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Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain generalization...

We study the residue current $R^f$ of Bochner-Martinelli type associated with a tuple $f=(f_1,\cdots ,f_m)$ of holomorphic germs at $0\in {\mathbf{C}}^n$, whose common zero set equals the origin. Our main results are a geometric description of $R^f$ in terms of the Rees valuations associated with the ideal $\left(f\right)$ generated by $f$ and a characterization of when the annihilator ideal of $R^f$ equals $\left(f\right)$.

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees ${\lambda }_k$ of $f$ satisfy ${\lambda }_k^2\>{\lambda }_{k-1}{\lambda }_{k+1}$. On the other hand, we give examples of monomial maps $f$, where this condition...