Kang Zuo

Base Spaces of Non-Isotrivial Families of Smooth Minimal Models

Given a polynomial h of degree n let M h be the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h. By [23] there exist a quasi-projective scheme M h together with a natural transformation n n

The cohomology of a variation of polarized Hodge structures over a quasi-compact Kähler manifold

Psi :mathcal{M}_h to Hom(_,M_h )

Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves

n nsuch that M h is a coarse moduli scheme for M h . For a complex quasi-projective manifold U we will say that a morphism ϕ U → M h factors through the moduli stack, or that ϕ is induced by a family, if ϕ lies in the image of Ψ(U), hence if ϕ = Ψ(ƒ: V → U).

The Oort Conjecture for Shimura Curves of Small Unitary Rank: Dedicated to celebrate the Sixtieth anniversary of USTC

Let Md,n be the moduli stack of hypersurfaces X ⊂ P n of degree d ≥ n +1 , and letM (1) be the sub-stack, parameterizing hypersurfaces obtained as a d-fold cyclic covering of P n�1 ramified over a hypersurface of degree d. Iterating this construction, one obtains M (ν)

Vanishing theorems for

A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kahler manifolds. Several isomorphisms between different cohomologies given.

L^2

Let S be a compact Riemann surface (holomorphic curve) of genus g. Let p1; p2; . . . ; ps be s > 0 points on it; these points define a divisor, and we denote the open Riemann surface Snfp1; . . . ; psg by S. When 3g 3 þ s > 0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points p1; p2; . . . ; ps then become cusps at infinity. Even in the remaining cases, that is, for a once or twice punctured sphere, we can equip S with a metric that is hyperbolic in the vicinity of the cusp(s), and for our purposes, the behavior of the metric there is all what counts, and we call such a metric Poincaré-like. In any case, our metric on S is denoted by o. Denote the inclusion map of S in S by j. Let r : p1ðSÞ ! Glðn;CÞ be a semisimple linear representation of p1ðSÞ which is unipotent near the cusps (for the precise definition, cf. §2.4). Corresponding to such a representation r, one has a local system Lr over S and a r-equivariant harmonic map h : S ! Glðn;CÞ=UðnÞ with a certain special growth condition near the divisor, which is especially of finite energy (for details, see §2.4). For the present case of complex dimension 1, this is actually elementary; it also follows from the general result of [11]. We also remark that if not imposing any growth condition, r-equivariant harmonic maps are not unique in general, even of infinite energy (cf. [13]). In order to see explicitly the behavior at the cusps of the harmonic map h used in this note, we explicitly construct an initial map, which has the required asymptotic behavior, and show why it has finite energy. Then the existence of the harmonic map h and its behavior are obtained in a standard manner presently. It should be pointed out that the construction here is essentially the same as that in [11]; but the target manifolds of [11] are very general, so the construction there was getting very complicated. The idea of the explicit construction here will also be used in the case of higher dimension. The harmonic map h obtained above can be considered as a Hermitian metric on Lr—harmonic metric—so that we have a so-called harmonic bundle ðLr; hÞ [23]. Such a bundle carries interesting structures, e.g. a Higgs bundle structure ðE; yÞ, where y 1⁄4 qh, and it has a log-singularity at the divisor.

-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry

This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of genus g>1 can not contain compact Shimura curves, and that it only contains a non-compact Shimura curve for g=3.

On the negativity of kernels of Kodaira–Spencer maps on Hodge bundles and applications

We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.