Wentao Chang | Searcharxiv

arXiv2026

Sharp Bounds for Totally Invariant Cycles of Projective Varieties

Let $X$ be a smooth projective variety, $f:X\to X$ an int-amplified endomorphism, and $L$ any ample line bundle on $X$. We prove that, in every codimension, the total degree of prime cycles that become totally invariant under an iterate of $f$ satisfies an explicit Hilbert-function bound on their total $L$-degree. In particular, when $X=\mathbf{P}^n$, the number of totally invariant prime $(n-r)$-cycle is bounded by $\binom{n+1}{r}$, and this bound is optimal.

math.AG↗

arXiv2025

Log Calabi-Yau structure of algebaic varieties admitting a polarized endomorphism

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. Then Broustet and Gongyo proposed the conjecture that $X$ is of Calabi-Yau type (CY for short), i.e., $(X,Δ)$ is lc for some effective $\mathbb{Q}$-divisor $Δ$ and $K_X+Δ\sim_{\mathbb{Q}}0$. We prove the conjecture when $X$ is a Gorenstein terminal 3-fold, extending the result of Sheng Meng for smooth threefolds. We then study the singularity type and CY property for $(X,Δ+\frac{R_Δ}{q-1})$ when $(X,Δ)$ is an $f$-pair, i.e., $K_X+Δ=f^*(K_X+Δ)+R_Δ$ with $Δ, R_Δ$ being effective. In particular, we show: (1) $K_X + \frac{R_f}{q-1}$ is $\mathbb{Q}$-Cartier and numerically trivial when $X$ is a $\mathbb{Q}$-factorial (or of klt type) $3$-fold; (2) $(X, \frac{R_{f}}{q-1})$ is log Calabi-Yau when $X$ is a surface with the Picard number $ρ(X)>1$ or $f^{-s}(P)=P$ for some prime divisor $P$ and $s>0$.

math.AG↗