In this paper, we observe several properties of an arithmetic divisor $\overline{D}$ on ${\mathbb{P}}_{\mathbb{Z}}^n$ and give the exact form of the Zariski decomposition of $\overline{D}$ on ${\mathbb{P}}_{\mathbb{Z}}^1$. Further, we show that, if $n\ge 2$ and $\overline{D}$ is big and non-nef, then for any birational morphism $f:X\to {\mathbb{P}}_{\mathbb{Z}}^n$ of projective, generically smooth, and normal arithmetic varieties, we cannot expect a suitable Zariski decomposition of $f^{\ast }\left(\overline{D}\right)$. We also give a concrete construction of Fujita’s approximation of $\overline{D}$.

Based on a variational representation for functionals of a general Poisson random measure plus an independent infinite-dimensional Brownian motion developed by Budhiraja, Dupuis, and Maroulas, the Freidlin-Wentzell large deviation principle is established for multivalued stochastic differential equations with Poisson jumps in this paper.

We construct toroidal partial compactifications of $p$-adic period domains.

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. This extends a previous result of Castelnuovo and Enriques. We classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo’s theorem. Then we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.

We discuss semigroups that preserve a convex set in a Banach space or a Hilbert space. We give sufficient conditions for which a semigroup preserves a convex set. Using this, we show that various issues can be treated in a unified way. We also discuss the problem in the Hilbert space setting, in which we use the sesquilinear form associated with a semigroup.

Let $H=X\otimes Y$ be a tensor product of separable Hilbert spaces $X$ and $Y$. We establish norm estimates for the resolvent and operator-valued functions of the operator $A={\sum }_{k=0}^m{B}_k\otimes S^k$, where $B_k$ $(k=0,\dots ,m)$ are bounded operators acting in $Y$, and $S$ is a self-adjoint operator acting in $X$. By these estimates we investigate spectrum perturbations of $A$. The abstract results are applied to the nonself-adjoint differential operators in Hilbert and Euclidean spaces. Our main tool is a combined use of some properties of operators on tensor products of Hilbert spaces and the recent estimates for the norm of the resolvent of a nonself-adjoint operator.

In this paper, we generalize construction of Seidel’s long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau manifolds. We use the framework of anchored Lagrangian submanifolds and some compactness theorem of smooth $J$-holomorphic sections of Lefschetz Hamiltonian fibration for a generic choice of $J$. The proof of the latter compactness theorem involves a study of proper pseudoholomorphic curves in the setting of noncompact symplectic manifolds with cylindrical ends.