Transcendental submanifolds of RP^n

Lagrangian submanifolds in Hyperkahler manifolds, Legendre transformation

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. We give a Plucker type formula for Lagrangian intersections under thi…

Ample subvarieties and rationally connected fibrations

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibr…

关于实代数簇的具有非奇异实轨迹的子簇

On the subvarieties with nonsingular real loci of a real algebraic variety

针对光滑射影实代数簇，研究了用子簇实轨迹逼近实流形及由非奇异或空实轨迹子簇生成的Chow群子群问题，给出了新的正反结果。

Local monomialization of transcendental extensions

Suppose that f is a dominant morphism from a k-variety X to a k-variety Y, where k is a field of characteristic 0 and v is a valuation of the function field k(X). We allow v to be an arbitary valuation, so it may not be discrete. We prove that there exist sequences of blowups of nonsingular subvarieties from X' to X and from Y' to Y such that X', Y' are nonsingular and X' to Y' is locally a monomial mapping near the center of v. This extends an earlier result of ours (in Asterisque 260) which proves the above result with the restriction that f is generically finite.

Projections from Subvarieties

Let $X\subset P^N$ be an n-dimensional connected projective submanifold of projective space. Let $p : P^N\to P^{N-q-1}$ denote the projection from a linear $P^q\subset P^N$. Assuming that $X\not\subset P^q$ we have the induced rational mapping $ψ:=p_X: X\to P^{N-q-1}$. This article started as an attempt to understand the structure of this mapping when $ψ$ has a lower dimensional image. In this case of necessity we have $Y := X\cap P^q$ is nonempty. We have in this article studied a closely related question, which includes many special cases including the case when the center of the projection …

Kodaira Dimension of Subvarieties

In this article we study how the birational geometry of a normal projective variety $X$ is influenced by a normal subvariety $A \subset X.$ One of the most basic examples in this context is provided by the following situation. Let $f:X\to Y$ be a surjective holomorphic map with connected fibers between compact connected complex manifolds. It is well known that given a general fiber $A$ of $f$ we have $$ κ(X)\le κ(A)+\dim Y. $$ This article grew out of the realization that this result should be true with $\dim Y$ replaced by the codimension $\cod_X A$ for a pair $(X,A)$ consisting of a normal s…