Homology of toric varieties

Question

In the proof of theorem 5.1.1 of this paper by Dotsenko, Shadrin, and Vallette, they define this (smooth, normal) toric variety $B(n)$, note that there is a surjective toric morphism to $\mathbb{P}^1$ and then claim that the inverse image of any two points under this map must be homologous. Is this obvious? Is this a particular theorem of toric geometry? What makes this map special compared to a generic continuous projection?

Answer 1

I suggest reading

Guillemin, Victor; Pollack, Alan, Differential topology, Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-5193-7/hbk). xviii, 222 p. (2010). ZBL1420.57001.

The relevant result is that if $X, Y$ are smooth compact manifolds, $f: X\to Y$ is a smooth map and $B\subset Y$ is a smooth (compact) submanifold with boundary such that $f$ is transversal to $B$ (e.g. if $B$ is contained in the set of regular values of $f$), then $A=f^{-1}(B)$ is also a smooth submanifold with boundary. Moreover, $\partial A= f^{-1}(\partial B)$.

Now, in your case, you have a surjective holomorphic mapping $f: X\to Y=CP^1$, with domain equal to a connected complex manifold $X$, which implies that it has finite set of critical values, I'll call it $C$. In particular, if $y_1, y_2\in Y\setminus C$ are distinct, there is a smooth simple arc $B$ in $Y\setminus C$ connecting $y_1, y_2$. The preimage $A=f^{-1}(B)$ will be a compact submanifold with boundary, $\partial A= D_1\sqcup D_2= f^{-1}(y_1) \sqcup f^{-1}(y_2)$. With a bit more thought (using the fact that $f$ is holomorphic), you will see that $A$ is orientable (use an orientation of the path $B$). Thus, after orienting $A$, it defines a chain, denoted $(A)$, in the singular homology group of $X$. Ditto $D_1, D_2$ (they are complex submanifolds, hence, carry a natural orientation); they define cycles $(D_1), (D_2)$. Therefore, for an appropriate orientation on $A$, in the singular chain complex of $X$, we get $$ \partial (A)= (D_1) - (D_2). $$ Hence, the cycles $(D_1), (D_2)$ are homologous in $X$.

In general, it is easy to find examples (even for holomorphic maps $P^1\to P^1$) such that preimages of different points are not homologous to each other. One would need to read the paper you linked to in great detail to find out if this happens in their setting. Most likely, they assumed that the points are chosen generically, just forgot to mention this.