No abstract

We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.

No abstract

Given an ideal I and a weight vector w which partially orders monomials we can consider the initial ideal in w (I ) which has the same Hilbert function. A well known construction carries this out via a one-parameter subgroup of a GL n+1 which can then be viewed as a curve on the corresponding Hilbert scheme. Galligo [A. Galligo, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier (Grenoble) 29 (2) (1979) 107-184, vii] proved that if I is in generic coordinates, and if w induces a monomial order up to a large enough degree, then in w (I ) is fixed by the action of the Borel subgroup of upper-triangular matrices. We prove that the direction the path approaches this Borel-fixed point on the Hilbert scheme is also Borel-fixed.

We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.