We consider the moment map m:PV_(n)→iu(n)for the action of GL(n)on V_(n)=■^(2)(C^(n))^(∗)■C^(n),and study the critical points of the functional Fn=||m||2:PV_(n)→R.Firstly,we prove that[μ]∈PV_(n) is a critical po...We consider the moment map m:PV_(n)→iu(n)for the action of GL(n)on V_(n)=■^(2)(C^(n))^(∗)■C^(n),and study the critical points of the functional Fn=||m||2:PV_(n)→R.Firstly,we prove that[μ]∈PV_(n) is a critical point if and only if Mμ=cμI+Dμfor some cμ∈R and Dμ∈Der(μ),where m([μ])=Mμ||μ||2.Then we show that any algebraμadmits a Nikolayevsky derivationϕμwhich is unique up to automorphism,and if moreover,[μ]is a critical point of Fn,thenϕμ=−1 cμDμ.Secondly,we characterize the maxima and minima of the functional Fn:A_(n)→R,where A_(n) denotes the projectivization of the algebraic varieties of all the n-dimensional associative algebras.Furthermore,for an arbitrary critical point[μ]of Fn:A_(n)→R,we obtain a description of the algebraic structure ofμ.Finally,we classify the critical points of Fn:A_(n)→R for n=2 and n=3,respectively.展开更多
Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,...Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,and so is the Cartan-Munzner polynomial F(x).This action is extended to a Hamiltonian action on C^(2 l).We give a new description of F(x)by the moment mapμ:C2 l→t^(*),where t≌o(m+1)is the Lie algebra of Spin(m+1).It also induces a Hamiltonian action on CP^(2 l-1).We consider the Gauss map g of N into the complex hyperquadric Q_(2 l-2)(C)■CP^(2 l-1),and show that g(N)lies in the zero level set of the moment map restricted to Q_(2 l-2)(C).展开更多
The study of the volume of big line bundles on a complex projective manifold M has been one of the main veins in the recent interest in the asymptotic properties of linear series. In this article, we consider an equiv...The study of the volume of big line bundles on a complex projective manifold M has been one of the main veins in the recent interest in the asymptotic properties of linear series. In this article, we consider an equivariant version of this problem, in the presence of a linear action of a reductive group on M.展开更多
We study the phase transition of Kähler Ricci-flat metrics on some open Calabi–Yau spaces with the help of the images of moment maps of natural torus actions on these spaces.
For any compact connected Lie group G,we study the Hamiltonian sum of two compact Hamiltonian group G-manifolds(X^(+),ω^(+),μ^(+))and(X^(−),ω^(−),μ^(−))along a common codimension 2 Hamiltonian submanifold Z with t...For any compact connected Lie group G,we study the Hamiltonian sum of two compact Hamiltonian group G-manifolds(X^(+),ω^(+),μ^(+))and(X^(−),ω^(−),μ^(−))along a common codimension 2 Hamiltonian submanifold Z with the opposite equivariant Euler classes of the normal bundles.We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds.We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of X±.展开更多
We study the long time behavior of J-flows on toric manifolds. By introducing the tran- sition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are ...We study the long time behavior of J-flows on toric manifolds. By introducing the tran- sition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are obtained.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12301032)Natural Science Foundation of Jiangsu Province(Grant No.BK20230803)the Fundamental Research Funds for the Central Universities(Grant No.4007012303)。
文摘We consider the moment map m:PV_(n)→iu(n)for the action of GL(n)on V_(n)=■^(2)(C^(n))^(∗)■C^(n),and study the critical points of the functional Fn=||m||2:PV_(n)→R.Firstly,we prove that[μ]∈PV_(n) is a critical point if and only if Mμ=cμI+Dμfor some cμ∈R and Dμ∈Der(μ),where m([μ])=Mμ||μ||2.Then we show that any algebraμadmits a Nikolayevsky derivationϕμwhich is unique up to automorphism,and if moreover,[μ]is a critical point of Fn,thenϕμ=−1 cμDμ.Secondly,we characterize the maxima and minima of the functional Fn:A_(n)→R,where A_(n) denotes the projectivization of the algebraic varieties of all the n-dimensional associative algebras.Furthermore,for an arbitrary critical point[μ]of Fn:A_(n)→R,we obtain a description of the algebraic structure ofμ.Finally,we classify the critical points of Fn:A_(n)→R for n=2 and n=3,respectively.
基金supported by Japan Society for the Promotion of Science(Grant No.15H03616)。
文摘Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,and so is the Cartan-Munzner polynomial F(x).This action is extended to a Hamiltonian action on C^(2 l).We give a new description of F(x)by the moment mapμ:C2 l→t^(*),where t≌o(m+1)is the Lie algebra of Spin(m+1).It also induces a Hamiltonian action on CP^(2 l-1).We consider the Gauss map g of N into the complex hyperquadric Q_(2 l-2)(C)■CP^(2 l-1),and show that g(N)lies in the zero level set of the moment map restricted to Q_(2 l-2)(C).
文摘The study of the volume of big line bundles on a complex projective manifold M has been one of the main veins in the recent interest in the asymptotic properties of linear series. In this article, we consider an equivariant version of this problem, in the presence of a linear action of a reductive group on M.
文摘We study the phase transition of Kähler Ricci-flat metrics on some open Calabi–Yau spaces with the help of the images of moment maps of natural torus actions on these spaces.
基金supported by National Natural Science Foundation of China(Grant No.12071322)the National Key R&D Program of China(Grant No.2020YFA0714000)+1 种基金the Sichuan Science and Technology Program(Grant No.2022JDTD0019)supported by the Guangdong Basic and Applied Basic Research Foundation of China(Grant No.2021A1515010379).
文摘For any compact connected Lie group G,we study the Hamiltonian sum of two compact Hamiltonian group G-manifolds(X^(+),ω^(+),μ^(+))and(X^(−),ω^(−),μ^(−))along a common codimension 2 Hamiltonian submanifold Z with the opposite equivariant Euler classes of the normal bundles.We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds.We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of X±.
基金Supported by National Natural Science Foundation of China(Grant No.11171143)
文摘We study the long time behavior of J-flows on toric manifolds. By introducing the tran- sition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are obtained.
