On asymptotic vanishing behavior of local cohomology

Let $R$ be a standard graded algebra over a field $k$, with irrelevant maximal ideal $\text{fm}$, and $I$ a homogeneous $R$-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $\{\text{HH}i\text{fm}R\mathrm{/}{I}^n{\}}_{n\in \text{NN}}$ for $i<\dim R\mathrm{/}I$. We show that, when $\text{chara}k=0$, $R\mathrm{/}I$ is Cohen-Macaulay, and $I$ is a complete intersection locally on $\text{Spec}R\setminus \{\text{fm}\}$, the lowest degrees of the modules $\{\text{HH}i\text{fm}R\mathrm{/}{I}^n{\}}_{n\in \text{NN}}$ are bounded by a linear function whose slope is controlled by the generating degrees of the dual of $I\mathrm{/}{I}^2$. Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal or the field $k$, we show that the complexity of the sequence of lowest degrees is at most polynomial, provided they are finite. Our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals.