International Journal of Mathematics

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group $G$. We first study the case when $G=\mathrm{\text{O}}(V)$ or $\mathrm{\text{Sp}}(V)$, interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich–Witten KdV tau function arise out of regular spectral curves and copies of the Brezin–Gross–Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve $(x^2-4)y^2=1$.

In this paper, we establish necessary and sufficient conditions for boundedness and compactness of weighted composition operators acting between Fock spaces ${{ℱ}}^{\infty }(ℂ)$ and ${{ℱ}}^p(ℂ)$. We also give complete descriptions of path connected components for the space of composition operators and the space of nonzero weighted composition operators in this context.

We will study certain boundary measures related to $m$-subharmonic functions on $m$-hyperconvex domains. These measures generalize the boundary measures studied by Wan and Wang (see [Complex Hessian operator and Lelong number for unbounded $m$-subharmonic functions, Potential. Anal. 44(1) (2016) 53–69]). For the case of plurisubharmonic functions ($m=n$) the boundary measure has been studied by Cegrell and Kemppe (see [Monge–Ampère boundary measures, Ann. Polon. Math. 96 (2009) 175–196]).

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${ℂ\mathbb{P}}^2\#6{\overline{ℂ\mathbb{P}}}^2$, ${ℂ\mathbb{P}}^2\#7{\overline{ℂ\mathbb{P}}}^2$, and $3{ℂ\mathbb{P}}^2\#k{\overline{ℂ\mathbb{P}}}^2$ for $k=16,17,18,19$ using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using $g$-nodal spherical deformations of certain singular fibers of genus $g\ge 3$ fibrations.