Yum-Tong Siu

Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance of Semipositively Twisted Plurigenera for Manifolds Not Necessarily of General Type

Let X be a holomorphic family of compact complex projective algebraic manifolds with fibers X t over the open unit 1-disk Δ. Let \( K_{x_t } \) and K X be respectively the canonical line bundles of X t and X. We prove that, if L is a holomorphic Une bundle over X with a (possibly singular) metric e −ϕ of semipositive curvature current on X such that e −ϕ|x 0 is locally integrable on X 0, then for any positive integer m, any s ∈ Γ(m \( K_{x_0 } \) + L) with |s|2 e −ϕ locally bounded on X 0 can be extended to an element of Γ (X, m K X + L). In particular, dim Γ (X t , m \( K_{x_t } \) + L) is independent of t for ϕ smooth. The case of trivial L gives the deformational invariance of the plurigenera. The method of proof uses an appropriately formulated effective version, with estimates, of the argument in the author’s earlier paper on the invariance of plurigenera for general type. A delicate point of the estimates involves the use of metrics as singular as possible for p \( K_{x_0 } \) + a p L on X 0 to make the dimension of the space of L 2 holomorphic sections over X 0 bounded independently of p, where a p is the smallest integer \( \geqslant \frac{{p - 1}} {m} \) . These metrics are constructed from s. More conventional metrics, independent of s, such as generalized Bergman kernels are not singular enough for the estimates.

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.

Hyperbolicity in Complex Geometry

A complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. We discuss the techniques and methods for the hyperbolicity problems for submanifolds and their complements in abelian varieties and the complex projective space. The discussion is focussed on Bloch’s techniques for the abelian variety setting, the recent confirmation of the longstanding conjecture of the hyperbolicity of generic hypersurfaces of high degree in the complex projective space, and McQuillan’s techniques for compact complex algebraic surfaces of general type.

Nonexistence of smooth Levi-∞at hypersurfaces in complex projective spaces of dimension‚ 3

In this paper we prove the following theorem. Main Theorem. Let n >= 3 and m >= 3n/2 +7. Then there exists no C^m Levi-flat real hypersurface M in P_n. The condition that M is Levi-flat means that when M is locally defined by the vanishing of a C^m real-valued function f, at every point of M the restriction of d d-bar f to the complex tangent space of M is identically zero. The case of the nonexistence of C^\infty Levi-flat real hypersurface in P_2 is motivated by problems in dynamical systems in P_2.

Strong Rigidity for Kähler Manifolds and the Construction of Bounded Holomorphic Functions

In this paper we shall give a survey of the known results and methods concerning the strong rigidity of Kahler manifolds and present some new related results. The important phenomenon of strong rigidity was discovered by Professor G.D. Mostow in the case of locally symmetric nonpositively curved Riemannian manifolds. He proved [18] that two compact locally symmetric nonpositively curved Riemannian manifolds are isometric up to normalization constants if they have the same fundamental group and neither one contains a closed one or two dimensional totally geodesic submanifold that is locally a direct factor. This last assumption is clearly necessary because of the existence of non-trivial holomorphic deformations of any compact Riemann surface of genus at least two. Mostow’s result says that if one can rule out the possibility of contribution to the change of metric structure from certain submanifolds of dimension two or lower, the metric structure is rigidly determined by the topology for compact locally symmetric nonpositively curved manifolds. Mostow’s result also holds for the non-compact complete case under the assumption of finite volume.

Strong rigidity of compact quotients of exceptional bounded symmetric domains

In this paper we complete the proof of the Main Theorem by doing the cases of the two exceptional bounded symmetric domains. The proof for the cases of the four types of classical bounded symmetric domains consists of using harmonic maps, deriving a Bochner type formula, and a tedious part involving complicated linear algebra manipulations to verify that the curvature conditions from the Bochner type formula are satisfied by the classical bounded symmetric domains. In this paper we reformulate the curvature conditions so that they can be verified directly from the root system without using any explicit expression of the curvature tensor in terms of the coordinates of a realization of the bounded symmetric domain (see 2 and 3). We prove the strong rigidity for the two exceptional cases by using this reformulation (see 4 and 5). This reformulation of the curvature condition, when applied to the cases of the first three classical types, can yield a proof of the strong rigidity for these cases which is simpler than that given in [5] (see the Remark in 4). (The strong rigidity proof given in [5] for the fourth classical type is already very simple.) We now know an alternative method of deriving the Bochner type formula for harmonic maps between compact Khler manifolds. This alternative method does not use the special trick of considering the wedge product instead of the square norm of the differential of a harmonic map. It is more natural and we give this alternative derivation in this paper (see 1). At the end of this paper ([}6) we discuss a conjecture concerning the analyticity of harmonic maps with sufficiently high rank into compact quotients of bounded symmetric domains. We would like to thank the referee for suggesting the present simple proof of Proposition (4.1) which replaces our original lengthy case-by-case verification and for suggesting the alternative proof of Proposition (5.1). In the meantime,

Multiplier ideal sheaves in complex and algebraic geometry

The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invariance of plurigenera for general compact algebraic manifolds is presented and discussed. Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.

All plane domains are Banach-Stein

AbstractIn this short note we show, by using the

Curvature characterization of hyperquadrics

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