Let X C P^NC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either m〉n/2 and X is a complete intersection or that m≥ N2.We show deg（X）｜deg（Y）and ...Let X C P^NC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either m〉n/2 and X is a complete intersection or that m≥ N2.We show deg（X）｜deg（Y）and codim Y Y ≥codimPN X,where Y is the linear span of Y.These bounds are sharp.As an application,we classify smooth projective n-dimensional quadratic varieties swept out by m≥[n/2]＋1 dimensional quadrics passing through one point.展开更多
We introduce an algebraicity criterion.It has the following form:Consider an analytic subvariety of some algebraic variety X over a global field K.Under certain conditions,if X contains many K-points,then X is algebra...We introduce an algebraicity criterion.It has the following form:Consider an analytic subvariety of some algebraic variety X over a global field K.Under certain conditions,if X contains many K-points,then X is algebraic over K.This gives a way to show the transcendence of points via the transcendence of analytic subvarieties.Such a situ-ation often appears when we have a dynamical system,because we can often produce infinitely many points from one point via iterates.Combining this criterion and the study of invariant subvarieties,we get some results on the transcendence in arithmetic dynamics.We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic.For this,we study the invariant subvarieties for products of endomorphisms.In particular,we partially generalize Medvedev-Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on(P^(1))^(N) in any characteristic.We also get some high dimensional partial generalization via introducing a notion of independence.We then study dominant endomorphisms f on A^(N) over a number field of algebraic degree d≥2.We show that in most cases(e.g.when such an endomor-phism extends to an endomorphism on P^(N)),there are many analytic curves centered at infinity which are periodic.We show that for most of them,it is algebraic if and only if it contains at least one algebraic point.We also study the periodic curves.We show that for most f,all periodic curves have degree at most 2.When N=2,we get a more precise classification result.We show that under a condition which is satisfied for a general f,if f has infinitely many periodic curves,then f is homogenous up to change of origin.展开更多
In this paper,we give a suficient condition under which an involution monoid generates a variety with continuum many subvarieties.According to this result,several involution J-trivial monoids are shown to generate var...In this paper,we give a suficient condition under which an involution monoid generates a variety with continuum many subvarieties.According to this result,several involution J-trivial monoids are shown to generate varieties with continuum many subvarieties.These examples include Rees quotients of free involution monoids,Lee monoids with involution,and Straubing monoids with involution.展开更多
In this paper, the author extends Peter Li and Tian Gang’s results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and prove...In this paper, the author extends Peter Li and Tian Gang’s results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and proves several properties. Moreover, the results are extended to vector bundles. The author also gets an upper bound of the heat operators of some Schrondinger type operators on vector bundles. As a corollary, an upper bound of the trace of the heat operators is obtained.展开更多
Some theorems concerning the projective normality and extrinsic geometry on the smooth variety are proved under some supposed conditions related to the arithmetic genus of the smooth variety.
文摘Let X C P^NC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either m〉n/2 and X is a complete intersection or that m≥ N2.We show deg（X）｜deg（Y）and codim Y Y ≥codimPN X,where Y is the linear span of Y.These bounds are sharp.As an application,we classify smooth projective n-dimensional quadratic varieties swept out by m≥[n/2]＋1 dimensional quadrics passing through one point.
文摘We introduce an algebraicity criterion.It has the following form:Consider an analytic subvariety of some algebraic variety X over a global field K.Under certain conditions,if X contains many K-points,then X is algebraic over K.This gives a way to show the transcendence of points via the transcendence of analytic subvarieties.Such a situ-ation often appears when we have a dynamical system,because we can often produce infinitely many points from one point via iterates.Combining this criterion and the study of invariant subvarieties,we get some results on the transcendence in arithmetic dynamics.We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic.For this,we study the invariant subvarieties for products of endomorphisms.In particular,we partially generalize Medvedev-Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on(P^(1))^(N) in any characteristic.We also get some high dimensional partial generalization via introducing a notion of independence.We then study dominant endomorphisms f on A^(N) over a number field of algebraic degree d≥2.We show that in most cases(e.g.when such an endomor-phism extends to an endomorphism on P^(N)),there are many analytic curves centered at infinity which are periodic.We show that for most of them,it is algebraic if and only if it contains at least one algebraic point.We also study the periodic curves.We show that for most f,all periodic curves have degree at most 2.When N=2,we get a more precise classification result.We show that under a condition which is satisfied for a general f,if f has infinitely many periodic curves,then f is homogenous up to change of origin.
基金supported by the National Natural Science Foundation of China(Nos.12271224,12171213,11771191)the Fundamental Research Funds for the Central Universities(No.lzujbky-2023-ey06)the Natural Science Foundation of Gansu Province(No.20JR5RA275).
文摘In this paper,we give a suficient condition under which an involution monoid generates a variety with continuum many subvarieties.According to this result,several involution J-trivial monoids are shown to generate varieties with continuum many subvarieties.These examples include Rees quotients of free involution monoids,Lee monoids with involution,and Straubing monoids with involution.
文摘In this paper, the author extends Peter Li and Tian Gang’s results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and proves several properties. Moreover, the results are extended to vector bundles. The author also gets an upper bound of the heat operators of some Schrondinger type operators on vector bundles. As a corollary, an upper bound of the trace of the heat operators is obtained.
文摘Some theorems concerning the projective normality and extrinsic geometry on the smooth variety are proved under some supposed conditions related to the arithmetic genus of the smooth variety.
