Necessary and sufficient conditions for the new concepts of (h0, h)-Lipschitz (local) semistability and (h0,h)-Lipschitz (locally weak) semistability are given, using Liapunov-like functions in this paper.
The notions of (Lipschitz) semistability of general control systems are introduced, and the necessary and sufficient conditions for (weak) semistability, Lipschitz (locally weak) semistability are given, using the ve...The notions of (Lipschitz) semistability of general control systems are introduced, and the necessary and sufficient conditions for (weak) semistability, Lipschitz (locally weak) semistability are given, using the versatile tools, Liapunov-like functions.展开更多
Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle ...Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle on X, and ρ：GLn（k） → GLm（k） a rational GLn（k）-representation of degree at most d such that ρ maps the radical R（GLn（k）） of GLn（k） into the radical R（GLm（k）） of GLm（k）. We show that if FXN＊（E） is semistable for some integer N ≥ max0 〈 r 〈 m （rm） · logp（dr）, then the induced rational GLm（k）-bundle E（GLm（k）） is semistable. As an application, if dim X=n, we get a sufficient condition for the semistability of Frobenius direct image FX＊（ρ＊（ΩX1））, where ρ＊（ΩX1） is the vector bundle obtained from ΩX1 via the rational representation ρ.展开更多
Let Q be a quiver with automorphism σ. We prove that semistable representations of Q over F_q give rise to semistable modules over the F_q-algebra A(Q, σ; q) associated with(Q, σ). As an application, we obtain a de...Let Q be a quiver with automorphism σ. We prove that semistable representations of Q over F_q give rise to semistable modules over the F_q-algebra A(Q, σ; q) associated with(Q, σ). As an application, we obtain a description of the semistable subcategories of A(Q, σ; q)-modules and determine the slopes of semistable A(Q, σ; q)-modules in the case that Q is a Dynkin or tame quiver.展开更多
Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set w = （ d →（1）,..., d→ （m）） of sequences of positive integers, denote by Lω the ample line bundle corresponding t...Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set w = （ d →（1）,..., d→ （m）） of sequences of positive integers, denote by Lω the ample line bundle corresponding to the polarization on the product X = Пi=1 m Flag（V, →n（i）） of flag varieties of type n→（i） determined by ω. We study the SL（V）-linearization of the diagonal action of SL（V） on X with respect to Lω. We give a sufficient and necessary condition on ω such that X ss（Lω） ≠ 0 （resp., Xs（Lω） ≠ 0）. As a consequence, we characterize the SL（V）-ample cone （for the diagonal action of SL（V） on X）,which turns out to be a polyhedral convex cone.展开更多
文摘Necessary and sufficient conditions for the new concepts of (h0, h)-Lipschitz (local) semistability and (h0,h)-Lipschitz (locally weak) semistability are given, using Liapunov-like functions in this paper.
文摘The notions of (Lipschitz) semistability of general control systems are introduced, and the necessary and sufficient conditions for (weak) semistability, Lipschitz (locally weak) semistability are given, using the versatile tools, Liapunov-like functions.
基金Supported by National Natural Science Foundation of China(Grant No.11501418)Shanghai Sailing Program(Grant No.15YF1412500)
文摘Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle on X, and ρ：GLn（k） → GLm（k） a rational GLn（k）-representation of degree at most d such that ρ maps the radical R（GLn（k）） of GLn（k） into the radical R（GLm（k）） of GLm（k）. We show that if FXN＊（E） is semistable for some integer N ≥ max0 〈 r 〈 m （rm） · logp（dr）, then the induced rational GLm（k）-bundle E（GLm（k）） is semistable. As an application, if dim X=n, we get a sufficient condition for the semistability of Frobenius direct image FX＊（ρ＊（ΩX1））, where ρ＊（ΩX1） is the vector bundle obtained from ΩX1 via the rational representation ρ.
基金supported by National Natural Science Foundation of China(Grant No.11271043)
文摘Let Q be a quiver with automorphism σ. We prove that semistable representations of Q over F_q give rise to semistable modules over the F_q-algebra A(Q, σ; q) associated with(Q, σ). As an application, we obtain a description of the semistable subcategories of A(Q, σ; q)-modules and determine the slopes of semistable A(Q, σ; q)-modules in the case that Q is a Dynkin or tame quiver.
文摘Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set w = （ d →（1）,..., d→ （m）） of sequences of positive integers, denote by Lω the ample line bundle corresponding to the polarization on the product X = Пi=1 m Flag（V, →n（i）） of flag varieties of type n→（i） determined by ω. We study the SL（V）-linearization of the diagonal action of SL（V） on X with respect to Lω. We give a sufficient and necessary condition on ω such that X ss（Lω） ≠ 0 （resp., Xs（Lω） ≠ 0）. As a consequence, we characterize the SL（V）-ample cone （for the diagonal action of SL（V） on X）,which turns out to be a polyhedral convex cone.
