c 2011 by Institut Mittag-Leﬄer. All rights reserved

### Chern classes of tropical vector bundles

Lars Allermann

**Abstract.** We introduce tropical vector bundles, morphisms and rational sections of these
bundles and deﬁne the pull-back of a tropical vector bundle and of a rational section along a
morphism. Most of the deﬁnitions presented here for tropical vector bundles will be contained
inTorchiani, C., *Line Bundles on Tropical Varieties, Diploma thesis, Technische Universit¨*at
Kaiserslautern, Kaiserslautern,2010, for the case of line bundles. Afterwards we use the bounded
rational sections of a tropical vector bundle to deﬁne the Chern classes of this bundle and prove
some basic properties of Chern classes. Finally we give a complete classiﬁcation of all vector
bundles on an elliptic curve up to isomorphisms.

**1. Tropical vector bundles**

In this section we will introduce our basic objects such as tropical vector bun- dles, morphisms of tropical vector bundles and rational sections.

*Deﬁnition* 1.1. (Tropical matrices) A tropical matrix is an ordinary matrix
with entries in the tropical semi-ring

(T=R∪{−∞}*,⊕,*),

where *a⊕b*=max*{a, b}* and *ab=a+b. We denote by Mat(m×n,*T) the set of
tropical*m×n*matrices. Let*A∈*Mat(m*×n,*T) and*B∈*Mat(n*×p,*T). We can form
a tropical matrix product*AB*:=(c*ij*)*∈*Mat(m*×p,*T), where *c**ij*=*m*

*k=1**a**ik**b**kj**.*
Moreover, let*G(r×s)⊆*Mat(r*×s,*T) be the subset of tropical matrices with at most
one ﬁnite entry in every row. Let *G(r) be the subset of* *G(r×r) containing all*
tropical matrices with exactly one ﬁnite entry in every row and every column.

This set*G(r) together with tropical matrix multiplication is a group whose neutral*
element is the tropical unit matrix, i.e. the matrix with zeros on the diagonal and
all other entries equal to*−∞*.

I would like to thank my advisor Andreas Gathmann for numerous helpful discussions.

*Notation* 1.2. For an element*σ* of the symmetric group*S**r* we denote by *A**σ*

the tropical matrix*A**σ*=(a*ij*)*∈*Mat(r*×r,*T) given by
*a**ij*:=

0, if*i=σ(j),*

*−∞,* otherwise.

Moreover, for *a*1*, ..., a**r**∈R* we denote by*D(a*1*, ..., a**r*) the tropical diagonal matrix
*D(a*1*, ..., a**r*)=(d*ij*)*∈*Mat(r*×r,*T) given by

*d**ij*:=

*a**i**,* if*i=j,*

*−∞,* otherwise.

**Lemma 1.3.** *Every elementM∈G(r)can be written asM*=A_{σ}*D(a*_{1}*, ..., a** _{r}*)

*for someσ∈S*

*r*

*and some numbers*

*a*1

*, ..., a*

*r*

*∈R*.

*Proof.* Let*M∈G(r). By deﬁnition there exists exactly one ﬁnite entry in every*
column. Let *a**i**∈R* be this ﬁnite entry in column *i, situated in row* *p**i*, *i=1, ..., r.*

Hence we can deﬁne a permutation *σ:{*1, ..., r*}*→{1, ..., r*}* by *σ(i):=p**i*, *i=1, ..., r,*
and obtain*A*_{σ}*D(a*_{1}*, ..., a** _{r}*)=M.

**Lemma 1.4.** *G(r)is precisely the set of invertible tropical matrices,i.e.*

*G(r) ={A∈*Mat(r*×r,*T)*|AA** ^{}*=

*A*

^{}*A*=

*E*

*for someA*

^{}*∈*Mat(r

*×r,*T)

*}.*

*Proof.*The inclusion

*G(r)⊆ {A∈*Mat(r*×r,*T)*|AA** ^{}*=

*A*

^{}*A*=

*E*for some

*A*

^{}*∈*Mat(r

*×r,*T)

*}*is obvious. Thus, let

*A, A*

^{}*∈*Mat(r

*×r,*T) be given such that

*AA*

*=A*

^{}

^{}*A=E.*

Assume that*A=(a**ij*) contains more than one ﬁnite entry in a row or column. For
simplicity we assume that *a*11*, a*12=*−∞*. As *AA** ^{}*=E we can conclude that the
ﬁrst two rows of

*A*

*look as*

^{}*A** ^{}*=

⎛

⎜⎝

*α* *−∞* *...* *−∞*

*β* *−∞* *...* *−∞*

### *

⎞

⎟⎠ for some*α, β∈*R*.*

As moreover *A*^{}*A=E, we can conclude from the second row of* *A** ^{}* and the ﬁrst
column of

*A*that

*a*11+β=*−∞,*
which is a contradiction to *a*11*, β∈R*.

*Remark* 1.5. Note that a matrix *A∈G(r×s) does, in general, not induce a*
map*f**A*:R* ^{s}*→R

*,*

^{r}*x →Ax, as the vectorAx*may contain entries that are

*−∞*. To obtain a map

*f*

*A*:R

*→R*

^{s}*anyway we use the following deﬁnition: For*

^{r}*x∈R*

*we set*

^{s}*f*

*(x):=(y*

_{A}_{1}

*, ..., y*

*), where*

_{r}*y**i*=
*r*

*k=1**a**ik**x**k**,* if *r*

*k=1**a**ik**>−∞,*

0, otherwise.

We have all requirements now to state our main deﬁnition.

*Deﬁnition* 1.6.(Tropical vector bundles) Let *X* be a tropical cycle (cf. [AR1,
Deﬁnition 5.12]). A*tropical vector bundle* over *X* of rank *r* is a tropical cycle *F*
together with a morphism *π:* *F*→*X* (cf. [AR1, Deﬁnition 7.1]) and a ﬁnite open
covering*{U*1*, ..., U**s**}* of *X* as well as a homeomorphism Φ*i*:*π*^{−}^{1}(U*i*)→^{∼}^{=}*U**i**×R** ^{r}* for
every

*i∈{*1, ..., s

*}*such that

(a) for all *i*we obtain a commutative diagram
*π*^{−}^{1}(U*i*)

Φ_{i}

*π*

*U**i**×R*^{r}

*P*^{(i)}

*U**i**,*

where*P*^{(i)}:*U**i**×R** ^{r}*→

*U*

*i*is the projection on the ﬁrst factor;

(b) for all*i*and*j* the composition*p*^{(i)}_{j}*◦*Φ* _{i}*:

*π*

^{−}^{1}(U

*)→Ris a regular invertible function (cf. [AR1, Deﬁnition 6.1]), where*

_{i}*p*

^{(i)}

*:*

_{j}*U*

*i*

*×R*

*→R, (x,(a1*

^{r}*, ..., a*

*r*))

*→a*

*j*;

(c) for every *i, j∈{*1, ..., s*}* there exists a *transition map* *M**ij*:*U**i**∩U**j*→*G(r)*
such that

Φ_{j}*◦*Φ^{−}_{i}^{1}: (U_{i}*∩U** _{j}*)

*×R*

^{r}*−*→(U

_{i}*∩U*

*)*

_{j}*×R*

^{r}is given by (x, a)* →*(x, M*ij*(x)*a) and the entries ofM**ij*are regular invertible func-
tions on*U**i**∩U**j* or constantly*−∞*;

(d) there exist representatives *F*_{0} of *F* and *X*_{0} of *X* such that *F*_{0}=*{π*^{−}^{1}(τ)*|*
*τ∈X*0*}*and*ω**F*0(π^{−}^{1}(τ))=ω*X*0(τ) for all maximal polyhedra *τ∈X*0.

An open set*U**i*together with the map Φ*i*:π^{−}^{1}(U*i*)→^{∼}^{=}*U**i**×R** ^{r}* is called a

*local trivial-*

*ization*of

*F*. Tropical vector bundles of rank one are called

*tropical line bundles.*

*Remark* 1.7. Let *V*1*, ..., V**t* be any open covering of *X*. Then the covering
*{U*_{i}*∩V*_{j}*}*together with the restricted homeomorphisms Φ_{i}*|**π** ^{−1}*(U

_{i}*∩*

*V*

*)and transition maps*

_{j}*M*

_{ij}*|*(U

_{i}*∩*

*V*

*)*

_{k}*∩*(U

_{j}*∩*

*V*

*) fulﬁlls all requirements of Deﬁnition 1.6 too, and hence*

_{l}deﬁnes again a vector bundle. As the open covering, the homeomorphisms and the transition maps are part of the data of Deﬁnition1.6this new bundle is (according to our deﬁnition) diﬀerent from our initial one even though they are “the same”

in some sense. Hence, if two vector bundles arise one out of the other by such a construction, we will identify those two vector bundles.

*Remark and Deﬁnition* 1.8. Let *π:* *F*→*X* together with the open covering
*U*1*, ..., U**s*, homeomorphisms Φ*i* and transition maps *M**ij* and *π*:*F*→*X* together
with the open covering *V*1*, ..., V**t*, homeomorphisms Ψ*i* and transition maps *N**ij*

be two tropical vector bundles according to Deﬁnition 1.6. We will identify these
vector bundles if the vector bundles *π*: *F*→*X* with open covering *{U**i**∩V**j**}* and
restricted homeomorphisms Φ*i**|**π** ^{−1}*(U

*i*

*∩*

*V*

*j*)respectively Ψ

*j*

*|*

*π*

*(U*

^{−1}*i*

*∩*

*V*

*j*) and transition maps

*M*

*ij*

*|*(U

*i*

*∩*

*V*

*)*

_{k}*∩*(U

*j*

*∩*

*V*

*) respectively*

_{l}*N*

*kl*

*|*(U

*i*

*∩*

*V*

*)*

_{k}*∩*(U

*j*

*∩*

*V*

*) are equal.*

_{l}*Remark* 1.9. Let*π*1:*F*1→*X* and*π*2:*F*2→*X* be two vector bundles on*X*. By
Deﬁnition1.8we can always assume that*F*_{1} and*F*_{2}satisfy Deﬁnition1.6with the
same open covering.

*Remark* 1.10. Let *π:* *F*→*X* be a vector bundle with open covering *U*_{1}*, ..., U** _{s}*
and transition maps

*M*

*ij*as in Deﬁnition 1.6. On the common intersection

*U*

*i*

*∩U*

*j*

*∩U*

*k*we obviously have

*M*

*ij*(x)=M

*kj*(x)

*M*

*ik*(x). This last equation is called the

*cocycle condition. Conversely, this data, the open covering*

*U*

_{1}

*, ..., U*

*together with transition maps*

_{s}*M*

*fulﬁlling the cocycle condition, is enough to construct a vector bundle as we will see in the following proposition.*

_{ij}**Proposition 1.11.** *Let* *U*1*, ..., U**s**be an open covering of* *X* *and let*
*M** _{ij}*:

*U*

_{i}*∩U*

_{j}*−*→

*G(r)*

*be maps such that the entries of* *M**ij*(x) *are regular invertible functions on* *U**i**∩U**j*

*or constantly* *−∞* *and the cocycle condition* *M** _{ij}*(x)=M

*(x)*

_{kj}*M*

*(x)*

_{ik}*holds on*

*U*

*i*

*∩U*

*j*

*∩U*

*k*.

*Then there exists a vector bundleπ*:

*F*→

*X*

*with open coveringU*1

*, ..., U*

*s*

*and transition functionsM**ij*.

*Proof.* We take the disjoint union *s*

*i=1*(U*i**×R** ^{r}*) and identify points (x, y)

*∼*(x, M

*ij*(x)

*y) to obtain the topological space*

*|F|*. We have to equip this space with the structure of a tropical cycle. As this construction is exactly the same as for tropical line bundles, we only sketch it here and refer to [T] for more de- tails. Let (((X0

*,|X*0

*|,{ϕ*

*σ*

*}*), ω

*X*

_{0}),

*{*Φ

*σ*

*}*) be a representative of

*X.*We deﬁne

*F*0:=

*{π*

^{−}^{1}(σ)

*|σ∈X*0

*}*and

*ω*

*F*

_{0}(π

^{−}^{1}(σ)):=ω

*X*

_{0}(σ) for all maximal polyhedra

*σ∈X*0. Our next step is to construct the polyhedral charts

*ϕ*

_{π}*−1*(σ) for

*F*

_{0}: Let

*σ∈X*

_{0}be given and let

*U*

_{i}_{1}

*, ..., U*

*i*

*t*be all open sets having non-empty intersection with

*σ.*

Moreover, let*{V**i**|i∈I}*be the set of all connected components of all*σ∩U**i** _{k}*. Every
such set

*V*

*i*comes from a set

*U*

*of the given open covering. Hence, for every pair*

_{j(i)}*k, l∈I*we have a restricted transition map

*N*

*kl*: =M

_{j(k),j(l)}*|*

*V*

_{k}*∩*

*V*

*. This implies that for all*

_{l}*k, l∈I*the entries of

*N*

_{kl}*◦*Φ

^{−}

_{σ}^{1}are (globally) integer aﬃne linear functions on

*V*

*k*

*∩V*

*l*. As

*σ*is simply connected, for every such entry

*h∈O*

*(V*

^{∗}*k*

*∩V*

*l*) of

*N*

*kl*there exists a unique continuation ˜

*h∈O*

*(σ). Hence we can extend all transition maps*

^{∗}*N*

*kl*:

*V*

*k*

*∩V*

*l*→

*G(r) to maps*

*N*

_{kl}*:*

^{}*σ*→

*G(r). Now we choose for every*

*i∈I*a point

*P*

_{i}*∈V*

*and for all pairs*

_{i}*k, l∈I*a path

*γ*

*: [0,1]→*

_{kl}*σ*from

*P*

*to*

_{k}*P*

*. Let*

_{l}*k, l∈I*be given. As the image of

*γ*

*kl*is compact there exists a ﬁnite covering

*V*

*μ*

_{1}

*, ..., V*

*μ*

*of*

_{c}*γ*

*kl*([0,1]). For

*x∈V*

*l*we set

*S(γ**kl*)(x) :=*N*_{μ}^{}_{1}_{,μ}_{2}(x)^{−}^{1}*...N*_{μ}^{}_{c−1}_{,μ}* _{c}*(x)

^{−}^{1}

*∈G(r).*

Now ﬁx some*k*0*∈I. For all* *l∈I* we deﬁne maps

*ϕ*^{(l)}_{π}_{−1}_{(σ)}:*V*_{l}*×R*^{r}*∼*=*π*^{−}^{1}(V* _{l}*)

*−*→R

^{n}

^{σ}^{+r}

*,*

(x, a)* −*→(ϕ* _{σ}*(x), S(γ

_{k}_{0}

*)(x)*

_{l}*a).*

These maps agree on overlaps and hence glue together to an embedding

*ϕ*_{π}*−1*(σ):*π*^{−}^{1}(σ)*−*→R^{n}^{σ}^{+r}*.*

In the same way we can construct the fan chartsΦ_{π}*−1*(σ). Then we deﬁne *F* to be
the equivalence class

*F*:= [(((F0*,|F*0*|,{ϕ**π**−1*(σ)*}*), ω*F*0),*{*Φ*π**−1*(σ)*}*)].

*Example* 1.12. Throughout the chapter, the curve*X*:=X_{2} from [AR1, Exam-
ple 5.5] will serve us as a central example. Recall that*X* arises by gluing open fans
as drawn in the following ﬁgure.

Moreover, recall from [AR1, Deﬁnition 5.4] that the transition functions between
these open fans composing*X* are integer aﬃne linear maps. This implies that the

curve*X* has a well-deﬁned lattice length*L. We can cover* *X* by open sets*U*1, *U*2

and*U*3 as drawn in the following ﬁgure.

The easiest way to construct a (non-trivial) vector bundle of rank*r*on*X* is ﬁxing a
(non-trivial) transition map*M*12:*U*1*∩U*2→*G(r) and deﬁningM*23:*U*2*∩U*3→*G(r)*
and *M*31:*U*3*∩U*1→*G(r) to be the trivial maps* *x →E* for all *x. We will see later*
that in fact every vector bundle of rank*r* on*X* arises in this way.

Knowing what tropical vector bundles are, there are a few notions related to this deﬁnition we want to introduce.

*Deﬁnition*1.13.(Direct sums of vector bundles) Let*π*_{1}:*F*_{1}→*X*and*π*_{2}:*F*_{2}→*X*
be two vector bundles of rank*r*and*r** ^{}*, respectively, with a common open covering

*U*1

*, ..., U*

*s*and transition maps

*M*

_{ij}^{(1)}and

*M*

_{ij}^{(2)}, respectively, satisfying Deﬁnition1.6 (see Remark1.9). We deﬁne the

*direct sum bundle*

*π:F*

_{1}

*⊕F*

_{2}→

*X*to be the vector bundle of rank

*r+r*

*we obtain from the gluing data*

^{}*U*1*, ..., U**s* and *M*_{ij}^{(1)}*×M*_{ij}^{(2)}:*U**i**∩U**j**−*→*G(r+r** ^{}*),

*x −*→

*M*_{ij}^{(1)}(x) *−∞*

*−∞* *M*_{ij}^{(2)}(x)

*.*

*Deﬁnition* 1.14. (Subbundles) Let *π:* *F*→*X* be a vector bundle with open
covering*U*1*, ..., U**s*and homeomorphisms Φ*i*according to Deﬁnition1.6. A subcycle
*E∈Z**l*(F) is called a*subbundle* of rank*r** ^{}* of

*F*if

*π|*

*E*:

*E*→

*X*is a vector bundle of rank

*r*

*such that we have for all*

^{}*i=1, ..., s:*

Φ*i**|*(π*|**E*)* ^{−1}*(U

*i*): (π

*|*

*E*)

^{−}^{1}(U

*i*)

*−−*

^{∼}^{=}→

*U*

*i*

*×e*

*j*

_{1}

*, ..., e*

*j*

*R*

_{r}for some 1*≤j*_{1}*<...<j*_{r}*≤r, where the* *e** _{j}* are the standard basis vectors inR

*.*

^{r}*Remark*1.15. If

*π*:

*F*→

*X*is a vector bundle of rank

*r*with subbundle

*E*of rank

*r*

*like in Deﬁnition 1.14this implies that there exists another subbundle*

^{}*E*

^{}of rank*r−r** ^{}* with

Φ*i**|*(π*|**E*)* ^{−1}*(U

*i*): (π

*|*

*E*

*)*

^{}

^{−}^{1}(U

*i*)

*−−*

^{∼}^{=}→

*U*

*i*

*×e*

*j*

*|j /∈ {j*1

*, ..., j*

*r*

^{}*}*R

and hence that*F*=E*⊕E** ^{}* holds.

*Deﬁnition* 1.16.(Decomposable bundles) Let *π:* *F*→*X* be a vector bundle of
rank *r. We say that* *F* is*decomposable* if there exists a subbundle *π|**E*:*E*→*X* of
*F* of rank 1*≤r*^{}*<r. Otherwise we callF* an*indecomposable vector bundle.*

As announced in the very beginning of this section we also want to talk about morphisms and, in particular, isomorphisms of tropical vector bundles.

*Deﬁnition* 1.17.(Morphisms of vector bundles) A*morphism of vector bundles*
*π*1:*F*1→*X*of rank*r*and*π*2:*F*2→*X*of rank*r** ^{}*is a morphism Ψ :

*F*1→

*F*2of tropical cycles such that

(a) *π*_{1}=π_{2}*◦*Ψ;

(b) there exist an open covering *U*1*, ..., U**s* according to Deﬁnition1.6for both
*F*1 and*F*2 (see Remark1.9) and maps*A**i*:*U**i*→*G(r*^{}*×r) for alli*such that

Φ^{F}_{i}^{2}*◦*Ψ*◦*(Φ^{F}_{i}^{1})^{−}^{1}:*U**i**×R*^{r}*−*→*U**i**×R*^{r}^{}

is given by (x, a)* →*(x, f_{A}_{i}_{(x)}(a)) (cf. Remark1.5) and the entries of*A** _{i}* are regular
invertible functions on

*U*

*i*or constantly

*−∞*.

An *isomorphism of tropical vector bundles* is a morphism of vector bundles
Ψ :*F*_{1}→*F*_{2} such that there exists a morphism of vector bundles Ψ* ^{}*:

*F*

_{2}→

*F*

_{1}with Ψ

^{}*◦*Ψ=id=Ψ

*◦*Ψ

*.*

^{}**Lemma 1.18.** *Let* *π*1:*F*1→*X* *andπ*2:*F*2→*X* *be two vector bundles of rank*
*rover* *X.* *Then the following are equivalent:*

(a) *There exists an isomorphism of vector bundles* Ψ : *F*1→*F*2;

(b) *There exist a common open covering* *U*_{1}*, ..., U*_{s}*of* *X* *and transition maps*
*M*_{ij}^{(1)} *forF*_{1} *and* *M*_{ij}^{(2)} *forF*_{2} *satisfying Deﬁnition* 1.6 (cf. *Remark* 1.9) *and maps*
*E** _{i}*:

*U*

*i*→

*G(r)fori=1, ..., ssuch that*

*−* *the entries of* *E**i* *are regular invertible functions on* *U**i* *or constantly* *−∞*;

*−* *E**j*(x)*M*_{ij}^{(1)}(x)=M_{ij}^{(2)}(x)*E**i*(x)*for all* *x∈U**i**∩U**j* *and alli* *andj.*

*Proof.* (a)*⇒*(b) We claim that the maps *A**i*:*U**i*→*G(r×r) of Deﬁnition* 1.17
are the wanted maps*E** _{i}*. As Ψ is an isomorphism we can conclude that

*A*

*(x) is an invertible matrix for all*

_{i}*x∈U*

*i*, i.e. that

*A*

*i*:

*U*

*i*→

*G(r). Hence it remains to check*that

*A*

*(x)*

_{j}*M*

_{ij}^{(1)}(x)=M

_{ij}^{(2)}(x)

*A*

*i*(x) holds for all

*x∈U*

*i*

*∩U*

*j*: Let

*i*and

*j*be given.

As Ψ :*F*1→*F*2 is an isomorphism, the diagram

(U*i**∩U**j*)*×R*^{r}^{Φ}

*F*2

*i* *◦*Ψ*◦*(Φ^{F}_{i}^{1})^{−1}

Φ^{F}_{j}^{1}*◦*(Φ^{F}_{i}^{1})^{−1}

(U*i**∩U**j*)*×R*^{r}

Φ^{F}_{j}^{2}*◦*(Φ^{F}_{i}^{2})^{−1}

(U*i**∩U**j*)*×R*^{r}

Φ^{F}_{j}^{2}*◦*Ψ*◦*(Φ^{F}_{j}^{1})^{−1}

(U*i**∩U**j*)*×R*^{r}

commutes. Hence *A**j*(x)*M*_{ij}^{(1)}(x)=M_{ij}^{(2)}(x)*A**i*(x).

(b)*⇒*(a) Conversely, let the maps *E** _{i}*:

*U*

*→*

_{i}*G(r) be given. The equation*

*E*

*j*(x)

*M*

_{ij}^{(1)}(x) =

*M*

_{ij}^{(2)}(x)

*E*

*i*(x)

for all*x∈U*_{i}*∩U** _{j}* ensures that the maps

*U*_{i}*×R*^{r}*−*→*U*_{i}*×R*^{r}*,*
(x, a)* −*→(x, E*i*(x)*a),*

on the local trivializations can be glued to a globally deﬁned map Ψ :*|F*1*|→|F*2*|*.
Moreover, this map is a morphism as *π*1 and *π*2 are morphisms and the maps
*p*^{(i)}_{j}*◦*Φ^{F}_{i}^{1}, *p*^{(i)}_{j}*◦*Φ^{F}_{i}^{2} and the ﬁnite entries of *E** _{i}* are regular invertible functions
(cf. Deﬁnition1.6). The equation

*E*

*j*(x)

*M*

_{ij}^{(1)}(x)=

*M*

_{ij}^{(2)}(x)

*E*

*i*(x) implies that

*E*_{j}^{−}^{1}(x)*M*_{ij}^{(2)}(x) =*M*_{ij}^{(1)}(x)*E*_{i}^{−}^{1}(x)

holds for all*x∈U**i**∩U**j*, where*E*^{−}_{k}^{1}(x):=E*k*(x)^{−}^{1}for all*x∈U**k*. As the ﬁnite entries
of*E*_{k}^{−}^{1}: *U**k*→*G(r) are again regular invertible functions we can also glue the maps*

*U**i**×R*^{r}*−*→*U**i**×R*^{r}*,*

(x, a)* −*→(x, E^{−}_{i}^{1}(x)*a),*

on the local trivializations to obtain the inverse morphism Ψ* ^{}*:

*|F*2

*|→|F*1

*|*, which proves that Ψ is an isomorphism.

The morphisms we have just introduced admit another important operation, namely the pull-back of a vector bundle.

*Deﬁnition* 1.19.(Pull-back of vector bundles) Let *π:* *F*→*X* be a vector bun-
dle of rank *r* with open covering *U*1*, ..., U**s* and transition maps *M**ij* as in Def-
inition 1.6, and let *f*:*Y*→*X* be a morphism of tropical cycles. Then the *pull-*
*back bundle* *π** ^{}*:

*f*

^{∗}*F*→

*Y*is the vector bundle we obtain by gluing the patches

*f*

^{−}^{1}(U1)

*×R*

^{r}*, ..., f*

^{−}^{1}(U

*s*)

*×R*

*along the transition maps*

^{r}*M*

*ij*

*◦f*. Hence we obtain the commutative diagram

*f*^{∗}*F*

*f*^{}

*π*^{}

*F*

*π*

*Y*

*f*

*X*
*,*

where*f** ^{}* and

*π*

*are locally given by*

^{}*f*

*:*

^{}*f*

^{−}^{1}(U

*i*)

*×R*

^{r}*−*→

*U*

*i*

*×R*

^{r}*,*(y, a)

*−*→(f(y), a),

and *π** ^{}*:

*f*

^{−}^{1}(U

*i*)

*×R*

^{r}*−*→

*f*

^{−}^{1}(U

*i*), (y, a)

*−*→

*y.*

To be able to deﬁne Chern classes in the second section we need the notion of a rational section of a vector bundle.

*Deﬁnition*1.20.(Rational sections of vector bundles) Let*π:F*→*X*be a vector
bundle of rank*r. Arational sections*:*X*→*F* of*F*is a continuous map*s*:*|X|→|F|*
such that

(a) *π(s(x))=x*for all*x∈|X|*;

(b) there exist an open covering *U*1*, ..., U**s*and homeomorphisms Φ*i* satisfying
Deﬁnition1.6(cf. Deﬁnition1.8) such that the maps*p*^{(i)}_{j}*◦*Φ*i**◦s:U**i*→Rare rational
functions on*U** _{i}* for all

*i*and

*j, wherep*

^{(i)}

*:*

_{j}*U*

_{i}*×R*

*→R, (x,(a*

^{r}_{1}

*, ..., a*

*))*

_{r}*→a*

*.*

_{j}A rational section *s:* *X*→*F* is called*bounded* if the above maps *p*^{(i)}_{j}*◦*Φ*i**◦s*are
bounded for all*i* and*j.*

*Remark* 1.21. Let*π*:*L*→*X* be a line bundle and *s:* *X*→*L* be a rational sec-
tion. By deﬁnition, the map*p*^{(i)}*◦*Φ_{i}*◦s*is a rational function on*U** _{i}* for all

*i. More-*over, on

*U*

*i*

*∩U*

*j*the maps

*p*

^{(i)}

*◦*Φ

*i*

*◦s*and

*p*

^{(j)}

*◦*Φ

*j*

*◦s*diﬀer by a regular invertible function only. Hence

*s*deﬁnes a Cartier divisor

*D*(s)

*∈*Div(X).

There is a useful statement on these Cartier divisors*D*(s) in [T] that we want
to cite here including its proof.

**Lemma 1.22.** *Let* *π*: *L*→*X* *be a line bundle and let* *s*1*, s*2: *X*→*L* *be two*
*bounded rational sections.* *ThenD*(s1)*−D*(s2)=h *for some bounded rational func-*
*tionh∈K** ^{∗}*(X),

*i.e.*

*D*(s1)

*andD*(s2)

*are rationally equivalents.*

*Proof.* Let *U*_{1}*, ..., U** _{s}* be an open covering of

*X*with transition maps

*M*

*and homeomorphisms Φ*

_{ij}*i*according to Deﬁnition 1.6 such that for all

*i*both

*s*

^{(i)}

_{1}:=

*p*^{(i)}_{1} *◦*Φ*i**◦s*1 and *s*^{(i)}_{2} :=p^{(i)}_{1} *◦*Φ*i**◦s*2 are rational functions on*U**i* (cf. Deﬁnition1.20).

We deﬁne *h**i*:=s^{(i)}_{1} *−s*^{(i)}_{2} *∈K** ^{∗}*(U

*i*). Since we have that

*s*

^{(i)}

_{1}

*−s*

^{(j)}

_{1}=s

^{(i)}

_{2}

*−s*

^{(j)}

_{2}=M

*ij*

*∈*

*O*

*(U*

^{∗}*i*

*∩U*

*j*) for all

*i*and

*j, these maps*

*h*

*i*glue together to

*h∈K*

*(X). Hence we have that*

^{∗}*D*(s1)*−D*(s2) = [*{*(U*i**, s*^{(i)}_{1} )*}*]*−*[*{*(U*i**, s*^{(i)}_{2} )*}*]

= [*{*(U_{i}*, s*^{(i)}_{1} *−s*^{(i)}_{2} )*}*]=[*{*(U_{i}*, h** _{i}*)

*}*]=[

*{*(

*|X|, h)}*].

*Remark* 1.23. Lemma1.22implies that we can associate with any line bundle
*L*admitting a bounded rational section*s*a Cartier divisor class*D*(F):=[*D*(s)] that
only depends on the bundle*L*and not on the choice of the rational section*s.*

Combining both the notion of morphism of vector bundles and the notion of rational section we can give the following deﬁnition.

*Deﬁnition*1.24.(Pull-back of rational sections) Let*π:* *F*→*X* be a vector bun-
dle of rank *r* and *f*:*Y*→*X* be a morphism of tropical varieties. Moreover, let
*s*: *X*→*F* be a rational section of *F* with open covering *U*1*, ..., U**s* and homeo-
morphisms Φ1*, ...,*Φ*s* as in Deﬁnition 1.20. Then we can deﬁne a rational section
*f*^{∗}*s:* *Y*→ *f*^{∗}*F* of*f*^{∗}*F*, the*pull-back section* of*s, as follows: Onf*^{−}^{1}(U* _{i}*) we deﬁne

*f*^{∗}*s*:*f*^{−}^{1}(U*i*)*−*→*f*^{−}^{1}(U*i*)*×R*^{r}*,*
*y −*→(y,(p*i**◦*Φ*i**◦s◦f*)(y)),

where *p**i*: *U**i**×R** ^{r}*→R

*is the projection on the second factor. Note that for*

^{r}*y∈*

*f*

^{−}^{1}(U

*i*)

*∩f*

^{−}^{1}(U

*j*) the points (y,(p

*i*

*◦*Φ

*i*

*◦s◦f*)(y)) and (y,(p

*j*

*◦*Φ

*j*

*◦s◦f*)(y)) are iden- tiﬁed in

*f*

^{∗}*F*if and only if (f(y),(p

_{i}*◦*Φ

_{i}*◦s◦f*)(y)) and (f(y),(p

_{j}*◦*Φ

_{j}*◦s◦f*)(y)) are identiﬁed in

*F*. But this is the case as

(f(y),(p_{i}*◦*Φ_{i}*◦s◦f*)(y)) = (Φ_{i}*◦s)(f*(y))*∼*(Φ_{j}*◦s)(f*(y)) = (f(y),(p_{j}*◦*Φ_{j}*◦s◦f*)(y)).

Hence we can glue our locally deﬁned map*f*^{∗}*s*to obtain a map *f*^{∗}*s*:*Y*→*f*^{∗}*F.*

We ﬁnish this section with the following statement on vector bundles on simply connected tropical cycles which will be of use for us later on.

**Theorem 1.25.** *Let* *π*:*F*→*X* *be a vector bundle of rank* *r* *on the simply*
*connected tropical cycleX*. *Then* *F* *is a direct sum of line bundles,* *i.e.* *there exist*
*line bundles* *L*1*, ..., L**r* *onX* *such that* *F*=L1*⊕...⊕L**r*.

*Proof.* We show that every vector bundle of rank *r≥*2 on *X* is decomposable.

Let*U*1*, ..., U**s* be an open covering of*X* and let

*M**ij*(x) =*D(ϕ*^{(1)}_{i,j}*, ..., ϕ*^{(r)}* _{i,j}*)(x)

*A*

*σ*

*(x) =:*

_{ij}*D*

*ij*(x)

*A*

*σ*

*(x),*

_{ij}*x∈U*

*i*

*∩U*

*j*

*,*with

*ϕ*

^{(1)}

_{i,j}*, ..., ϕ*

^{(r)}

_{i,j}*∈O*

*(U*

^{∗}*i*

*∩U*

*j*) and

*σ*

*ij*(x)

*∈S*

*r*being transition functions according to Deﬁnition 1.6. We only have to show that it is possible to track the ﬁrst co- ordinate of the R

*-factor in*

^{r}*U*1

*×R*

*consistently along the transition maps: Let*

^{r}*γ*: [0,1]→|

*X|*be a closed path starting and ending in

*P∈U*1. Decomposing

*γ*into several paths if necessary, we may assume that

*γ*has no self-intersections, i.e. that

*γ|*[0,1) is injective. As

*γ([0,*1]) is compact we can choose an open covering

*V*

_{1}

*, ..., V*

*of*

_{t}*γ([0,*1]) such that for all

*j*we have

*V*

*j*

*⊆U*

*i*for some index

*i=i(j),P∈V*1=V

*t*

*⊆U*1, all sets

*V*

*j*and all intersections

*V*

*j*

*∩V*

*j+1*are connected and all intersections

*V*

*j*

*∩V*

*j*

^{}for non-consecutive indices are empty. For sets*V** _{j}* and

*V*

*with non-empty intersec- tion we have restricted transition maps*

_{j}*M*

*V*

*j*

*,V*

*(x)=*

_{j}*D*

*V*

*j*

*,V*

*(x)*

_{j}*A*

*σ*

*induced by the transition maps between*

_{Vj ,Vj}*U*

_{i(j)}*⊇V*

*and*

_{j}*U*

*)*

_{i(j}*⊇V*

*. Note that the permuta- tion parts*

_{j}*A*

*σ*

*of the transition maps do not depend on*

_{Vj ,Vj}*x*as all intersections

*V*

_{j}*∩V*

*are connected and the permutations have to be locally constant. We deﬁne*

_{j}*I*

*γ*:=σ

*V*

_{t−1}*,V*

_{t}*◦...◦σ*

*V*

_{1}

*,V*

_{2}(1). We have to check that

*I*

*γ*=1 holds. First we show that

*I*

*γ*does not depend on the choice of the covering

*V*1

*, ..., V*

*t*. Hence, let

*V*

_{1}

^{}*, ..., V*

_{t}*be another covering as above. We may assume that all intersections*

^{}*V*

_{j}*∩V*

_{j}

^{}*are con- nected, too. Between any two sets*

_{}*A, B∈{V*1

*, ..., V*

*t*

*, V*

_{1}

^{}*, ..., V*

_{t}

^{}*}*with non-empty in- tersection we have restricted transition maps

*M*

*A,B*(x)=

*D*

*A,B*(x)

*A*

*σ*

*as above.*

_{A,B}Moreover, let 0=α0*<...<α**p*=1 be a decomposition of [0,1] such that for all *i* we
have*γ([α*_{i}*, α** _{i+1}*])

*⊆V*

_{j}*∩V*

_{j}

^{}*for some indices*

_{}*j*and

*j*

*. Let*

^{}*i*

_{0}be the maximal index such that

*γ([α*

*i*0

*, α*

*i*0+1])

*⊆V*

*a*

*∩V*

_{b}*and*

^{}*σ**V*_{a−1}*,V*_{a}*◦...◦σ**V*_{1}*,V*_{2}=*σ**V*_{b}^{}*,V*_{a}*◦σ**V*_{b−1}^{}*,V*_{b}^{}*◦...◦σ**V*_{1}^{}*,V*_{2}^{}

is still fulﬁlled. Assume that *i*0*<p−*1. Let *γ([α**i*_{0}+1*, α**i*_{0}+2])*⊆V**a*^{}*∩V*_{b}* ^{}*. Hence

*γ(α*

*i*

_{0}+1)

*∈V*

*a*

*∩V*

_{b}

^{}*∩V*

*a*

*∩V*

_{b}*and we can conclude using the cocycle condition that*

^{}*σ**V*_{a}*,V*_{a}*◦σ**V*_{a−1}*,V*_{a}*◦...◦σ**V*_{1}*,V*_{2}=*σ**V*_{a}*,V*_{a}*◦σ*_{V}*b**,V**a**◦σ*_{V}

*b−1**,V*_{b}^{}*◦...◦σ** _{V}*
1

*,V*

_{2}

^{}=*σ**V*_{a}*,V*_{a}*◦σ*_{V}

*b**,V**a**◦σ*_{V}*b**,V*^{}

*b**◦σ*_{V}

*b**−*1*,V*_{b}^{}*◦...◦σ** _{V}*
1

*,V*

_{2}

^{}=*σ*_{V}

*b**,V*_{a}*◦σ*_{V}*b**,V*^{}

*b**◦σ*_{V}

*b−1**,V*_{b}^{}*◦...◦σ** _{V}*
1

*,V*

_{2}

^{}*,*

a contradiction to our assumption. Hence *i*_{0}=p*−*1 and we conclude that *I** _{γ}* is
independent of the chosen covering.

If*γ* and *γ** ^{}* are paths that pass through exactly the same open sets

*U*

*i*in the same order, then we can conclude that

*I*

*γ*=I

*γ*

*holds as exactly the same transition functions are involved. Hence, a continuous deformation of*

^{}*γ*does not change

*I*

*γ*. As

*X*is simply connected we can contract

*γ*to a point. This implies that

*I*

*=I*

_{γ}

_{γ}_{0}, where

*γ*0 is the constant path

*γ*0(t)=P for all

*t. ThusI*

*γ*=I

*γ*

_{0}=1. This proves the claim.

There is a related theorem in [T] which we want to state here. As we will not need the result in this work, we will omit the proof and refer to [T] instead.

**Theorem 1.26.** *Letπ:* *L*→*X* *be a line bundle on the simply connected tropical*
*cycleX*. *ThenL* *is trivial,* *i.e.L∼*=X*×R* *as a vector bundle.*

Combining Theorems1.25and1.26we can conclude the following result.

**Corollary 1.27.** *Let* *π*:*F*→*X* *be a vector bundle of rank* *r* *on the simply*
*connected tropical cycleX*. *ThenF* *is trivial,i.e.* *F∼*=X*×R*^{r}*as a vector bundle.*

**2. Chern classes**

In this section we will introduce Chern classes of tropical vector bundles and prove some basic properties. To be able to do this we need some preparation.

*Deﬁnition*2.1. Let*π:F*→*X*be a vector bundle of rank*r*and let*s*:*X*→*F* be a
rational section with open covering*U*1*, ..., U**s*as in Deﬁnition1.20. We ﬁx a natural
number 1*≤k≤r* and a subcycle *Y∈Z**l*(X). By deﬁnition, *s**ij*:=p^{(i)}_{j}*◦*Φ*i**◦s*:*U**i*→R
is a rational function on *U**i* for all *i* and *j*. Hence, for all *i* we can take local
intersection products

(s^{(k)}*·Y*)*∩U**i*:=

1*≤**j*_{1}*<...<j*_{k}*≤**r*

*s**ij*_{1}*...s**ij*_{k}*·*(Y*∩U**i*).

Since *s**i*^{}*j*=s* _{iσ(j)}*+ϕ

*j*on

*U*

*i*

*∩U*

*i*

*for some*

^{}*σ∈S*

*r*and some regular invertible map

*ϕ*

*j*

*∈O*

*(U*

^{∗}*i*

*∩U*

*i*), the intersection products (s

^{(k)}

*·Y*)

*∩U*

*i*and (s

^{(k)}

*·Y*)

*∩U*

*i*coincide on

*U*

_{i}*∩U*

*and we can glue them to obtain a global intersection cycle*

_{i}*s*

^{(k)}

*·Y∈*

*Z*

*l*

*−*

*k*(X).

**Lemma 2.2.** *Letπ*:*F*→*X* *be a vector bundle of rankr,ﬁxk∈{*1, ..., r*}and let*
*s*:*X*→*F* *be a rational section.* *Moreover,letY∈Z**l*(X)*be a cycle and letϕ∈K** ^{∗}*(Y)

*be a bounded rational function onY*.

*Then*

*s*^{(k)}*·*(ϕ*·Y*) =*ϕ·*(s^{(k)}*·Y*).

*Proof.* The claim follows immediately from the deﬁnition of the product
*s*^{(k)}*·Y*.

**Lemma 2.3.** *Let* *π*:*F*→*X* *andπ** ^{}*:

*F*

*→*

^{}*X*

*be two isomorphic vector bundles*

*of rank*

*r*

*with isomorphismf*:

*F*→

*F*

*.*

^{}*Moreover,*

*ﬁx*

*k∈{*1, ..., r

*}*,

*lets*:

*X*→

*F*

*be*

*a rational section and letY∈Z*

*l*(X)

*be a cycle.*

*Then*

*s*^{(k)}*·Y* = (f*◦s)*^{(k)}*·Y∈Z**l**−**k*(X).

*Proof.* Let*U*_{1}*, ..., U** _{s}*be an open covering of

*X*satisfying Deﬁnition1.6for both

*F*and

*F*

*and let*

^{}*s*

*:=p*

_{ij}^{(i)}

_{j}*◦*Φ

_{i}*◦s*:

*U*

*→Rand (f*

_{i}*◦s)*

*:=p*

_{ij}^{(i)}

_{j}*◦*Φ

_{i}*◦f◦s*:

*U*

*→Ras in Deﬁnition 2.1. By Lemma 1.18 the isomorphism*

_{i}*f*can be described on

*U*

*i*

*×R*

*by (x, a)*

^{r}*→*(x, E

*i*(x)

*a) withE*

*i*(x)=D(ϕ1

*, ..., ϕ*

*r*)

*A*

*σ*for some regular invertible functions

*ϕ*

_{1}

*, ..., ϕ*

_{r}*∈O*

*(U*

^{∗}*) and a permutation*

_{i}*σ∈S*

*. Hence (f*

_{r}*◦s)*

*=s*

_{ij}*+ϕ*

_{iσ(j)}*on*

_{j}*U*

*and thus*

_{i}

1*≤**j*_{1}*<...<j*_{k}*≤**r*

*s*_{ij}_{1}*...s*_{ij}_{k}*·*(Y*∩U** _{i}*) =

1*≤**j*_{1}*<...<j*_{k}*≤**r*

(f*◦s)*_{ij}_{1}*...(f◦s)*_{ij}_{k}*·*(Y*∩U** _{i}*),

which proves the claim.

To be able to prove the next theorem, which will be essential for deﬁning Chern classes, we ﬁrst need some generalizations of our previous deﬁnitions:

*Deﬁnition*2.4.(Inﬁnite tropical cycle) We deﬁne an*inﬁnite tropical polyhedral*
*complex* to be a tropical polyhedral complex according to [AR1, Deﬁnition 5.4] but
we do not require the set of polyhedra*X* to be ﬁnite. In particular, all open fans
*F**σ* still have to be open tropical fans according to [AR1, Deﬁnition 5.3]. Then an
*inﬁnite tropical cycle*is an inﬁnite tropical polyhedral complex modulo reﬁnements
analogous to [AR1, Deﬁnition 5.12].

*Remark* 2.5. Deﬁnition2.4implies that an inﬁnite tropical polyhedral complex
is locally ﬁnite, i.e. there are only ﬁnitely many polyhedra adjacent to any single
polyhedron. We can therefore think of inﬁnite tropical cycles to be inﬁnitely many
ordinary tropical fans glued together.

*Deﬁnition* 2.6. (Inﬁnite rational functions and inﬁnite Cartier divisors) Let
*C* be an inﬁnite tropical cycle and let *U* be an open set in *|C|*. As in [AR1,
Deﬁnition 6.1] an*inﬁnite rational function* on*U* is a continuous function*ϕ:U*→R
such that there exists a representative (((X,*|X|,{m*_{σ}*}**σ**∈**X*), ω* _{X}*),

*{M*

_{σ}*}*

*σ*

*∈*

*X*) of

*C,*which may now be an inﬁnite tropical polyhedral complex, such that for each face

*σ∈X* the map*ϕ◦m*^{−}_{σ}^{1} is locally integer aﬃne linear (where deﬁned). Analogously
it is possible to deﬁne*inﬁnite regular invertible functions* on*U*.

A *representative of an inﬁnite Cartier divisor* on *C* is a set *{*(U*i**, ϕ**i*)*|i∈I}*,
where*{U**i**}**i* is an open covering of*|C|*and*ϕ**i*is an inﬁnite rational function on*U**i*.
An *inﬁnite Cartier divisor* on *C* is a representative of an inﬁnite Cartier divisor
modulo the equivalence relation given in [AR1, Deﬁnition 6.1].

*Remark* 2.7. Using these basic deﬁnitions it is possible to generalize many
other concepts to the inﬁnite case. In particular, as our inﬁnite objects are locally
ﬁnite, it is possible to perform intersection theory as before.

*Deﬁnition*2.8.(Tropical vector bundles on inﬁnite cycles) Let*X* be an inﬁnite
tropical cycle. A *tropical vector bundle* over *X* of rank *r* is an inﬁnite tropical
cycle*F* together with a morphism *π:F*→*X* such that properties (a)–(d) given in
Deﬁnition 1.6 are fulﬁlled with the diﬀerence that the open covering *{U*_{i}*}**i* of *X*
may now be inﬁnite.

Now we are ready to prove the following announced theorem.

**Theorem 2.9.** *Let* *π*:*F*→*X* *be a vector bundle of rank* *r* *and* *s*1*, s*2:*X*→*F*
*be two bounded rational sections.* *Thens*^{(k)}_{1} *·Y* *ands*^{(k)}_{2} *·Y* *are rationally equivalent,*
*i.e.*

[s^{(k)}_{1} *·Y*] = [s^{(k)}_{2} *·Y*]*∈A** _{∗}*(X)

*for all subcycles*

*Y∈Z*

*(X).*

_{l}*Proof.* Let*p:|X|→|X|*be the universal covering space of*|X|*. We can locally
equip*|X|*with the tropical structure inherited form*X*and obtain an inﬁnite tropical
cycle *X* according to Deﬁnition2.4. Moreover, pulling back *F* along *p, we obtain*
a tropical vector bundle *p*^{∗}*F* on *X* according to Deﬁnition 2.8. As *X* is simply
connected we can conclude by Lemma1.25that*p*^{∗}*F*=L1*⊕...⊕L**r*for some inﬁnite
tropical line bundles *L*1*, ..., L**r* on *X*. Hence, the bounded rational sections *p*^{∗}*s*1

and*p*^{∗}*s*_{2} correspond to*r*inﬁnite tropical Cartier divisors as in Deﬁnition2.6each,
which we will denote by*ϕ*1*, ..., ϕ**r* and *ψ*1*, ..., ψ**r*, respectively. By Lemma 1.22we
can conclude that for all*i* these Cartier divisors diﬀer by bounded inﬁnite rational
functions only, i.e.*ϕ**i**−ψ**i*=h*i* for some bounded inﬁnite rational function*h**i*on*X*.
In particular,

1*≤**j*1*<...<j**k**≤**r*

*ϕ**j*1*...ϕ**j*_{k}*−*

1*≤**j*1*<...<j**k**≤**r*

*ψ**j*1*...ψ**j*_{k}

*·X*= ˜*h·ξ*˜2*...ξ*˜*k**·X*