to Y$ above the relative Plücker
embedding.

######  Proposition 3.5.

Let $\pi:X\to B$ be a holomorphic fibration over a connected base whose fibers
are Grassmannian manifolds. If $B$ admits a metric with positive holomorphic
sectional curvature, then so does $X$. This metric is Kähler if the metric on
$B$ is Kähler.

###### Proof.

Consider the relative Plücker embedding $j:X\to Y$ as above. Kodaira [11,
Section 4(V)] showed that there exists a closed Hermitian form on $Y$ whose
restriction to any fiber is the Fubini–Study metric. The pullback of this
metric to $X$ gives a form $h_{X/B}$ on $X$ whose restriction to each fiber is
the Kähler–Einstein metric on the fiber. Theorem 3.1 now applies. ∎

######  Corollary 3.6.

Let $\pi:X\to B$ be a holomorphic fibration over a compact Kähler manifold
whose fibers are flag manifolds. Suppose $B$ admits a Kähler metric with
positive holomorphic sectional curvature. Then there exists a Kähler metric on
$X$ that has positive holomorphic sectional curvature.

###### Proof.

Denote the fibers by $P(d_{0},\ldots,d_{k})$. The forgetful map
$P(d_{0},\ldots,d_{k})\to P(d_{0},\ldots,\hat{d_{j}},\ldots,d_{k})$ is a
holomorphic fibration whose fiber is a Grassmannian. Similar to before, this
extends to a morphism $f$ of holomorphic fibrations

${X}$${X_{1}}$${B}$${B}$$\scriptstyle{f}$

whose fibers are Grassmannians. Continuing this process we end up with a
sequence $X\to X_{1}\to\cdots\to X_{k}\to B$ of fibrations with Grassmannian
fibers, and Theorem 3.1 applies to each step in turn. ∎

## References

  * [1] Angelynn Reario Álvarez.  On the positive holomorphic sectional curvature of projectivized vector bundles over compact complex manifolds.  PhD thesis, 2016.
  * [2] Angelynn Reario Álvarez, Gordon Heier, and Fangyang Zheng.  On projectivized vector bundles and positive holomorphic sectional curvature.  Proc. AMS, 146(7):2877–2882, 2018.
  * [3] IV Bel’ko.  Degenerate riemannian metrics.  Mathematical notes of the Academy of Sciences of the USSR, 18(5):1046–1049, 1975.
  * [4] Yang Bo and Fangyang Zheng.  Hirzebruch manifolds and positive holomorphic sectional curvature.  In Ann. Institut Fourier, volume 69, pages 2589–2634, 2019.
  * [5] Eugenio Calabi.  Métriques kählériennes et fibrés holomorphes.  Ann. sci. ÉNS, 12(2):269–294, 1979.
  * [6] Ananya Chaturvedi and Gordon Heier.  Hermitian metrics of positive holomorphic sectional curvature on fibrations.  Mathematische Zeitschrift, 295(1):349–364, 2020.
  * [7] Michael J Cowen.  Automorphisms of grassmannians.  Proceedings of the American Mathematical Society, 106(1):99–106, 1989.
  * [8] Jean-Pierre Demailly.  Complex analytic and differential geometry.  2021\.
  * [9] Wolfgang Fischer and Hans Grauert.  Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten.  Vandenhoeck & Ruprecht, 1965.
  * [10] Phillip A Griffiths.  Hermitian differential geometry and the theory of positive and ample holomorphic vector bundles.  J. Math. Mechanics, 14(1):117–140, 1965.
  * [11] Kunihiko Kodaira.  On Kähler varieties of restricted type (An intrinsic characterization of algebraic varieties).  Ann. Math., 60(1):28–48, 1954.
  * [12] Gunnar Þór Magnússon.  A natural hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class.  Comptes Rendus Math., 350(1-2):63–66, 2012.
  * [13] Cristi Stoica.  Cartan’s structural equations for degenerate metric.  arXiv preprint math.DG/1111.0646, 7, 2011.
  * [14] Ovidiu Cristinel Stoica.  On singular semi-riemannian manifolds.  International Journal of Geometric Methods in Modern Physics, 11(05):1450041, 2014.
  * [15] Fangyang Zheng.  Complex differential geometry.  Number 18. Amer. Math. Soc., 2000.


# Power Fluctuations of An Irreversible Quantum Otto Engine

Guangqian Jiao1 Shoubao Zhu1 Jizhou He1 Yongli Ma2 Jianhui Wang1,2
<EMAIL_ADDRESS>${}^{1}\,$ Department of Physics, Nanchang University,
Nanchang 330031, China
${}^{2}\,$ State Key Laboratory of Surface Physics and Department of Physics,
Fudan University, Shanghai 200433, China

###### Abstract

We derive the general probability distribution function of stochastic work for
quantum Otto engines in which both the isochoric and driving processes are
irreversible due to finite time duration. The time-dependent power
fluctuations, average power, and thermodynamic efficiency are explicitly
obtained for a complete cycle operating with an analytically solvable two-
level system. We show that, there is a trade-off between efficiency (or power)
and power fluctuations.

PACS number(s): 05.70.Ln

##  I Introduction

The second law of thermodynamics tells us that any heat engines working
between a hot and a cold thermal bath of constant inverse temperatures
$\beta_{h}$ and $\beta_{c}$ (with $\beta=1/T$ and $k_{B}\equiv 1$) are not
able to run more efficiently than a reversible Carnot cycle with its
efficiency $\eta_{C}=1-\beta_{h}/\beta_{c}$. When a cyclic heat engine runs at
a Carnot efficiency, its cycle operation consisting of consecutive
thermodynamic processes must be reversible and thus the power output becomes
null. Practically heat engines must proceed in a finite time period accounting
for irreversibility and produce finite power output 1 ; 2 ; 3 ; 4 ; 5 ; 6 .
The performance in finite time was intensively studied for both quantum7 ; 8
and classical4  engines. For an adequate description of heat engines, the
effects induced by finite-time cycle operation on the machine performance have
to be considered by involving both heat-transfer and adiabatic processes.

Unlike in macroscopic systems where both work and heat are deterministic, the
work 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17  and heat 18 ; 19 ; 20  for
microscale systems are random due to thermal and quantum fluctuations8 ; 10 ;
21 ; 22 ; 23 ; 24 . The statistics of either work or efficiency for heat
engines at microscale were examined experimentally and theoretically8 ; 16 ;
22 ; 23 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33 ; 34 , but under the
assumption that either adiabatic8 ; 23  or isothermal strokes22  are
reversible. To our knowledge, the power fluctuations have not been examined so
far for an irreversible quantum heat engine where the isothermal and adiabatic
strokes are finite-time and thus both of them are away from reversible limit.

In this paper, we derive the general expression for the distribution function
of quantum work and heat along irreversible quantum Otto engines composed of
two finite-time isochoric and two driving strokes8 ; 22 ; 23 ; 34 ; 35 ; 36 ;
37 ; 38 . This distribution function allows us to obtain the quantum work
statistics explicitly depending on the time evolution dynamics of the two
isochoric and two driving processes. We then analytically examine the power
statistics of an Otto cycle working with an exactly, solvable two-level
system. The effects of irreversibility induced by finite-time duration of
either driving or isochoric strokes on machine performance and power
statistics are discussed. We finally demonstrate that irreversibility results
in deterioration of efficiency and power but increasing the power
fluctuations, thereby indicating that the price for improving machine
efficiency is increasing power fluctuations.

##  II Model

The model of a quantum Otto engine is sketched in Fig. 1. This machine
consists of two isochoric branches, one with a hot and another with a cold
thermal bath where the system frequency is kept constant, and two driving
strokes, where the system (which is isolated from the two heat reservoirs and
driven by the external field) undergoes unitary transformation. The four
branches can be described as follows.

Figure 1: Schematic diagram of a two-level quantum Otto cycle operating
between a hot and a cold thermal bath of inverse constant inverse temperatures
($\beta_{h}$ and $\beta_{c}$) in $(\omega,\langle n\rangle)$ plane. The cycle
consists of the two driving strokes (connecting states $A$ and $B$, and $C$
and $D$), where the system isolated from the two thermal baths evolves unitary
transformation and the respective time durations are $\tau_{ch}$ and
$\tau_{hc}$, and two isochoric strokes (connecting states $B$ and $C$, and $D$
and $A$), where the system is kept in thermal contact with the hot and the
cold reservoir, respectively, and these respective time durations are
$\tau_{h}$ and $\tau_{c}$. The average populations at the four instants
$A,B,C,D$ can be expressed as $\langle n_{A}\rangle=\langle
n(0)\rangle,\langle n_{B}\rangle=\langle n(\tau_{ch})\rangle,\langle
n_{C}\rangle=\langle n(\tau_{ch}+\tau_{h})\rangle,\langle n_{D}\rangle=\langle
n(\tau_{cyc}-\tau_{c})\rangle$. The mean population $\langle
n_{A}\rangle(\langle n_{C}\rangle)$ at the end of the cold (hot) isochoric
stroke would approach its asymptotic value $\langle n_{c}\rangle^{eq}$
($\langle n_{h}\rangle^{eq}$), when and only when the thermalization is
completed after infinite long duration of the system-bath interaction interval
$\tau_{c}$ ($\tau_{h}$).

During the adiabatic compression $A\rightarrow B$, the system is isolated from
the two thermal baths but driven by the external field, and it undergoes a
unitary expansion from time $t=0$ to $t=\tau_{ch}$. Initially, the system of
inverse temperature $\beta_{A}$ is assumed to be (local) thermal equilibrium 2
; 3 , which means that the thermal occupation probability at instant $A$ takes
the canonical form

$p_{n}^{0}=\frac{e^{-\beta_{A}E^{c}_{n}}}{Z_{A}},$ (1)

with the partition function $Z_{A}=\sum_{n}e^{-\beta_{A}E_{n}^{c}}$. The
transition probability from eigenstate $|n\rangle$ to $|m\rangle$ is given by

$p_{n\rightarrow m}^{\tau_{ch}}=|\langle n|U_{\mathrm{com}}|m\rangle|^{2},$
(2)

where $U_{\mathrm{com}}$ denotes the unitary time evolution operator along the
compression. In the special case when the system Hamiltonian evolves slowly
enough by satisfying quantum adiabatic condition 38 , the system remains in
the same state and the transition probability becomes $p_{n\rightarrow
m}^{\tau_{ch}}=\delta_{nm}$, with the Kronecker delta function $\delta$.

The next step is the hot isochore $B\rightarrow C$, where the system with
constant frequency $\omega=\omega_{h}$ is in contact with the hot thermal bath
in a time duration $\tau_{h}$. The probability density of the stochastic heat
absorbed from the hot bath can be determined by the conditional probability to
obtain

$p(q_{h})=\sum_{k,l}\delta[q_{h}-(E_{l}^{h}-E_{k}^{h})]p_{k\rightarrow
l}^{\tau_{h}}.$ (3)

After this system-bath interaction interval, the system is assumed to be at
the local thermal equilibrium state of inverse temperature
$\beta_{C}(\geq\beta_{h})$, which means that $p_{k\rightarrow
l}^{\tau_{h}}={e^{-\beta_{C}E^{h}_{l}}}/{Z_{C}}$ with partition function
$Z_{C}=\sum_{l}{e^{-\beta_{C}E^{h}_{l}}}$.

During the expansion $C\rightarrow D$, the system is isolated from these two
thermal baths in a time period $\tau_{hc}$ while its frequency changes back to
$\omega_{c}$ from $\omega_{h}$. Like in the driving compression, there is a
transition probability from initial state $i$ to final one $j$, which reads

$p_{i\rightarrow j}^{\tau_{hc}}=|\langle i|U_{\mathrm{exp}}|j\rangle|^{2},$
(4)

where $U_{\mathrm{exp}}$ is the time evolution operator along the expansion.
At the initial instant $C$ the system approaches the thermal equilibrium and
the occupation probabilities take the form
$p_{i}^{\tau_{ch}+\tau_{h}}=\delta_{il}p_{k\rightarrow l}^{\tau_{h}}$, where
$p_{k\rightarrow l}^{\tau_{h}}$ was defined in Eq. (3).

On the fourth branch $D\rightarrow A$, the system with constant frequency
$\omega=\omega_{c}$ is coupled to a cold reservoir of inverse temperature
$\beta_{c}$ in a time period $\tau_{c}$. After a cycle with the total period
$\tau_{cyc}=\tau_{ch}+\tau_{h}+\tau_{hc}+\tau_{c}$ the system returns to its
initial state $A$, we can easily determine the quantum heat $q_{c}$ similar to
that of the quantum heat $q_{h}$. As emphasized, at instant $A$ the system
tends to be (local) thermal equilibrium after system-bath interaction
interval, which is consistent with the assumption of canonical form Eq. (1).

In view of the fact that the work per cycle is produced only in the two
driving strokes $A\rightarrow B$ and $C\rightarrow D$, the quantum work output
can be obtained by determining the total work produced along these two
microscopic trajectories to arrive at

$\mathrm{w}[n(0)\rightarrow m(\tau_{ch});i(\tau_{ch}+\tau_{h})\rightarrow
j(\tau_{cyc}-\tau_{c})]=(E_{i}^{h}-E_{j}^{c})-(E_{m}^{h}-E_{n}^{c}),$ (5)

where $E_{n}^{c}$ and $E_{m}^{h}$ ($E_{i}^{h}$ and $E_{j}^{c}$) denote the
respective energy eigenvalues at the initial and final instants of the
compression (expansion). Here $n(t)$ are quantum numbers indicating the states
occupied by the system at time $t$.

The states $|n(0)\rangle$ and $|m(\tau_{ch})\rangle$ can be assumed to be
independent of $i(\tau_{ch}+\tau_{h})$ and $j(\tau_{cyc}-\tau_{c})$, since the
system would relax to thermal equilibrium in an isochoric process if its time
duration is long enough. This leads to the main result of this paper, namely,
the following expression for the probability density of the work $\mathrm{w}$:

$p(\mathrm{w})=\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-\mathrm{w}[n(0)\rightarrow
m(\tau_{ch});i(\tau_{ch}+\tau_{h})\rightarrow j(\tau_{cyc}-\tau_{c})]\\},$ (6)

where $p_{n\rightarrow m}^{\tau_{ch}}$ and $p_{i\rightarrow j}^{\tau_{hc}}$
were defined in Eq. (2) and Eq. (4), respectively. The probability
distribution function Eq. (6) for the irreversible Otto engine allows us to
determine all moments of quantum work:
$\langle\mathrm{w}^{k}\rangle=\int\mathrm{w}^{k}p(\mathrm{w})d\mathrm{w}$
($k=1,2,\dots).$ This result reduces to that previously obtained in a
quasistatic 27  or an endoreversible 8  Otto engine when $\xi\rightarrow 0$ or
$\tau_{c,h}\rightarrow\infty$. We present it here in a broader context by
arguing that irreversibility is unavoidable in a finite-time cyclic engine
where both the driving and system-bath interaction steps are away from
quasistatic limit. For an endoreversible model where the two driving strokes
are isentropic, we recover the work fluctuations 8  by setting
$p_{n\rightarrow m}^{\tau_{ch}}=\delta_{nm}$ and $p_{i\rightarrow
j}^{\tau_{hc}}=\delta_{ij}$,
$p(\mathrm{w})=\sum_{n,i}p_{n}^{0}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-\mathrm{w}[n(0);i(\tau_{ch}+\tau_{h})]\\}$.

##  III Quantum Otto engine working with a two-level system

We now consider a quantum Otto engine that works with a two-level system of
the eigenenergies $E_{+}=\hbar\omega/2$ and $E_{-}=-\hbar\omega/2$. As the
occupation probabilities at these two eigenstates
$p_{+}=e^{-\beta\hbar\omega_{c}/2}/Z_{A}$ and
$p_{-}=e^{\beta\hbar\omega_{c}/2}/Z_{A}$, where the partition function
$Z_{A}=e^{-\beta_{A}\hbar\omega_{c}/2}+e^{\beta_{A}\hbar\omega_{c}/2}=2\cosh\left(\frac{\beta_{A}\hbar\omega_{c}}{2}\right),$
the mean population at instant $A$ (with time $t=0$) can be determined by
using $\langle n\rangle=\sum_{n}np_{n}$ to arrive at

$\langle
n(0)\rangle=-\frac{1}{2}\tanh\left(\frac{\beta_{A}\hbar\omega_{c}}{2}\right).$
(7)

The mean population at final instant $B$ can then be determined according to
$\langle n(\tau_{ch})\rangle=\sum_{n,m}np_{m\rightarrow
n}^{\tau_{ch}}p_{m}^{0}$, which together with Eqs. (1) and (2), leads to

$\langle n(\tau_{ch})\rangle=\left(1-2\xi\right)\langle n(0)\rangle,$ (8)

where $\xi=|\langle\pm|U_{\mathrm{com}}|\mp\rangle|^{2}$. Using $\langle
n(\tau_{cyc}-\tau_{c})\rangle=\sum_{n,m}np_{m\rightarrow
n}^{\tau_{hc}}p_{m}^{\tau_{ch}+\tau_{h}}$, for the unitary expansion
$C\rightarrow D$ there is the relation:

$\langle n(\tau_{cyc}-\tau_{c})\rangle=(1-2\xi)\langle
n(\tau_{ch}+\tau_{h})\rangle,$ (9)

where we have assumed
$\xi=|\langle\pm|U_{\mathrm{exp}}|\mp\rangle|^{2}=|\langle\pm|U_{\mathrm{com}}|\mp\rangle|^{2}$.
$\xi$ is called the adiabaticity parameter representing the probability of
transition between state $|+\rangle$ and $|-\rangle$ during the compression or
expansion, and the probability of no state transition is accordingly
$|\langle\pm|U_{\mathrm{exp}}|\pm\rangle|^{2}=|\langle\pm|U_{\mathrm{com}}|\pm\rangle|^{2}=1-\xi$.
Since the population $\langle n\rangle$ at any instant along the cycle is
negative, $\xi$ must be situated between $0\leq\xi<1/2$. The adiabaticity
parameter $\xi$ depends on the speed at which the driving process is
performed7 ; 22 ; 36 , and it is thus a monotonic decreasing function of time
duration $\tau_{h,c}$. Unlike in an adiabatic phase where the time scale of
state change must be much larger than the dynamical one and the probability of
state change is vanishing ($\xi=0$), during the nonadiabatic driving stroke
the rapid change in frequency $\omega$ leads to inner friction and results in
possible state transitions ($\xi>0$)22 ; 36 ; 39 ; 40 ; 41 ; 42 . Such
nonadiabatic internal dissipation accounts for irreversible entropy production
and an increase in mean population $\langle n\rangle$ (see Fig. 1).

We are interested in the finite-time operation of the Otto engine in which the
isochoric processes are far away from quasistatic limit and complete
thermalization can thus not be achieved for the system. Using the master
equation of stochastic thermodynamics , one can find that the mean populations
$\langle n(0)\rangle$ and $\langle n(\tau_{ch}+\tau_{h})\rangle$ can be
expressed in terms of the corresponding asymptotic equilibrium values $\langle
n_{c}\rangle^{eq}$ and $\langle n_{h}\rangle^{eq}$ (see Appendix A),

$\langle n(0)\rangle=\langle n_{c}\rangle^{eq}+[\langle
n(\tau_{cyc})\rangle-\langle n_{c}\rangle^{eq}]e^{-\gamma_{c}\tau_{c}},$ (10)
$\langle n(\tau_{ch}+\tau_{h})\rangle=\langle n_{h}\rangle^{eq}+[\langle
n(\tau_{ch})\rangle-\langle n_{h}\rangle^{eq}]e^{-\gamma_{h}\tau_{h}}.$ (11)

Using Eqs. (8),(9),(10) and (11), $\langle n(0)\rangle$ and $\langle
n(\tau_{ch}+\tau_{h})\rangle$ can be rewritten as

$\langle n(0)\rangle=\langle n_{c}\rangle^{eq}+\Delta_{c},\langle
n(\tau_{ch}+\tau_{h})\rangle=\langle n_{h}\rangle^{eq}+\Delta_{h}$ (12)

where

$\Delta_{h}=\frac{\left(2\xi-1\right)\left[\left(2\xi-1\right)\langle
n_{h}\rangle^{eq}+\langle n_{c}\rangle^{eq}\right]-x_{c}\left[\langle
n_{h}\rangle^{eq}+\left(2\xi-1\right)\langle
n_{c}\rangle^{eq}\right]}{x_{h}x_{c}-\left(2\xi-1\right)^{2}},$ (13)
$\Delta_{c}=\frac{\left(2\xi-1\right)\left[\left(2\xi-1\right)\langle
n_{c}\rangle^{eq}+\langle n_{h}\rangle^{eq}\right]-x_{h}\left[\langle
n_{c}\rangle^{eq}+\left(2\xi-1\right)\langle
n_{h}\rangle^{eq}\right]}{x_{h}x_{c}-\left(2\xi-1\right)^{2}}.$ (14)

Here $x_{h}\equiv e^{\gamma_{h}\tau_{h}}$ and $x_{c}\equiv
e^{\gamma_{c}\tau_{c}}$ denote the effective time durations along the hot and
cold isochoric branches, respectively. Here $\langle n_{c}\rangle^{eq}$ and
$\langle n_{h}\rangle^{eq}$ are achieved in the reversible, quasistatic limit
when $x_{h},x_{c}\rightarrow\infty$ leads to $\Delta_{h,c}\rightarrow 0$,
whether $\xi$ is zero or not. However, $\Delta_{c}$ and $\Delta_{h}$ are still
positive for finite values of $x_{c}$ and $x_{h}$ even for two reversible
driving processes with $\xi\rightarrow 0$. These corrections $\Delta_{c}$ and
$\Delta_{h}$ indicate that how far the heat-transfer processes deviate from
the reversible limit, and imply that irreversibility is exclusively caused by
heat transferred between the system and the thermal bath. Such a deviation is
quite natural in both quantum heat engines and classical context when the
heat-transfer processes are irreversible due to finite-time duration.

For this two-level engine, the probability density of quantum work Eq. (6) can
be analytically obtained as,

$\displaystyle p(\mathrm{w})=$ $\displaystyle\left[\frac{1}{2}+2\langle
n(0)\rangle\langle
n(\tau_{ch}+\tau_{h})\left(1-2\xi\right)-\left(1-\xi\right)\xi\right]\delta(\mathrm{w})$
(15) $\displaystyle+\frac{1}{2}\left[\left(1-2\langle
n(0)\rangle\right)\left(1-\xi\right)\xi\right]\delta(\mathrm{w}+\hbar\omega_{c})$
$\displaystyle+\frac{1}{2}\left[\left(2\langle
n(\tau_{ch}+\tau_{h})\rangle+1\right)\left(1-\xi\right)\xi\right]\delta(\mathrm{w}-\hbar\omega_{h})$
$\displaystyle+\frac{1}{4}\left[\left(2\langle
n(\tau_{ch}+\tau_{h})\rangle-1\right)\left(2\langle
n(0)\rangle-1\right)\xi^{2}\right]\delta(\mathrm{w}+\hbar\omega_{h}+\hbar\omega_{c})$
$\displaystyle+\frac{1}{4}\left[1-\left(2\langle
n(\tau_{ch}+\tau_{h})\rangle\right)\left(2\langle
n(0)\rangle+1\right)\left(1-\xi\right)^{2}\right]\delta(\mathrm{w}+\hbar\omega_{h}-\hbar\omega_{c})$
$\displaystyle+\frac{1}{4}\left[\left(2\langle
n(\tau_{ch}+\tau_{h})\rangle+1\right)\left(1-2\langle
n(0)\rangle\right)\left(1-\xi\right)^{2}\right]\delta(\mathrm{w}+\hbar\omega_{c}-\hbar\omega_{h})$
$\displaystyle+\frac{1}{2}\left[\left(2\langle
n(0)\rangle+1\right)\left(1-\xi\right)\xi\right]\delta(\mathrm{w}-\hbar\omega_{c})$
$\displaystyle+\frac{1}{4}\left[\left(2\langle
n(0)\rangle+1\right)\left(2\langle
n(\tau_{ch}+\tau_{h})\rangle+1\right)\xi^{2}\right]\delta(\mathrm{w}-\hbar\omega_{c}-\hbar\omega_{h})$
$\displaystyle+\frac{1}{2}\left[\left(1-2\langle
n(\tau_{ch}+\tau_{h})\rangle\right)\left(1-\xi\right)\xi\right]\delta(\mathrm{w}+\hbar\omega_{h}).$

The stochastic work can take nine different discrete values as shown Fig. 2.
The following should be noted in respect of the stochastic work per cycle: (i)
$\mathrm{w}=0$ indicates that the stochastic work by the system along the
expansion is fully counterbalanced by that during the compression. (ii)
$\mathrm{w}=\hbar\omega_{c}-\hbar\omega_{h}$
($\mathrm{w}=\hbar\omega_{h}-\hbar\omega_{c}$) corresponds to the case when
the system jumps down from high-energy (low-energy) state to low-energy (high-
energy) one along the hot isochore, namely, $n=m=1/2$ but $i=j=-1/2$ (
$n=m=-1/2$ and $i=j=1/2$) in Eq. (6). These values exist in the adiabatic or
nonadiabatic driving. (iii) There are more values of stochastic work in the
nonadiabatic driving than in the adiabatic case due to quantum fluctuations
induced by nonadiabatic transitions. (iv) Finally the distribution
$p(\mathrm{w})$ is expected to be normalized to one either for adiabatic or
nonadiabatic driving.

Figure 2: The probability distribution $p(\mathrm{w})$ of the quantum work for
the adiabatic (with $\xi=0,$ black squares) and nonadiabatic (with $\xi=0.3,$
red dots) steps. The parameters are $x_{c}=x_{h}=8$,
$\omega_{c}=0.4\omega_{h}=0.4$, $\beta_{h}=0.2\beta_{c}=0.2$, and $\hbar=2$.

Using Eq. (3), the average heat injection, $\langle q_{h}\rangle=\int
q_{h}p(q_{h})dq_{h}$, can be obtained as

$\langle q_{h}\rangle=\hbar\omega_{h}[\langle
n(\tau_{ch}+\tau_{h})\rangle-(1-2\xi)\langle n(0)\rangle].$ (16)

By using simple algebra (see Appendix B for details), the average work
($\langle\mathrm{w}\rangle$) and the work fluctuations
($\delta\mathrm{w}^{2}=\langle\mathrm{w}^{2}\rangle-\langle\mathrm{w}\rangle^{2}$)
can be obtained as

$\langle\mathrm{w}\rangle=\hbar[\omega_{c}-(1-2\xi)\omega_{h}]\langle
n(0)\rangle+\hbar[\omega_{h}-(1-2\xi)\omega_{c}]\langle
n(\tau_{ch}+\tau_{h})\rangle,$ (17)

and

$\displaystyle\delta\mathrm{w}^{2}=$
$\displaystyle\hbar^{2}\omega_{h}^{2}[\frac{1}{2}-\langle
n(0)\rangle^{2}(1-2\xi)^{2}-\langle n(\tau_{ch}+\tau_{h})\rangle^{2}]$ (18)
$\displaystyle+\hbar^{2}\omega_{c}^{2}[\frac{1}{2}-\langle
n(0)\rangle^{2}-(1-2\xi)^{2}\langle n(\tau_{ch}+\tau_{h})\rangle^{2}]$
$\displaystyle-\hbar^{2}\omega_{c}\omega_{h}(1-2\xi)[1-2\langle
n(0)\rangle^{2}-2\langle n(\tau_{ch}+\tau_{h})\rangle^{2}].$

From Eqs. (16) and (17), the machine efficiency defined by
$\eta=\langle\mathrm{w}\rangle/\langle q_{h}\rangle$, can be expressed as

$\eta=1-\frac{\omega_{c}}{\omega_{h}}\frac{\langle n(0)\rangle-(1-2\xi)\langle
n(\tau_{ch}+\tau_{h})\rangle}{(1-2\xi)\langle n(0)\rangle-\langle
n(\tau_{ch}+\tau_{h})\rangle}.$ (19)

In case both isochoric and driving processes proceed in finite time, internal
dissipation and uncomplete thermalization occur in the system resulting in the
thermodynamic efficiency Eq. (19) that depends on the time evolution along
each cycle except if these four processes may be done infinitely long, making
the efficiency reduce to the one for cycles with complete thermalization,
$\eta=1-\frac{\omega_{c}}{\omega_{h}}\frac{(1-2\xi)\langle
n_{h}\rangle^{eq}-\langle n_{c}\rangle^{eq}}{\langle
n_{h}\rangle^{eq}-(1-2\xi)\langle n_{c}\rangle^{eq}}$7 ; 34 ; 36 , or the one
for models without internal dissipation 2 ; 3 ; 35 ,
$\eta=1-\frac{\omega_{c}}{\omega_{h}}$.

Figure 3: (Color online) The work fluctuations $\delta\mathrm{w}^{2}$ (black
solid line ) and the efficiency $\eta$ (red dashed line) as a function of the
inverse temperature of cold bath $\beta_{c}=5\beta_{h}$ for $\xi=0.01$ (a) and
$\xi=0.03$ (b). The parameters are $x_{c}=x_{h}=8$,
$\omega_{h}=2\omega_{c}=1$, and $\hbar=2$.

The efficiency and work fluctuations as a function of the inverse temperature
$\beta_{c}$ of cold bath is shown in Fig. 3(a) and Fig. 3(b). When increasing
temperature, the efficiency $\eta$ increases but the work fluctuations
$\delta\mathrm{w}^{2}$ decrease. The efficiency at high temperatures is larger
than that at low temperatures, showing that the quantum effects which are of
significance in low temperature regime can improve the machine efficiency.
Because the quantum fluctuations characterizing the low temperature domain are
smaller than the thermal fluctuations dominating the high temperature region,
the work fluctuations $\delta\mathrm{w}^{2}$ get increased while the
temperature is increased and _vice versa_. The increase in the adiabacity
parameter $\xi$ yields in a decrease (an increase) in the machine efficiency
(work fluctuations) as it should, thereby showing that there is a trade-off
between efficiency and work fluctuations.

Since the stochastic power is the work divided by the cycle period
$\tau_{cyc}$, namely,
$\dot{\mathrm{w}}[|n(0)\rangle;|n(\tau_{ch}+\tau_{h})\rangle]=\mathrm{w}[|n(0)\rangle;|n(\tau_{ch}+\tau_{h})\rangle]/\tau_{cyc}$,
the relative fluctuations of the power $f_{\dot{\mathrm{w}}}$ are equivalent
to corresponding those of work $f_{\mathrm{w}}$, leading to
$f_{\dot{\mathrm{w}}}=f_{\mathrm{w}}=\sqrt{\delta\mathrm{w}^{2}}/\langle\mathrm{w}\rangle$.
For nonadiabatic driving branches, the relative power fluctuations are
increasing with decreasing effective time durations $x_{c}$ and $x_{h}$, see
Fig. 4(a) and Fig. 4(b). Comparison between these two figures shows that
irreversible nonadiabatic friction related to $\xi$, which is a monotonic
decreasing function of $\tau_{h}$ as well as $\tau_{c}$, leads to an increase
in relative power fluctuations. This can be understood by the fact that
irreversibility either induced by finite heat flux between the system and the
bath or caused by internal irreversible dissipation yields larger fluctuations
than those in the reversible cycle operation. In view of the fact that the
power is a monotonic decreasing function of time durations $\tau_{h}$ and
$\tau_{c}$, we see that the price one should pay for increasing power is
increasing power fluctuations.

Figure 4: Contour plot of the relative fluctuations of the power
$f_{\dot{\mathrm{w}}}$ in the effective time duration $(x_{h},x_{c})$ plane
for a nonadiabatic driving, with $\xi=0.01$ ($a$) and $\xi=0.03$ ($b$). The
values of the parameters are $\beta_{h}=0.2\beta_{c}=0.2$,
$\omega_{h}=2\omega_{c}=1$, and $\hbar=2$.

##  IV Conclusions

In summary, we derived the probability distribution of stochastic work of
quantum Otto engines working within a cycle period of finite time that leads
to irreversibility in both the two isochoric and two driving processes.
Employing a two-level system as the working substance of these engines, we
find that, although the average work is positive, the quantum work may be
negative due to quantum indeterminacy. Afterwards, we derived the analytical
expressions for the power output, efficiency, and work fluctuations, all of
which are dependent on the time allocations the four thermodynamic processes.
We finally determined statistics of work as well as power at any finite
temperatures, and showed that there is a trade-off between efficiency (or
power) and power fluctuations.

##  Appendix A Time evolution of population along an isochoric process

The dynamics of the system with energy quantization along the system-bath
interaction interval can be described by changes in the occupation
probabilities $p_{n}$ at states $n=0,1,2,\cdots$. This reduced description in
which the dynamical response of the heat reservoir is cast in kinetic terms.
The master equation is given by 21 ; 43

$\dot{p}_{n}=\sum_{n^{\prime}}W_{n,n^{\prime}}p_{n^{\prime}},$ (A.1)

where the transition rate matrix $W_{n,n^{\prime}}$ must satisfy
$\sum_{n}W_{n,n^{\prime}}=0$. For the system in contact with a heat bath of
constant temperature $\beta$, we assume that the transition rates from state
$n^{\prime}$ to $n$, $W_{nn^{\prime}}$, fulfill the detailed balance
$W_{nn^{\prime}}e^{-\beta E_{n^{\prime}}}=W_{n^{\prime}n}e^{-\beta E_{n}}$,
ensuring that the system can achieve asymptotically the thermal state after an
infinitely long system-bath interaction duration. At thermal state, the
occupation probabilities $p_{n}$ achieve their asymptotic stationary values
$\pi_{n}$. These $\pi_{n}$ can be determined from the steady-state solution of
Eq. (A.1) and given by the Boltzmann distribution: $\pi_{n}(\beta)={e^{-\beta
E_{n}}}/{Z},$ where $Z=\sum_{n}e^{-\beta E_{n}}$ is the canonical partition
function.

For the two-level system where the energy spectrum reads
$E_{+}=\frac{1}{2}\hbar\omega$ and $E_{-}=-\frac{1}{2}\hbar\omega$, the master
equation Eq. (A.1) becomes

$\left(\begin{array}[]{c}\dot{p}_{+}\\\
\dot{p}_{-}\end{array}\right)=\left(\begin{array}[]{cc}-W_{-+}&W_{+-}\\\
W_{-+}&-W_{+-}\end{array}\right)\left(\begin{array}[]{c}{p}_{+}\\\
{p}_{-}\end{array}\right),$ (A.2)

where these two transition rates $W_{-+}$ and $W_{+-}$ obey the detailed
balance, $W_{-+}/W_{+-}=e^{-\beta\hbar\omega}$. From Eq. (A.2), the motion for
the average population can be obtained as

$\langle\dot{n}\rangle=-\gamma(\langle n\rangle-\langle n\rangle^{eq}),$ (A.3)

where $\gamma=W_{-+}+W_{+-}$ and

$\langle
n\rangle^{eq}=-\frac{1}{2}\frac{W_{-+}-W_{+-}}{W_{-+}+W_{+-}}=-\frac{1}{2}\tanh\left(\frac{1}{2}\beta\hbar\omega\right).$
(A.4)

From Eq. (A.3), we find that instantaneous population $\langle n(t)\rangle$
along the thermalization process (staring at initial time $t=0$) can be
written in terms of the population $\langle n(0)\rangle$,

$\langle{n}(t)\rangle=\langle n\rangle^{eq}+[\langle n(0)\rangle-\langle
n\rangle^{eq}]e^{-\gamma t}.$ (A.5)

##  Appendix B Relation between populations at the begin and the end of a
unitary driving process

We consider the unitary time evolution along the driving compression
$A\rightarrow B$ from $t=0$ to $t=\tau_{ch}$ to determine $\langle
n^{2}(0)\rangle$ and $\langle n^{2}(\tau_{ch})\rangle$ at $A$ and $B$,
respectively. Using $p_{n}^{0}=e^{-\beta_{A}n\hbar\omega_{c}}/Z_{A}$ with
$Z_{A}=e^{-\beta_{A}\hbar\omega_{c}/2}+e^{\beta_{A}\hbar\omega_{c}/2}$, it
follows that $\langle n^{2}(0)\rangle=\sum_{n}n^{2}p_{n}^{0}=1/4$. Meanwhile,
$\langle n^{2}(\tau_{ch})\rangle$ can be calculated as

$\displaystyle\langle n^{2}(\tau_{ch})\rangle=$
$\displaystyle\sum_{n,m}n^{2}p_{m\rightarrow
n}^{\tau_{ch}}p_{m}^{0}(\beta_{A})$ (B.1) $\displaystyle=$
$\displaystyle\sum_{n,m}n^{2}|\langle
m|U_{\mathrm{com}}|n\rangle|^{2}p_{m}^{0}(\beta_{A})$ $\displaystyle=$
$\displaystyle\frac{1}{4Z_{A}}\big{[}e^{-\beta_{A}\hbar\omega_{c}/2}\left(|\langle+|U_{\mathrm{com}}|+\rangle|^{2}+\langle+|U_{\mathrm{com}}|-\rangle|^{2}\right)$
$\displaystyle+e^{\beta_{A}\hbar\omega_{c}/2}\left(|\langle-|U_{\mathrm{com}}|+\rangle|^{2}+\langle-|U_{\mathrm{com}}|-\rangle|^{2}\right)\big{]}$
$\displaystyle=$ $\displaystyle\langle n^{2}(0)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4},$

and $\langle n(0)n(\tau_{ch})\rangle$ reads

$\displaystyle\langle n(0)n(\tau_{ch})\rangle=$
$\displaystyle\sum_{n,m}nmp_{n\rightarrow m}^{\tau_{ch}}p_{n}^{0}(\beta_{A})$
(B.2) $\displaystyle=$ $\displaystyle\sum_{n,m}nm|\langle
n|U_{\mathrm{com}}|m\rangle|^{2}p_{n}^{0}(\beta_{A})$ $\displaystyle=$
$\displaystyle\frac{1}{4Z_{A}}\big{[}e^{-\beta_{A}\hbar\omega_{c}/2}\left(|\langle+|U_{\mathrm{com}}|+\rangle|^{2}-\langle+|U_{\mathrm{com}}|-\rangle|^{2}\right)$
$\displaystyle+e^{\beta_{A}\hbar\omega_{c}/2}\left(-|\langle-|U_{\mathrm{com}}|+\rangle|^{2}+\langle-|U_{\mathrm{com}}|-\rangle|^{2}\right)\big{]}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(1-2\xi),$

where
$\xi=|\langle\pm|U_{\mathrm{exp}}|\mp\rangle|^{2}=|\langle\pm|U_{\mathrm{com}}|\mp\rangle|^{2}$
and
$1-\xi=|\langle\pm|U_{\mathrm{exp}}|\pm\rangle|^{2}=|\langle\pm|U_{\mathrm{com}}|\pm\rangle|^{2}$
have been used. For the unitary expansion $C\rightarrow D$ of the two-level
engine, we therefore have

$\langle n^{2}(\tau_{cyc}-\tau_{c})\rangle=\langle
n^{2}(\tau_{ch}+\tau_{h})\rangle=\frac{1}{4},$ (B.3)

and

$\langle
n(\tau_{ch}+\tau_{h})n(\tau_{cyc}-\tau_{c})\rangle=\frac{1}{4}(1-2\xi).$ (B.4)

Integrating over the probability density function Eq. (15)in Main text, the
first two central moment of quantum work can be calculated as

$\displaystyle\langle\mathrm{w}\rangle=$
$\displaystyle\int\mathrm{w}p(\mathrm{w})d\mathrm{w}$ (B.5) $\displaystyle=$
$\displaystyle\int\mathrm{w}d\mathrm{w}\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-\mathrm{w}[n(0)\rightarrow
m(\tau_{ch});i(\tau_{ch}+\tau_{h})\rightarrow j(\tau_{cyc}-\tau_{c})]\\}$
$\displaystyle=$
$\displaystyle\int\mathrm{w}d\mathrm{w}\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-[(E_{i}^{h}-E_{j}^{c})-(E_{m}^{h}-E_{n}^{c})]\\}$
$\displaystyle=$
$\displaystyle\int\mathrm{w}d\mathrm{w}\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-\hbar[(i\omega_{h}-j\omega_{c})-(m\omega_{h}-n\omega_{c})]\\}$
$\displaystyle=$ $\displaystyle\hbar[(\omega_{c}-(1-2\xi)\omega_{h}]\langle
n(0)\rangle+\hbar[(\omega_{h}-(1-2\xi)\omega_{c}]\langle
n(\tau_{ch}+\tau_{h})\rangle,$

and

$\displaystyle\langle\mathrm{w}^{2}\rangle=$
$\displaystyle\int\mathrm{w}^{2}p(\mathrm{w})d\mathrm{w}$ (B.6)
$\displaystyle=$
$\displaystyle\int\mathrm{w}^{2}d\mathrm{w}\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-\mathrm{w}[n(0)\rightarrow
m(\tau_{ch});i(\tau_{ch}+\tau_{h})\rightarrow j(\tau_{cyc}-\tau_{c})]\\}$
$\displaystyle=$
$\displaystyle\int\mathrm{w}^{2}d\mathrm{w}\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-[(E_{i}^{h}-E_{j}^{c})-(E_{m}^{h}-E_{n}^{c})]\\}$
$\displaystyle=$
$\displaystyle\int\mathrm{w}^{2}d\mathrm{w}\sum_{m,n,i,j}p_{n\rightarrow
m}^{\tau_{ch}}p_{n}^{0}p_{i\rightarrow
j}^{\tau_{hc}}p_{i}^{\tau_{ch}+\tau_{h}}\delta\\{\mathrm{w}-\hbar[(i\omega_{h}-j\omega_{c})-(m\omega_{h}-n\omega_{c})]\\}$
$\displaystyle=$ $\displaystyle\hbar^{2}\omega_{h}^{2}\langle
n^{2}(\tau_{ch}+\tau_{h})\rangle-2\hbar^{2}\omega_{c}\omega_{h}\langle
n(\tau_{ch}+\tau_{h})n(\tau_{cyc}-\tau_{c})\rangle-2\hbar^{2}\omega_{c}\omega_{h}\langle
n(0)n(\tau_{ch})\rangle$ $\displaystyle+\hbar^{2}\omega_{c}^{2}\langle
n^{2}(0)\rangle+2\hbar^{2}[(\omega_{c}-(1-2\xi)\omega_{h}]\langle
n(0)\rangle[(\omega_{h}-(1-2\xi)\omega_{c}]\langle
n(\tau_{ch}+\tau_{h})\rangle$ $\displaystyle+\hbar^{2}\omega_{c}^{2}\langle
n^{2}(\tau_{cyc}-\tau_{c})\rangle+\hbar^{2}\omega_{h}^{2}\langle
n^{2}(\tau_{ch})\rangle.$

With the above results, the second moment of stochastic work can be simplified
as

$\displaystyle\langle\mathrm{w}^{2}\rangle=$ $\displaystyle
2\hbar^{2}\big{\\{}\frac{1}{4}[\omega_{h}^{2}+\omega_{c}^{2}-2\omega_{c}\omega_{h}(1-2\xi)]+[\omega_{c}-(1-2\xi)\omega_{h}]$
(B.7) $\displaystyle\times\langle
n(0)\rangle[\omega_{h}-(1-2\xi)\omega_{c}]\langle
n(\tau_{ch}+\tau_{h})\rangle\big{\\}}.$

Combining this with Eq. (B.5), the work fluctuations,
$\delta\mathrm{w}^{2}=\langle\mathrm{w}^{2}\rangle-\langle\mathrm{w}\rangle^{2}$,
we can obtain Eq. (18) in Main text.

Acknowledgements We acknowledge the finical support by National Natural
Science Foudantion (Grant Nos. 11875034 and 11505091), and the Major Program
of Jiangxi Provincial Natural Science Foundation (No. 20161ACB21006). Y.L.M.
acknowledges the financial support from the State Key Programs of China under
Grant (No. 2017YFA0304204).

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