It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric [6, 22, 38]. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page-Bérard-Bergery's ansatz [41, 14]. We also construct the interesting case of Hermitian metrics of zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the problem of the existence in a conformal class of critical metrics of the total Chern scalar curvature, studied by Gauduchon in [26, 27].

Let $\mathscr{A}$ be an abelian category with enough projective objects and $\mathscr{C}$ an additive and full subcategory of $\mathscr{A}$, and let $\mathscr{G}(\mathscr{C})$ be the Gorenstein category of $\mathscr{C}$. We study the properties of the $\mathscr{C}$-derived category $D_\mathscr{C}^b(\mathscr{A})$, $\mathscr{C}$-singularity category $D_{\mathscr{C}-sg}(\mathscr{A})$ and $\mathscr{G}(\mathscr{C})$-defect category $D_{\mathscr{G(C)}-defect}(\mathscr{A})$ of $\mathscr{A}$. Let $\mathscr{C}$ be admissible in $\mathscr{A}$. We show that $D_{\mathscr{G(C)}-defect}(\mathscr{A})\simeq D_{\mathscr{C}-sg}(\mathscr{A})$ if and only if $\mathscr{C}=\mathscr{G(C)}$; and $D_{\mathscr{G(C)}-defect}(\mathscr{A})=0$ if and only if the stable category $\underline{\mathscr{G}(\mathscr{C})}$ of $\mathscr{G}(\mathscr{C})$ is triangle-equivalent to $D_{\mathscr{C}-sg}(\mathscr{A})$, and if and only if every object in $\mathscr{A}$ has finite $\mathscr{C}$-proper $\mathscr{G}(\mathscr{C})$-dimension. Then we apply these results to module categories. We prove that under some condition, the Gorenstein derived equivalence of artin algebras induces the Gorenstein singularity equivalence. Finally, for an artin algebra $A$, we establish the stability of Gorenstein defect categories of $A$.

We introduce local cohomology complexes and local homology complexes with respect to a pair of ideals $(I,J)$ of a commutative noetherian ring $R$, characterize them using Čech complexes and establish "MGM" equivalence between the local cohomology functor $\mathrm{R}\Gamma_{I,J}$ and local homology functor $\mathrm{L}\Lambda_{I,J}$ on $\mathrm{D}(R)$.

We classify four-dimensional homogeneous manifolds satisfying some generalized Einstein conditions.

We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification of the mapping classes of hyperelliptic periodic diffeomorphisms. As an application, we obtain the Dehn twist presentations of hyperelliptic periodic mapping classes, which are closely related to the ones obtained by Ishizaka.

In this paper, we give some results on the algebraic dependence and finiteness of meromorphic mappings from $\mathbf{C}^m$ into $\mathbf{P}^n(\mathbf{C})$ sharing $2n+1$ hyperplanes in general position with truncated multiplicities to level $n$, where all zeros with multiplicities more than certain values do not need to be counted.

We apply the complex analysis over the double numbers $\mathbf{D}$ to study the minimal time-like surfaces in $\mathbf{R}^4_2$. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit special geometric (canonical) parameters. Then the geometry of the minimal time-like surfaces of general type is determined by the Gauss curvature $K$ and the curvature of the normal connection $\varkappa$, satisfying the system of natural equations for these surfaces. We prove the following: If $(K, \varkappa),\, K^2- \varkappa^2 > 0 $ is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the first type and exactly one minimal time-like surface of the second type with invariants $(K, \varkappa)$; if $(K, \varkappa),\, K^2- \varkappa^2$ < 0 is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the third type with invariants $(K, \varkappa)$.

We introduce the notion of pseudo-cones of metric spaces as a generalization of both of the tangent cones and the asymptotic cones. We prove that the Assouad dimension of a metric space is bounded from below by that of any pseudo-cone of it. We exhibit an example containing all compact metric spaces as pseudo-cones, and examples containing all proper length spaces as tangent cones or asymptotic cones.

In this article, we study the Heller relative $K_{0}$ group of the map $\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r}$, where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0}$, provided $X$ is flat over $S$. As a corollary, we get a description of the relative group $K_{0}(\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S$.