In this paper, we investigate the geometry of base complex manifold an effectively parametrized holomorphic family stable Higgs bundles over a fixed compact K\"{a}hler manifold. The starting point our study is Schumacher-Toma/Biswas-Schumacher's curvature formulas for Weil-Petersson-type metrics, in Sect. 2, give some applications their on geometric properties 3, calculate higher direct image b...

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a spinc Dirac operator twisted by a line bundle with curvature of a mixed sign. In t...

where (h) is the inverse of the matrix (hij), ∆M = ∑ i,j h ∂ij and Γ s tγ denote the Christoffel symbols of the Hermitian metric g on N . It follows from (1.1) that if u is holomorphic, then u must be harmonic. Thus, it is natural to ask under what circumstances a harmonic map is holomorphic or antiholomorphic. Under the assumption that both M and N are compact, Siu [31] demonstrated that if th...

Following [KNTY] we study a so-called bc-ghost system of zero conformal dimension from the viewpoint of [TUY] and [U2]. We show that the ghost vacua construction results in holomorphic line bundles with connections over holomorphic families of curves. We prove that the curvature of these connections are up to a scale the same as the curvature of the connections constructed in [TUY] and [U2]. We...

Let X be a compact complex manifold and L → X a positive holomorphic line bundle. Assume that Aut(X,L)/C× is discrete, where Aut(X,L) is the group of holomorphic automorphisms of the pair (X,L). Donaldson [D] has recently proved that if X admits a metric ω ∈ c1(L) of constant scalar curvature, then (X,L) is Hilbert-Mumford stable for k sufficiently large. Since Kähler-Einstein metrics have cons...

We obtain a vanishing theorem for the half-kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the half-kernel of a spinc Dirac operator twisted by a line bundle with curvature of a mixed ...

The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain YII(r, p;K) is obtained in this paper when the parameter K equals p 2 + 1 p+1 . The estimate of holomorphic sectional curvature under this metric is also given which intervenes between −2K and − 2K p and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalen...