Related papers: A remark on generalized abundance for surfaces

Let $X$ be a smooth projective rationally connected threefold with nef anticanonical divisor. We give a classification for the case when $-K_X$ is not semi-ample.

Let $(X, \Delta)$ be a four-dimensional log variety that is projective over the field of complex numbers. Assume that $(X, \Delta)$ is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of "log…

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k length arithmetic progression and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result…

Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2…

We prove that if $(X,\Delta)$ is a threefold pair with mild singularities such that ${-}(K_X+\Delta)$ is nef, then the numerical class of ${-}(K_X+\Delta)$ is effective.

We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in…

A $k$-very ample line bundle L on a Del Pezzo Surface is numerically characterized. We find that the set of the exceptional curves and effective divisors with selfintersection zero, $\cal S$, plays a very important rule in checking the…

We develop a theory of $P$-trivial MMP whose each step is $P$-trivial for a given nef divisor $P$. As an application, we prove that, given a projective generalised klt pair $(X,B+M)$ with data $M'$ being just a nef $\mathbb{R}$-divisor, if…

We reduce Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension to a special case of the generalized abundance conjecture by establishing an Iitaka type inequality for Nakayama's numerical Kodaira dimension. Our proof…

In this paper we prove a generalization of a theorem of Schneider, which gives a criterion for a projective surface over the complex numbers to have an ample cotangent bundle. After reviewing different notions of positivity, we introduce a…

Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal…

Let $(X, \Delta)$ be a compact K\"ahler klt pair, where $K_X + \Delta$ is ample or numerically trivial, and $\Delta$ has standard coefficients. We show that if equality holds in the orbifold Miyaoka-Yau inequality for $(X, \Delta)$, then…

Let $f: X \to Z$ be a fibration from a normal projective variety $X$ of dimension $n$ onto a normal curve $Z$ over a perfect field of characteristic $p>2$. Let $(X, B)$ be a dlt pair such that the induced pair on a general fibre is log…

Let $X$ be a smooth complex projective variety of dimension three and let $L$ be an ample line bundle on $X$. In this paper, we provide a lower bound of the dimension of the global sections of $m(K_{X}+L)$ under the assumption that…

We prove that for a normal projective variety $X$ in characteristic 0, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\Cl(X)\to \Cl(Y)$ is an isomorphism for a general member $Y\in |L|$…

Fix $c\in (1,23/22)$. Let $\alpha$ and $\beta$ be two distinct non-zero real numbers with $|\alpha|\neq |\beta|$. It is shown that for any measure preserving system $(X,\mathcal{X},\mu,T)$ and any $f,g\in L^{\infty}(\mu)$, the limit…

The following generalization of a result of S. Nemirovski is proved: if $X$ is either a projective or a Stein manifold and $K\subset X$ is a compact sublevel set of a strictly plurisubharmonic function $\varphi$ defined in a neighborhood of…

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to $\det(E)$). This result is a higher-rank version of a theorem…

We show that for every morphism f between nonsingular hypersurfaces of dimension at least 3 and of general type in projective space, there is an everywhere defined endomorphism F of projective space that restricts to f. As a corollary, we…

The cotangent bundle of a non-uniruled projective manifold is generically nef, due to a theorem of Miyaoka. We show that the cotangent bundle is actually generically ample, if the manifold is of general type and study in detail the case of…