Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory

Hilbert-Mumford criterion for nodal curves

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a question raised by Mumford and Gieseker to prove the Chow asymptotic stability of stable nodal curves by Hilbert-Mumford criterion.

广义蒙日–安培方程的希尔伯特–芒福德判据

A Hilbert--Mumford criterion for generalised Monge--Ampère equations

本文提出一个符合GIT精神的数值判据，用Paul意义下的配对稳定性刻画逆黑塞方程（含J-方程）解的存在性，并证明广义Chow线丛与混合Deligne配对的等距性。

A study of the Hilbert-Mumford criterion for the stability of projective varieties

We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties $(X,L)$; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope $μ$ for varieties and their subschemes; if $(X,L)$ is semistable then $μ(Z)\leμ(X)$ for all $Z\subset X$. We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of th…

Scalar curvature and asymptotic Chow stability of projective bundles and blowups

The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme. These invariants are calculated in two special cases. The first is a projective bundle over a curve of genus at least two, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the underlying vector bundle is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarizati…

Facets of secondary polytopes and Chow stability of toric varieties

Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. Th…

nilsoliton（幂零孤立子）的Hilbert–Mumford准则

A Hilbert--Mumford criterion for nilsolitons

本文提出幂零实李代数的代数准则作为对应李群存在nilsoliton的充要条件，推广了Nikolayevsky准则并证明修正版Taketomi–Tamaru猜想给出存在性障碍。