In this paper,we study the K-stability theory of nonlinear delay systems.In the more general case,we establish two nonlinear delay differential inequalities.Therefore,to study the X-stability,a powerful method is prov...In this paper,we study the K-stability theory of nonlinear delay systems.In the more general case,we establish two nonlinear delay differential inequalities.Therefore,to study the X-stability,a powerful method is provided.By making use of the foregoing inequalities,we analyse and investigate some K-stabiiity conditions of nonlinear delay systems.Finally,some examples are given to illustrate our theory.展开更多
In this paper, we first introduce the concept of k-globally asymptotic stability and present a differential-difference inequality with infinite delay. By combining nonlinear inequality and nonlinear variation-of-param...In this paper, we first introduce the concept of k-globally asymptotic stability and present a differential-difference inequality with infinite delay. By combining nonlinear inequality and nonlinear variation-of-parameters formula, we derive the k-globally asymptotic stability criteria for nonlinear neutral system with infinite delay. In the end of this paper, an example is given to illustrate our theory.展开更多
Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariant...Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariants of their equivariant normal R-test configurations in terms of the combinatory data.Based onHan and Li“Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties”,we compute the semistable limit of aK-unstable FanoG-compactification.As an application,we show that for the two smooth K-unstable Fano SO4(C)-compactifications,the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.展开更多
Given a klt singularity x∈(X,D),we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized vol-ume function vol_((X,D),x),if and only if v induces a de...Given a klt singularity x∈(X,D),we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized vol-ume function vol_((X,D),x),if and only if v induces a degeneration to a K-semistable log Fano cone singularity.Moreover,such a minimizer is unique among all quasi-mono-mial valuations up to rescaling.As a consequence,we prove that for a klt singular-ity x∈X on the Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds,the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of x∈X,hence confirming a conjecture by Donaldson-Sun.展开更多
We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties w...We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties with discrete automorphism groups.We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments.This perturbation approach depends on the valuative criterion and non-Archimedean estimates,and is motivated by our previous paper.展开更多
基金Project supported by the National Natural Science Foundation of China
文摘In this paper,we study the K-stability theory of nonlinear delay systems.In the more general case,we establish two nonlinear delay differential inequalities.Therefore,to study the X-stability,a powerful method is provided.By making use of the foregoing inequalities,we analyse and investigate some K-stabiiity conditions of nonlinear delay systems.Finally,some examples are given to illustrate our theory.
基金the National Natural Science Foundation of China (10461006)the Younger Foundation of Yantai University (SX06Z9)
文摘In this paper, we first introduce the concept of k-globally asymptotic stability and present a differential-difference inequality with infinite delay. By combining nonlinear inequality and nonlinear variation-of-parameters formula, we derive the k-globally asymptotic stability criteria for nonlinear neutral system with infinite delay. In the end of this paper, an example is given to illustrate our theory.
文摘Let G be a connected,complex reductive group.In this paper,we classify G×G equivariant normal R-test configurations of a polarized G-compactification.Then,for Q-Fano G-compactifications,we express the H-invariants of their equivariant normal R-test configurations in terms of the combinatory data.Based onHan and Li“Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties”,we compute the semistable limit of aK-unstable FanoG-compactification.As an application,we show that for the two smooth K-unstable Fano SO4(C)-compactifications,the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.
文摘Given a klt singularity x∈(X,D),we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized vol-ume function vol_((X,D),x),if and only if v induces a degeneration to a K-semistable log Fano cone singularity.Moreover,such a minimizer is unique among all quasi-mono-mial valuations up to rescaling.As a consequence,we prove that for a klt singular-ity x∈X on the Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds,the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of x∈X,hence confirming a conjecture by Donaldson-Sun.
基金supported by NSF(Grant No.DMS-1810867)research fellowship.G.Tian is partially supported by NSF(Grant No.DMS-1607091)and NSFC(Grant No.11331001)partially supported by NSFC(Grant No.11501501).
文摘We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties with discrete automorphism groups.We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments.This perturbation approach depends on the valuative criterion and non-Archimedean estimates,and is motivated by our previous paper.
