Sai-Kee Yeung

Fake projective planes

A fake projective plane is a complex surface different from but has the same Betti numbers as the complex projective plane. It is a complex hyperbolic space form and has the smallest Euler Poincare characteristic among smooth surfaces of general type. The first example was constructed by Mumford. Later on two more examples were found by Ishida and Kato. A fourth possible one was recently proposed by Keum. In a recent joint work with Gopal Prasad, we showed that there are seventeen non-empty classes of fake projective planes and there can at most be four more specific classes. Higher dimensional analogues and examples were also obtained. The main purpose of the talk is to explain the joint work with Prasad and other related results such as arithmeticity of the lattices involved obtained earlier by Klingler and Yeung independently.

DEFECTS FOR AMPLE DIVISORS OF ABELIAN VARIETIES, SCHWARZ LEMMA, AND HYPERBOLIC HYPERSURFACES OF LOW DEGREES

We prove that the defect vanishes for a holomorphic map f from the affine complex line to an abelian variety A and for an ample divisor D in A. The proof uses the translational invariance of the Zariski closure of the k-jet space of the image of f and the theorem of Riemann Roch to construct a nonidentically zero meromorphic k-jet differential whose pole divisor is dominated by a divisor equivalent to pD and which vanishes along the k-jet space of D to order q with p/q smaller than a prescribed small positive number. Then estimates involving the theta function with divisor D and the logarithmic derivative lemma are used. We also prove a pointwise Schwarz lemma which gives the vanishing of the pullback, by a holomorphic map from the affine complex line to a compact complex manifold, of a holomorphic jet differential vanishing on an ample divisor. This pointwise Schwarz lemma is a slight modification of a statement whose proof Green and Griffiths sketched in their alternative treatment of Blochs theorem on entire curves in abelian varieties. The log-pole case of the pointwise Schwarz lemma is also given. We construct examples of hyperbolic hypersurface whose degree is only 16 times the square of its dimension.

ARITHMETIC FAKE PROJECTIVE SPACES AND ARITHMETIC FAKE GRASSMANNIANS

In a recent paper we have classified fake projective planes. Natural higher dimensional generalization of these surfaces are arithmetic fake

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

{\bf P}^{n-1}_{C}

Effective Isometric Embeddings for Certain Hermitian Holomorphic Line Bundles

, and arithmetic fake

EFFECTIVE ESTIMATES ON THE VERY AMPLENESS OF THE CANONICAL LINE BUNDLE OF LOCALLY HERMITIAN SYMMETRIC SPACES

{\bf Gr}_{m,n}

Geometric Realizations of Uniformization of Conjugates of Hermitian Locally Symmetric Manifolds

. In this paper we show that arithmetic fake

A surface of maximal canonical degree

{\bf P}^{n-1}_{C}

Compactification of complete Kähler surfaces with negative Ricci curvature

can exist only if

Classification of surfaces of general type with Euler number 3

n = 3,\, 5