International Journal of Mathematics

For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder–Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder–Narasimhan stratification. In this paper, we show how to endow each Harder–Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder–Narasimhan filtration with a given Harder–Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder–Narasimhan type. The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder–Narasimhan type form an algebraic stack in the sense of Artin.

In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c² f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).

Let X be a smooth projective surface such that linear and numerical equivalence of divisors on X coincide and let σ ⊆ |D| be a linear pencil on X with integral general fibers. A fiber of σ will be called special if either it is not integral or it has nongeneric multiplicity at some of the base points (including the infinitely near ones) of the pencil. In this paper, we provide a procedure to compute the integral components of the special fibers of σ.

In this paper, we define a kind of KdV (Korteweg–de Vries) geometric flow for maps from a real line ℝ or a circle S¹ into a Kähler manifold (N, J, h) with complex structure J and metric h as the generalization of the vortex filament dynamics from a real line or a circle. By Hasimoto transformation, we find that the KdV geometric flow on a Riemann surface of constant Gauss curvature is just classical complex-valued mKdV equation. From the view point of geometric analysis we show that the Cauchy problems of KdV flow on a Kähler manifold admits a unique local solution in suitable Sobolev spaces. In the case the target manifold (N, J, h) with complex structure J and metric h is a certain type of locally Hermitian symmetric space, we show that the KdV flow exists globally by exploiting the conservation laws and semi-conservation law of KdV flow.

Let X be a projective manifold containing a quasi-line l. An important difference between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper, we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration on X that maps the general leaf of the foliation onto a quasi-line in a rational variety.