AN UPPER BOUND FOR POLYNOMIAL VOLUME GROWTH OF AUTOMORPHISMS OF ZERO ENTROPY

The paper proves the sharp upper bound plov(f) ≤ (k/2 + 1)d for an automorphism f with quasi-unipotent action on N^1(X)R (k+1 = max Jordan block size) via a combinatorial/representation-theoretic analysis of intersection-number identities, formalized through a weighted incidence matrix Ak,d,n and an sl2(C)-type hard Lefschetz argument (see Theorem 1.1 and the surrounding sketch) . The candidate solution reaches the same inequality by a different route: Khovanskii–Teissier reduction to a quadratic form, convolution estimates for correlations a_m, and polynomial growth bounds from the Jordan form, after constructing an f*-invariant nef class to make the quadratic form f*-invariant. All steps align with standard tools and the core identity Vol(Γ_n) = (∑_{i=0}^{n−1}(f^i)^*h)^d used in the paper’s preliminary section . However, the candidate’s Step 2 (existence of an f*-fixed nef class θ) is under-justified as written: the claimed norm estimate ∥(U−I)v_M∥ = O(M^k) for the polynomially weighted averages is not established and appears too optimistic if taken with a generic operator norm. This can be repaired by a standard compact-slice fixed-point normalization (e.g., x ↦ T x / ℓ(T x) on {x ∈ Nef : ℓ(x)=1}) which, under quasi-unipotency, yields a nef fixed vector, or by refining the averaging argument. With that clarification, the model’s proof delivers the same exponent as the paper. The paper’s argument is self-contained and complete, and its optimality discussion (e.g., Example 5.6) corroborates the sharpness of the bound .