LetM be a compact, holomorphic symplectic Kähler manifold, and L a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of L is effective. This result is related to the hyperkähler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.

Let M be a complete noncompact Kähler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by Od(M) the space of holomorphic functions of polynomial growth of degree at most d on M. In this paper we prove that dimCOd(M) ≤ dimCO[d](C), for all d > 0, with equality for some positive integer d if and only if M is holomorphically isometric to C. We also obtain ...

In [Y], Yau proposed to study the uniformization of complete Kähler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete Kähler manifold M with positive bisectional curvature is biholomorphic to C. See also [GW], [Si]. For this sake, it was further asked in [Y] whether or not the ring of the holomorphic functions with polynomial growth, which we...

In a previous paper, [1], we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted L-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of L-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sect...

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...

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...