Using the degeneration formula for Donaldson-Thomas invariants [L-W,MNOP2], we proved formulae for blowing up a point, simple flops, and extremal transitions.

No abstract uploaded!

In this paper, we prove that for any Kähler metrics ω and χ on M, there exists a Kähler metric ω_φ = ω_0 + √-1 ∂ \bar∂ φ > 0 satisfying the J-equation tr_{ω_φ} χ = c if and only if (M, [ω0], [χ]) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kähler metrics with c_1 < 0. Using the same method, we also prove a similar result for the supercritical deformed Hermitian–Yang–Mills equation.

No abstract uploaded!

We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.