EXPONENTIAL ACTIONS DEFINED BY VECTOR CONFIGURATIONS, GALE DUALITY, AND MOMENT-ANGLE MANIFOLDS

The paper proves: (i) freeness on U(K) iff AI is linearly independent for all I∈K, and properness iff {K,A} is a (simplicial) fan (Theorem 5.1) , with the fan condition stated equivalently as pairwise disjoint relative interiors of the cones cone AI (Theorem 4.4) ; (ii) compactness of U(K)/V iff the fan is complete (Theorem 6.1) and a description via the invariant ℓ(x)=∑i ai log|xi| used in the proof ; (iii) identification for complete simplicial fans: U(K)/V ≅ RK (Theorem 7.3) and, in the holomorphic case with even dim V and a complex structure, UC(K)/Ṽ ≅ ZK (Theorem 10.6) ; (iv) in the rational setting, proper algebraic C×_L-actions, Cox quotients UC(K)/C×_L ≅ XΣ̂, and relations to partial quotients by TL and V/Q (Section 12; Proposition 12.3) . The candidate solution reproduces these results with the same core ideas: stabilisers via {γj : xj≠0} (cf. Proposition 2.1) and Gale dual linear independence (Proposition 3.1) for freeness; a sequence/compactness argument leading to opposite index sets I+, I− and disjointness of relint cones for properness (matching the paper’s proof of Theorem 5.1) ; the ℓ-invariant and solving ⟨γi,v⟩ equations to land in RK or ZK (as in the proofs of Theorems 6.1 and 10.6) ; and Cox-style rational quotients (Section 12) . Minor phrasing differences aside (e.g., the structure group of the holomorphic bundle over XΣ is a complex torus C×_L/Ṽ rather than the real torus TL), the arguments are essentially the same. Hence: Both correct (substantially same proof).