Related papers: On k-ampleness equivalence

Let $F$ be a totally real field in which $p$ is unramfied and let $S$ denote the integral model of the Hilbert modular variety with good reduction at $p$. Consider the usual automorphic line bundle $\mathcal{L}$ over $S$. On the generic…

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$…

We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat K\"ahler metric. Here the degree of an affine bundle means the negative of the self-intersection number of…

Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact K\"ahler submanifold of $X$…

Let $M$ be a closed manifold and $\alpha : \pi_1(M)\to U_n$ a representation. We give a purely $K$-theoretic description of the associated element $[\alpha]$ in the $K$-theory of $M$ with $\R/\Z$-coefficients. To that end, it is convenient…

Let $G$ be the group scheme $\operatorname{SL}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $\alpha_{2},\cdots,\alpha_{d-1}$. Then $G/Q$ is the $\mathbb{Z} $-scheme of partial flags…

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface…

If the log canonical divisor on a projective variety with only Kawamata log terminal singularities is numerically equivalent to some semi-ample $\mathbf{Q}$-divisor, then it is semi-ample.

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of…

We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kahler space X: if L is a nef line bundle with numerical dimension at least equal to 2, then the q-th cohomology group of K_X+L vanishes for q at least equal to the dimension…

Let X be a projective hypersurface in P_k^n of degree d <= n. In this paper we study the relation between the class [X] in K_0(Var_k) and the existence of k-rational points. Using elementary geometric methods we show, for some particular X,…

Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X,…

Fujino gave a proof in [Fuj03] for the semi-ampleness of the moduli part in the canonical bundle formula in the case when the general fibers are K3 surfaces or Abelian varieties. We show a similar statement when the general fibers are…

This short note includes three results: $(1)$ If a reflexive sheaf $\mathcal{E}$ on a log terminal K\"{a}hler-Einstein variety $(X,\omega)$ is slope stable with respect to a singular K\"{a}hler-Einstein metric $\omega$, then $\mathcal{E}$…

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex…

Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds:…

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an…

Let $(\mathcal{E}, \phi)$ be a rank two co-Higgs vector bundles on a K\"ahler compact surface $X$ with $\phi\in H^0(X,End(\mathcal{E})\otimes T_X)$ nilpotent. If $(\mathcal{E}, \phi)$ is semi-stable, then one of the following holds up to…

We study the relation between semipositivity, nefness, and bigness of line bundles on compact K\"ahler manifolds. Every nef and big line bundle on a compact K\"ahler manifold $X$ is positive when ${\rm dim}\,X = 1$. Kim constructed an…

Chow's Theorem and GAGA are renowned results demonstrating the algebraic nature of projective manifolds and, more broadly, projective analytic varieties. However, determining if a particular manifold is projective is not, generally, a…