Generalized Minkowski weights and Chow rings of \(T\)-varieties (Q6572076)

In general, for algebraic variety \(X\), the Kronecker duality homomorphism\N\begin{align*}\NA^k(X) \longrightarrow \text{Hom} \left( A_k(X), \mathbb{Z} \right), \ \ \ z \longmapsto \left( a \mapsto \text{deg} (z \cap a) \right)\N\end{align*}\Nis not an isomorphism. One of the results of this paper is that for a large class of \(T\)-varieties of complexity one, the Kronecker duality homomorphism, after tensoring with \(\mathbb{Q}\), is indeed an isomorphism. The proof follows from Poincare duality and a combinatorial property related to \(T\)-varieties called shellability.\N\NThe author, inspired by [\textit{W. Fulton} and \textit{B. Sturmfels}, `Topology 36, No. 2, 335--353 (1997; Zbl 0885.14025)], associate to a given Chow cohomology class a certain function on the set of polyhedra (and the set of cones) of \(T\)-variety \(X\). She calls these functions generalized Minkowski weights and defines the tropicalization map\N\begin{align*}\N\text{trop}: A^k(X) \longrightarrow M_k(X)\N\end{align*}\Nbetween the Chow cohomology groups \(A^k(X)\) and the group of \(k\)-codimensional generalized Minkowski weights \(M_k(X)\). Then she proves for the case of projective, \(\mathbb{Q}\)-factorial, contraction-free \(T\)-variety the induced tropicalization \(A^k(X)_\mathbb{Q} \longrightarrow M_k(X)_\mathbb{Q}\) is a bijection.\N\NAnother result of this article is defining an intersection pairing between Cartier divisorial support functions and generalized Minkowski weights\N\begin{align*}\N\text{CaSF}(\mathcal{S}) \times M_k(X) \longrightarrow M_{k+1}(X), \ \ \ (h, c) \longmapsto h \cdot c\N\end{align*}\NThe author shows that this pairing is compatible with the tropicalization isomorphism in the sense that for any \(z \in A^k(X)_\mathbb{Q}\) we have\N\begin{align*}\N\text{trop} \left( [D_h] \cup z \right) = h \cdot \text{trop}(z)\N\end{align*}\N\NThis paper is a first step towards a convex-geometrical description of the intersection theory of nef b-divisors (b stands for birational) on complexity-one \(T\)-varieties in the spirit of \textit{A. M. Botero} and \textit{J. I. B. Gil} [Math. Z. 300, No. 1, 579--637 (2022; Zbl 1511.14011)]. The author expect that nef b-divisors on such a \(T\)-variety can be characterized in terms of some ``concave'' functions on the divisorial fan. She plans to pursue these ideas in the future.