Linear maps on nonnegative symmetric matrices preserving a given independence number

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace, and let $J_n$ be the $n\times n$ matrix with all entries equal to one. Given any integers $n,t$ with $2\le t\le n-1$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $\phi(J_n-I_n)=J_n-I_n$ and $$\alpha(\phi(X))=t~\iff \alpha(X)=t {~~~~\rm for~ all~}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=P^TXP{~~~~\rm for~ all~}X\in S_n^+.$

Flat λ\lambda-Connections, Mochizuki Correspondence and Twistor Spaces

In this paper, we first collect some basic results for $\lambda$-flat bundles, and then get an estimate for the norm of $\lambda$-flat sections, which leads to some vanishing theorem. Mochizuki correspondence provides a homeomorphism between the moduli space of (poly-)stable $\lambda$-flat bundles and that of (poly-)stable Higgs bundles, and provides a dynamical system on the later moduli space (the Dolbeault moduli space). We investigate such dynamical system, in particular, we discuss the corresponding first variation and asymptotic behavior. We generalize the Deligne's twistor construction for any element $\gamma$ of the outer automorphism group of the fundamental group of Riemann surface to obtain the $\gamma$-twistor space, and we apply the twistor theory to study a Lagrangian submanifold of the de Rham moduli space. As an application, we prove a Torelli-type theorem for the twistor spaces, and meanwhile, we prove that the oper stratum in the oper stratification of the de Rham moduli space is the unique closed stratum of minimal dimension, which partially confirms a conjecture by Simpson.Comment: Simpson pointed out a mistake on the Moishezon property for the twistor space in the last version, we delete it and add a section on the study of oper stratification of the de Rham moduli space as an applicatio

The Hitchin--Kobayashi Correspondence for Quiver Bundles over Generalized K\"ahler Manifolds

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian--Einstein metric.Comment: To appear in The Journal of Geometric Analysi