Related papers: Extending rationally connected fibrations from amp…

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.

We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected…

Let $B$ be a simply-connected projective variety such that the first cohomology groups of all line bundles on $B$ are zero. Let $E$ be a vector bundle over $B$ and $X={\mathbb P} (E)$. It is easily seen that a power of any endomorphism of…

We discuss the local freeness and the numerical semipositivity of direct images of relative pluricanonical bundles for surjective morphisms between smooth projective varieties with connected fibers. We give a desirable semipositivity…

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it…

We continue to study birational geometry of Fano fibrations $\pi\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of…

We prove the irredcibility (and the rational connectedness) of the moduli spaces of (free) morphisms from a projective line to a successive blowing-up of a product of projective spaces if a suitable numerical condition on morphisms is…

We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem…

We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to…

We study fibered partially hyperbolic diffeomorphisms. We show that as long as certain topological obstructions vanish and as long as homological minimum expansion dominates the distortion on the fibers that a fibered partially hyperbolic…

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity…

Let $X$ be a smooth algebraic variety over $k$. We prove that any flat quasicoherent sheaf on $\operatorname{Ran}(X)$ canonically acquires a D-module structure. In addition, we prove that, if the geometric fiber $X_{\overline{k}}$ is…

The Hodge conjecture is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric…

In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that: 1. the Albanese map…

In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there…

Let $\mathcal{S}$ be a small category, and suppose that we are given a full subcategory $\mathcal{U}$ such that every object of $\mathcal{S}$ can be embedded into some object of $\mathcal{U}$ in the same way as every quasi-projective…

We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of…