International Journal of Mathematics

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is shown that $x+yz$ is a moderate expander on $n\times n$ matrices with exponent $\frac{n+1}{6}$. These results generalize the main theorems in the recent work [29] of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory.

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme X = Y_red is smooth, irreducible, and that Y can be locally embedded in a smooth variety of dimension $dim(X)+1$. If n is the multiplicity of Y, there is a canonical filtration $X=X_1\subset X_2\subset \cdots \subset X_n=Y$, such that $X_i$ is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle $L^{\ast }$ by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that $Y=X_n$ is of multiplicity n, and can be extended to $X_{n+1}$ of multiplicity $n+1$, and let $M_n$ be a fine moduli space of vector bundles on $X_n$. With suitable hypotheses, we construct a fine moduli space $M_{n+1}$ for the vector bundles on $X_{n+1}$ whose restriction to $X_n$ belongs to $M_n$. It is an affine bundle over the subvariety $N_n\subset M_n$ of bundles that can be extended to $X_{n+1}$. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes Y such that the dualizing sheaf ${\omega }_Y$ is trivial.

In this paper, we consider the Boltzmann equation modeling the motion of a polyatomic gas where the integration collision operator in comparison with the classical one involves an additional internal energy variable $I\in {ℝ}_{+}$ and a parameter $\delta \ge 2$ standing for the number of internal degrees of freedom. In perturbation framework, we establish the global well-posedness for bounded mild solutions near global equilibria on torus. The proof is based on the $L^2\cap L^{\infty }$ approach. Precisely, we first study the $L^2$ decay property for the linearized equation, then use the iteration technique for the linear integral operator to get the linear weighted $L^{\infty }$ decay, and in the end obtain $L^{\infty }$ bounds as well as exponential time decay of solutions for the nonlinear problem with the help of the Duhamel’s principle. Throughout the proof, we present a careful analysis for treating the extra effect of internal energy variable I and the parameter $\delta$.

In this paper, we give a complete classification of Ricci–Bourguignon soliton on real hypersurfaces in the complex hyperbolic space $ℂH^n=S{U}_1,n/S(U_1{U}_n)$. Next, as an application, we give a complete classification of gradient Ricci–Bourguignon soliton on Hopf real hypersurfaces in the complex hyperbolic space $ℂH^n$.

We prove that if $(X,B,M)$ is a generalized klt pair with $K_X+B+M_X$ nef and abundant, then $K_X+B+M_X$ is semiample.

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of $t$-Gauduchon Ricci-flat metrics on the suspension of a compact Sasaki–Einstein manifold, for all $t\in (-\infty ,1)$; in particular, for the Bismut, Minimal and Hermitian conformal connection. A monotonicity theorem is obtained for the Gauduchon holomorphic sectional curvature, illustrating a maximality property for the Chern connection and furnishing insight into known phenomena concerning hyperbolicity and the existence of rational curves. Moreover, we show a rigidity result for Hermitian metrics which have a pair of Gauduchon holomorphic sectional curvatures that are equal, elucidating a duality implicit in the recent work of Chen–Nie.