Related papers: A view on contractions of higher dimensional varie…

In this paper we study a local structure of extrmal contractions $f\colon X\to S$ from threffolds $X$ with only terminal singularities onto a surface $S$. If the surface $S$ is non-singular and $X$ has a unique non-Gorenstein point on a…

Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f…

We consider birational projective contractions f:X -> Y from a smooth symplectic variety X over the complex numbers. We first show that exceptional rational curves on X deform in a family of dimension at least 2n-2. Then we show that these…

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and…

Let $f\colon X\to S$ be an extremal contraction from a threefold with only terminal singularities to a surface. We study local analytic structure such contractions near degenerate fiber $C$ in the case when $C$ is irreducible and $X$ has on…

Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold $X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is $\f$-ample then $\f$ is called a good contraction. In the present paper we study good…

Let $G$ be a connected algebraic group. We study $G$-equivariant extremal contractions whose centre is a codimension three $G$-simply connected orbit. In the spirit of an important result by Kawakita in 2001, we prove that those…

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q >…

Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (resp.~quasi-amplified) if $f^*D-D$ is ample (resp.~big) for some Cartier divisor $D$. We show that after…

We consider $K_X$-negative extremal contractions $f\colon X\to (Z,o)$, where $X$ is an algebraic threefold with only $\epsilon$-log terminal Q-factorial singularities and $(Z,o)$ is a two (resp., one)-dimensional germ. The main result is…

Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the…

An abelian cover is a finite morphism $X\to Y$ of varieties which is the quotient map for a generically faithful action of a finite abelian group $G$. Abelian covers with $Y$ smooth and $X$ normal were studied in…

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 real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$.…

Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the…

In this paper we study singularities defined by the action of Frobenius in characteristic $p > 0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein…

Let C be a smooth curve on an index 1 terminal 3-fold. We investigate the existence of extremal terminal divisorial contractions Y-->X that contract an irreducible surface E to C. We consider cases in respect to the singularities of the…

An extremal curve germ is a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve~$C$ such that there exists a $K_X$-negative contraction $f : X \to Z$ with~$C$ being a fiber. We give a rough…