均匀 Turán 估计以及次数与混合轨道增长的尖锐界
Uniform Turán estimates and sharp bounds for degree and mixed orbit growth
- School of Mathematics, Nanjing University（南京大学数学系）
- Shanghai Center for Mathematical Sciences & School of Mathematical Sciences, Fudan University（上海数学中心与复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明射影自态射迭代次数的多项式修正指数受Hodge指标定理约束，得到相邻余维数间的耦合不等式及尖锐界，并推广到零熵自同构的混合轨道函数和Kähler流形上的算子范数增长。
AI中文摘要：
射影自态射迭代的次数以两层方式增长：由动力次数测量的指数速率，以及由外围 Jordan 块承载的多项式修正。对数凹性约束第一层；我们证明 Hodge 指标定理已经约束第二层，在所有余维数和任意特征下均成立。设 $f$ 是代数闭域上正规射影 $d$ 维簇的满射自态射，其动力次数为 $\lambda_i$，使得 $\deg_i(f^n)\asymp\lambda_i^n n^{\mu_i}$，其中 $\mu_i\ge0$ 为唯一整数。我们证明这些指数在三个相邻余维数之间耦合：$\lambda_i$ 在 $i$ 处的严格对数凹性迫使 $\mu_i=0$，而等式给出 \\[ 0\\ \le\\ 2\mu_i-\mu_{i-1}-\mu_{i+1}\\ \le\\ 4. \\] 因此，$\mu_i\le2i(d-i)$，并且对每个 $i$ 这是尖锐的。对于零熵自同构，我们证明由 $d$ 的任意弱组合 $\gamma$ 索引的每个混合轨道函数是总次数至多 $d^2-\sum_j\gamma_j^2$ 的偶次多元拟多项式。对于紧 Kähler $d$ 维簇 $Y$ 的零熵全纯自同构 $g$，相同方法给出 \\[\left\\|(g^n)^*|_{H^{p,q}(Y,\mathbb{C})}\right\\| =O\bigl(n^{p(d-p)+q(d-q)}\bigr).\\] 椭圆曲线的幂达到所有指数界以及常数 $4$。
英文摘要：
The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every codimension and in arbitrary characteristic. Let $f$ be a surjective endomorphism of a normal projective $d$-fold over an algebraically closed field, with dynamical degrees $λ_i$, so that $\textrm{deg}_i(f^n)\asympλ_i^n n^{μ_i}$ for a unique integer $μ_i\ge0$. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of $(λ_i)_i$ at $i$ forces $μ_i=0$, while equality gives \[ 0\ \le\ 2μ_i-μ_{i-1}-μ_{i+1}\ \le\ 4 . \] Consequently, $μ_i\le2i(d-i)$, and this is sharp for every $i$. For a zero-entropy automorphism, we prove that every mixed orbit function indexed by any weak composition $γ$ of $d$ is a multivariate quasipolynomial of even total degree at most $d^2-\sum_jγ_j^2$. For a zero-entropy holomorphic automorphism $g$ of a compact Kähler $d$-fold $Y$, the same method gives \[\left\|(g^n)^*|_{H^{p,q}(Y,\mathbb{C})}\right\| =O\bigl(n^{p(d-p)+q(d-q)}\bigr).\] Powers of elliptic curves attain all exponent bounds as well as the constant $4$.
