Some Problems Of K(?)hler Geometry On Toric Manifolds

Toric manifolds is an interesting and important kind of algebraic manifolds,its position in algebraic manifolds similar to rational in real numbers.For its algebra-ic geometry properties,we already know a lot,but for Kahler geometry,such as the problem of existence of constant scalar curvature Kahler metric,a little is known.The thesis discusses three topics about toric manifolds:long time behavior of J-flow,the a-invariant of general Kahler class and a necessary condition for Chow poly-stability.we also discussed related issues for general Kahler manifolds:whether the solvabili-ty of Donaldson's equation depends on the choice of reference metric in its class,the properness of K-energy.Donaldson's equation c??= ?^??n-1 is first considered in[D2],its solution is the zeros of moment map which be defined on infinite dimensional space.Sub-sequently,in Xiuxiong Chen[Ch2],it arises as the critical equation of J-functionals,and J-functionals is one term of K-energy,thus the solvability of equation is associ-ated with the propemess of K-energy.In Jian Song and Weinkove[SW1],through study the negative gradient flow of J-functional(J-flow),they prove that the equation is solvable if and only if a inequality holds which is similar to the assumption there exists a subsolution,but this condition is difficult to verify,in particular,we still do not know whether the solvability of equation just depends on the Kahler classes.For this problem,Szekelyhidi[Sz]from the perspective of geometric stability conjecture the solvability of equation is equivalent to a series of inequality which only involves the Kahler classes.The starting point of this thesis is try to prove this conjecture for toric manifolds.By the symmetry reduction,J-flow can be reduced to a quasi-linear parabolic equa-tion defined on polytope,the feature is degenerate and satisfies the Guillemin boundary condition that is difficult to handle.We find that the transit on map Ut between two moment maps induced by ??t and ? has many well properties,it is a diffeomorphism between two polytope and preserve the combinatorial structure,this allows us avoid the Guillemin boundary condition.Moreover the transform Aji= ?jkgik on T1,0 M in-duced by ??t and ? has the same characteristic polynomial with the tangent map of Ut,this makes many important quantities defined on M can be expressed by Ut,including the inequality proposed by Szekelyhidi.So the problem is transformed to the study of long time behavior of Ut,which satisfies a degenerate quasilinear parabolic system with divergence structure.Then we made a preliminary study of it,get a partial C0 estimate of DUt,we conjecture that DUt is uniformly bounded along time and Ut converges,and if the limit map is degenerate will violate the Szekelyhidi's inequality.For the general Kahler manifolds,we try to use continuation method to prove e-quation still have solution when a change to ?f.Found that when we change a,J-functional can be well controlled and when the equation is solvable,J-functional is proper.Similar to the results of Chen[Ch2]that is a sufficient condition ensure the properness of K-energy,we also get a sufficient condition,but no longer need c1(M)<0,and it involves the a-invariant.Then we applied it to a class of Del Pezzo surfaces,found a family of Kahler class in which K-energy is proper.Jian Song[So]use the asymptotic expansion of Bergmann kernel to compute the a-invariant of Fano toric manifolds,we use the log canonical threshold formula due to Demailly to get the formula of a-invariant of general Kahler class on toric manifolds.Ono[Ono]proved that if a polarized toric manifold is Chow semi-stable,then the Delzant polytope must satisfies a harsh barycenter identity.His proof mainly used the results of Gelfand etc.on the weight polytope of Chow form.For the polystable case,from the perspective of balanced embedding,combined with the work of Fine on balancing flow,we give a simple differential-geometry proof to Ono's identity.