Chow Stability Criterion in Terms of Log Canonical Threshold

Stability and Singularities of Relative Hypersurfaces

“…Using a result due to Lee [17], we can can prove via Theorem 1.3 a condition on the singularities of the effective divisors in |O P (k) ⊗ π * M −1 | with y/k ≥ µ (see Theorem 3.7); this condition involves the log canonical threshold (lct) of the couple (Σ, X |Σ ), where Σ is a general fibre of π.…”

“…If y/k > µ then any fibre of f is singular. We now use Theorem 3.2 combined with a result of Lee [17] relating the Chow stability of a variety with the log canonical threshold (lct) of its Chow variety.…”

Stability Conditions and Positivity of Invariants of Fibrations

“…Almost nothing is known about this conjecture in dimension higher than 2. In [8] we prove this inequality for families of hypersurfaces whose general fibres satisfy a very weak singularity condition expressed in terms of its log canonical threshold and depending upon the degree of the hypersurfaces (see [32]).…”

“…A hypersurface F ⊂ P r of degree d ≥ r + 2 and only log terminal singularities is Hilbert semistable [51]. • Higher codimensional varieties: Lee [32] proved that a subvariety F ⊂ P r of degree d is Chow semistable as far as the log canonical threshold of its Chow form is greater or equal to r+1 d (resp. > for stability).…”

Slopes of trigonal fibred surfaces and of higher dimensional fibrations

“…For instance it is not known if a "general" surface of general type satisfies it or not. The Hilbert stability of hypersurfaces with log-canonical singularities can be derived by a result of Kim and Lee [14] (see also [17]), which relates the stability of a hypersurface X ⊂ P n to the log-canonical threshold of the couple (P n , X ). The case of log-terminal singularities was established by Tian [30] using methods of differential geometry.…”