We first study the log map
\[
\Log_t:= \log_t|-|: \CC\to \RR\cup\{-\infty\}
\]
where \(t>1\).
and try to define a ``ring structure'' on \(\RR\cup\{-\infty\}\) which makes this log map a homomorphism.
The first guess that one would take is to define
\begin{align*}
``\times" \text{ given by the operation } q_1 ``\times" q_2 =
&\Log_t(\Log_t^{-1}(q_1) \cdot \Log_t^{-1}(q_2)) =q_1+q_2 \\
``+" \text{ given by the operation } q_1 ``+" q_2 = &\Log_t(\Log_t^{-1}(q_1) + \Log_t^{-1}(q_2))
\end{align*}
While the first operation is well defined, the second is not! In order to make this well defined we take the limit of the second equation as \(t\to \infty\), from which we obtain
\[\lim_{t\to\infty} \Log_t(\Log_t^{-1}(q_1) + \Log_t^{-1}(q_2)) =
\max(q_1, q_2).\]
This gives us a definition for our new operations.
definition 1.0.1
The semi-field of tropical numbers is the set \(\TT:= \RR\cup_\infty\) equipped with the operations (called tropical plus and tropical times):
\begin{align*}
q_1\oplus q_2 =& \max(q_1, q_2)\\
q_1\odot q_2 =& q_1+q_2
\end{align*}
where \(q_1, q_2\in \TT\).The goal is now to understand how algebraic geometry over \(\CC\) relates to ``algebraic geometry'' over the semi-field \(\TT\). The first thing to do it to exchange polynomials for tropical polynomials. For instance, given \(f(z_1, z_2): \CC^2\to \CC\) a polynomial of two variables, we declare the tropicalization of the polynomial to be
\[f(z_1, z_2)=\sum a_{ij} z_1^iz_2^j \leftrightarrow \TropB(f)(q_1, q_2):= \max (a_{ij}+iq_1 + jq_2 )\]
example 1.0.2
If \(f=q_1+q_2+1\), the tropicalization is given by \(\TropB(f) = \max(1+q_1, 1+q_2, 1)\)
2: the zero locus of a tropical polynomial
The next question is to understand the zero set of a tropical function. When \(f\) is a polynomial, we have the associated variety \(V(f):= \{q\st f(q)=0\}\). Naïvely, one might try define the tropical zero set to be the set where \(\TropB(f)=-\infty\) (as \(-\infty\) is the tropical additive identity). However, it is clear to see that this will have no solutions.
example 2.0.1
Take \(f=q_1+q_2+1\), where \(q_1,q_2 \in (\CC^*)^2\). Drawing the image of \(V(f)\) under \(\Log_t\) gives us the following sequence of images:
We call the image of \(V(f)\) under the \(\Log_t\) map the amoeba of \(V(f)\), which (in this example) appears to converge to some piecewise linear object.
This mimics the following behavior. A tropical polynomial \(\phi: \RR^n\to \RR\) by definition, is a convex piecewise integral affine function on \(\RR^2\). We can associate to every such function the locus where \(\phi\) where \(\phi\) is not differentiable. This will be a union of polyhedra of dimension \(n-1\).
definition 2.0.2
Let \(\phi: \RR^n\to \RR\) be a tropical polynomial. The tropical zero set is set
\[V(f):=\{q\in \RR^n \text{ such that \(f\) is not differentiable at \(q\)}\}.\]
We now give some more details to the previous section.
definition 2.0.3
Fix a lattice structure \(\ZZ^n\subset \RR^n\) and \(\ZZ\subset \RR\).
We say that a function \(\underline \phi: \RR^n\to \RR\) is integral affine if \(\underline \phi= \phi_{l}+c\) where \(\phi_{l}\) is \(\ZZ\)-linear and \(c\in \RR\) is some constant.
definition 2.0.4
For \(\underline \phi: \RR^n\to \RR\) a integral affine structure, define \(H_{\underline \phi}:= \{q\in \RR^n \st \underline\phi(q) \geq 0 \}.\)
definition 2.0.5
We say that \(\sigma\subset \RR^n\) is a rational polyhedron if \(\sigma = \bigcap_{i\in I} H_{\underline \phi_i}\) for some collection of integral affine functions.
We say that it is a cone if a translate of it is closed under multiplication by \(\RR_{\geq 0}\)
We say that \(\sigma\) is a polytope if it is compact.
definition 2.0.6
A set of polyhedra is called a polyhedral complex if
if \(\sigma\in P\), then every face \(\tau< \sigma\) is also contained in \(P\),
For \(\sigma_1, \sigma_2\in P\): if \(\sigma_1\cap\sigma_2= \tau \neq \emptyset\),then \(\tau\) is a face of both \(\sigma\) and \(\sigma'\).
figure 2.0.7:Two examples of unions of polyhedra. On the left, a polyhedral complex. The right figure is not an example of a polyhedral complex
lemma 2.0.8
Every set of polyhedra has a polyhedral complex subdivision.
Later, we will impose some additional criteria on these complexes:
Pure dimension \(k\) --- every maximal cell is of dimension \(k\),
``Weighted'': there exists a weight function \(\weight: \{\text{Maximal Cells}\} \to \ZZ\)
``Balanced'': for a weighted pure complex: every codimension 1 face \(\tau\) is we have \(\sum_{\sigma} \weight(\sigma) v_{\sigma/\tau}=0\) in \(\RR^n /\Span(\tau)\).
definition 2.0.9
A tropical \(k\)-cycle in \(\RR^n\) is a balanced weighted polyhedral complex.
3: a sketch of intersection theory
Given \(Z_k(\RR^n)\), the set of tropical \(k\)-cycles, has a group structure, whose additive structure is given by taking unions and subdividing. This also comes with a ring structure
\begin{align*}
Z_k(\RR^n)\otimes Z_l(\RR^n)\to \ZZ_{n+l-n}(\RR^n)\\
X\tensor Y \mapsto \lim_{\epsilon\to 0} (X\cap (T+\eps v))
\end{align*}
where is chose so that the intersection is transverse for all \(\epsilon>0\) sufficiently small.
figure 3.0.1:We compute the intersection between the two tropical curves drawn on the left hand side by first taking a slight perturbation so that they intersect transversely. The set obtain in the limit as the perturbations go to zero is independent of the choice of transverse perturbation. In this setting, the intersection of two tropical lines is a point.
The key step to proving that this ring structure is well defined is the moving lemma.
lemma 3.0.2
The content of the moving lemma goes here
From this information, it is natural to define a tropical version of the Chow group. This means that we need a substitute for rational/algebraic/numerical equivalence.
definition 3.0.3
A function \(\psi: \RR^n\to \RR\) is a tropical rational function is a piecewise integral affine function.
example 3.0.4
The function \(\psi(q_1, q_2):= \text{min}(1+q_1, 1+q_2, 1)\) is an example of a rational function. Observe that this is not a convex function, and therefore is not a tropical polynomial
To a rational function \(f\), we associate a tropical \((n-1)\) cycle in \(\RR^n\). First, pick a minimal polyhedral subdivision \(P\) of \(\RR^n\) so that \(\psi_\sigma\) is integral affine for every \(\sigma\in P\). This is not uniquely determined. Then take the \((n-1)\) skeleton for this polyhedral structure; then assign weights.
definition 3.0.5
\begin{align*}
R_k:= &\{\div(\psi)_S\cdot c : \psi\in PA(\RR^n), c\in Z_{k+1}(\RR^n)\}\\
R_k^b:=&\{\div(\psi)_S\cdot c : \psi\in PA(\RR^n), \text{bounded} , c\in Z_{k+1}(\RR^n)\}
\end{align*}
We then can define :
\begin{align*}
CH_k(\RR^n):=Z_k/R_k \\
CH_k^b(\RR^n):= Z_k/R^b_k
\end{align*}
4: a sketch of compactification in tropical geometry
Note that we have some ideas of compactification already: after all we can define tropical polynomials on \(\RR^n\), and \(\TT^n\) is a compactification of \(\RR^n\).
The space \(\TT^n\) has a decomposition into pieces \(\RR_I\), which we call the torus orbits. The key notion that we will need to study is how many ``-'infinity'''s does a point \(x\) belong to.
definition 4.0.1
Given \(x\in \TT^n\), we define the sedentarity of \(x\) to be the number of \(-\infty\)'s belonging to \(q\) in its coordinate description.
We say that \(\sigma\in \TT^n\) is a polyhedron if it is the closure of a polyhedron in one of the torus orbits of \(\TT^n\).
From these compact pieces, we can define a tropical variety: a locally open subset of polyhedra satisfying a balancing condition.
5: the tropical Hodge conjecture
Associated to \(X\) a tropical variety, we want to some geometry. We have a short exact sequence
\[ 0 \to \RR_X\to PA_X\to \Omega_X\to 0\]
where \(\Omega_X\) gives the tropical cotangent sheaf, whose sections are called tropical 1-forms. Another characterization is: given a fan \(\Sigma\subset \RR^n\), want to define \(F_k(\Sigma)\subset \bigwedge^k\ZZ^n\) the set of polyvectors generated by \(v_1\wedge \cdots \wedge v_k\), where \(v_1, \ldots, v_k\) belong to a single cone of \(\Sigma\). This associates to \(X\) a tropical variety \(F_k\) a constructible sheaf on \(X\) so that \(F_{k, x}=F_k(\Sigma(x))\), where \(\Sigma(x)\) is the tangent cone at \(x\). This defines for us a chain complex
\[C_\bullet(X, F_k)\]
which comes with a tropical \((p, q)\) homology which is the homology \(H_q(X, F_p)\). When \(X\) can be geometrically realized, this compute some limit of a mixed Hodge structure, and satisfies the Kähler/Hodge package. Finally, there exists a cycle class map
\begin{align*}
CH_k(\RR^n)\to H_{p}(\RR^n, F_p)&&
Z\mapsto \Sigma_\subset K_{cell} \weight(\Delta)\cdot v_\Delta \otimes \Delta
\end{align*}
where \(\Delta\) is the volume of the \(k\)-cell. The tropical Hodge conjecture is that this is an isomorphism.
