Related papers: Minimal model theorem for toric divisors

The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$\mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant…

In this paper, we prove the cone theorem and the contraction theorem for pairs $(X, B)$, where $X$ is a normal variety and $B$ is an effective $\mathbb R$-divisor on $X$ such that $K_X+B$ is $\mathbb R$-Cartier.

We add further notions to Lehmann's list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to…

Given $d\in \mathbb{Z}_{\geq 2}$, for every $\kappa=(k_1,\dots,k_n) \in \mathbb{Z}^{n}$ such that $k_i\geq 1-d$ and $k_1+\dots+k_n=-2d$, denote by $\Omega^d\mathcal{M}_{0,n}(\kappa)$ and $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$ the…

We prove the following result of Bondal's: that there is a fully faithful embedding $\kappa$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this…

Let $M$ be a $2$-space form. Let $P$ be a convex polygon in $M$. For these polygons, we define (and justify) a curvature $\kappa_i$ at each vertex $A_i$ of the polygon and and prove the following Blaschke's type theorem: If $P$ is a convex…

Let $X$ be an irreducible algebraic variety over $\mathbb{C}$, endowed with an algebraic foliation ${\cal{F}}$. In this paper, we introduce the notion of minimal invariant variety $V({\cal{F}},Y)$ with respect to $({\cal{F}},Y)$, where $Y$…

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci…

We propose a conjectural theory of $p$-integral models of Shimura varieties with level structure at $p$ given by a class of normal subgroups of parahoric subgroups with abelian quotient group. The role of the theory of local models is…

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating…

We compute the successive minima of the projective toric variety $X_\cA$ associated to a finite set $ \cA \subset \Z^n$. As a consequence of this computation and of the results of S.-W. Zhang on the distribution of small points, we derive…

We complete the proof of the Nisnevich conjecture in equal characteristic: for a smooth algebraic variety $X$ over a field $k$, a $k$-smooth divisor $D \subset X$, and a reductive $X$-group $G$ whose base change $G_D$ is totally isotropic,…

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the…

Given a minimal set of generators $\bold{x}$ of an ideal $I$ of height d in a regular local ring ($R, m, k$) we prove several cases for which the map $K_d(\bold{x}; R) \otimes k \to \Tor_d^R (R/I, k)$ is the 0-map. As a consequence of the…

We prove that the non-vanishing conjecture and the log minimal model conjecture for projective log canonical pairs can be reduced to the non-vanishing conjecture for smooth projective varieties such that the boundary divisor is zero.

We establish two consequences of the Kawamata--Morrison--Totaro cone conjecture, and prove them unconditionally in all dimensions. First, for a K-trivial variety, the natural action of its automorphism group on the set of ample divisor…

In this paper we will prove a uniformity result for the Iitaka fibration $f:X \rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $\kappa(X)=\rm{dim}(X)-1$.

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a klt pair $(X,\Delta)$ can be detected on the base of the $(K_{X}+\Delta)$-trivial reduction map. Thus we show that…

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite \'etale cover, admit a toric fibration over…

Let $f:X\to Y$ be a fibration from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p >5$. We prove that if the generic fiber $X_{\eta}$ has big canonical divisor…