We show that positivity of the CM line associated to a family of polarised varieties is intimately related to the stability of its members. We prove that the CM line is nef on any curve which meets the stable locus, and that it is pseudoeffective (i.e. in the closure of the effective cone) as long as there is at least one stable fibre. We give examples showing that the CM line can be strictly negative or strictly positive on curves in the unstable locus.
We give a short, geometric proof of Graham's theorem on positivity in the equivariant cohomology of a flag variety, based on a transversality argument.
The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate--Shafarevich group and the Tate conjecture of surfaces over finite fields.
We give a criterion for a projectivized toric vector bundle to be a Mori dream space and describe its Cox ring using generators and relations. Both of these results are in terms of the matroids of all symmetric powers of the bundle. We also give a criterion for a toric vector bundle to be big and describe several interesting examples of toric vector bundles which highlights how positivity properties for toric vector bundles are more complicated than for toric line bundles.
We prove a generalization of the Fujita-Kawamata-Zuo semi-positivity Theorem for filtered regular meromorphic Higgs bundles and tame harmonic bundles. Our approach gives a new proof in the cases already considered by these authors. We give also an application to the geometry of smooth quasi-projective complex varieties admitting a semisimple complex local system with infinite monodromy.
We give the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano, and we prove that these morphisms are Fano conic bundles. We also review some known results about the interaction between the positivity properties of anticanonical divisors of varieties of contractions.
We study the $(k,s)$-positivity for holomorphic vector bundles on compact complex manifolds. $(0,s)$-positivity is exactly the Demailly $s$-positivity and a $(k,1)$-positive line bundle is just a $k$-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for $(k,s)$-positive vector bundles are proved and the vanishing theorems for $k$-ample vector bundles on projective algebraic manifolds are generalized to $k$-positive vector bundles on compact Kähler manifolds.
We give equivalent descriptions for the augmented and diminished base loci of vector bundles in characteristic zero. We show that these base loci behave well under pullback, tensor product, and direct sum. Pathological behavior is observed on some nonsplit exact sequences.
Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian metric" on vector bundles (together with an appropriate definition of positivity of the associated curvature) plays a major role here, and its properties are studied via a version of Ohsawa-Takegoshi extension theorem. Part of this article is based on the joint work of the first named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.
This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.
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 slightly weaker notion of ampleness, which we call quasi-ample, and then are able to extend Schneider's result to higher dimensions.
The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the l…
Let $X$ be a smooth projective variety of dimension $n$, and let $E$ be an ample vector bundle over $X$. We show that any non-zero Schur class of $E$, lying in the cohomology group of bidegree $(n-1, n-1)$, has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton-Lazarsfeld for certain degrees.
Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ general points $p_1,\ldots,p_r \in \mathbb{P}^2$. We study line bundles on $X$ given by plane curves of degree $d$ passing through $p_i$ with multiplicity $m_i$. We establish conditions for ampleness, very ampleness and global generation of such line bundles.
Shiv Nadar University（希夫纳达尔大学）；Indian Institute of Technology (Indian School of Mines) Dhanbad（印度技术学院（印度矿业学院）丹巴德）；Indraprastha Institute of Information Technology, Delhi（德里因德拉普拉斯塔信息技术学院）
Let $C \subset \mathbb{P}^2$ be an irreducible and reduced curve of degree $e > 0$. Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ distinct smooth points $p_1,\ldots,p_r \in C$. We study line bundles on $X$ and establish conditions for ampleness and $k$-very ampleness.
We prove that the normalized Poincaré bundle on the moduli space of stable rank $r$ vector bundles with a fixed determinant on a smooth projective curve $X$ induces a family of nef vector bundles on the moduli space. Two applications follow. We show that when the genus of $X$ is large, the derived category of $X$ is embedded into the derived category of the moduli space for arbitrary rank and coprime degree, which extends the results of Narasimhan, Fonarev-Kuznetsov, and Belmans-Mukhopadhyay. As the second application, we construct a family of ACM bundles on the moduli space. A key ingredient …
We give an algebraic proof valid in arbitrary characteristic for the known equivalence between (strongly) slope semistable vector bundles with vanishing discriminant and vanishing determinant and numerically flat bundles. We also address a related question of Misra.
We prove the bigness of the Chow-Mumford line bundle associated to a $\mathbb{Q}$-Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a recent theorem of Codogni and Patakfalvi to the logarithmic setting.
We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs.
We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point, and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles M_L that arise as the kernel of the evaluation map on sections of L, when L is an ample line bundle. We give examples of twi…
In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to a lc pair is big as soon as there exists a generically injective morphism from an ample line bundle to some symmetric power of the cotangent bundle associated to the orbifold pair.
We study the relationship between positivity of line bundles restricted to complete intersection subvarieties and the vanishing of higher cohomology groups. Based on this connection we prove generalizations of the vanishing theorems of Kawamata-Viehweg and Fujita that are valid for not necessarily big line bundles.
We investigate the interaction between the existence of reproducing kernels on infinite-dimensional Hermitian vector bundles and the positivity properties of the corresponding bundles. The positivity refers to the curvature form of certain covariant derivatives associated to reproducing kernels on the vector bundles under consideration. The values of the curvature form are Hilbert space operators, and its positivity is thus understood in the usual sense from operator theory.
Let $X$ be a smooth complex projective variety. A recent conjecture of S. Kovács states that if t\ he $p^{\text{th}}$-exterior power of the tangent bundle $T_X$ contains the $p^{\text{th}}$-exterior power of an ample vector bundle, then $X$ is either a projective space or a smooth quadric hypersurface. This conjecture is appealing since it is a common generalization of Mori's, Wahl's, Andreatt\ a-Wísniewski's, and Araujo-Druel-Kovács's characterizations of these spaces. In this paper I give a proof affirming this conjecture for varieties with Picard number 1.
Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot π^*L_S - α\cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $π:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide a criterion for $k$-very ampleness of $M$. Also, we deal with the case when $S$ is an arbitrary surface of Picard number one with a numerically trivial canonical divisor.
For any non-negative integer $k$ the $k$-th osculating dimension at a given point $x$ of a variety $X$ embedded in projective space gives a measure of the local positivity of order $k$ at that point. In this paper we show that a smooth toric embedding having maximal $k$-th osculating dimension, but not maximal $(k+1)$-th osculating dimension, at every point is associated to a Cayley polytope of order $k$. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly $k$ a…
The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth n-dimensional complex projective varieties by considering the sum of at least n general sufficiently ample hypersurfaces.
We study the following question: Given a vector bundle on a projective variety $X$ such that the restriction of $E$ to every closed curve $C \,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we answer the question in the affirmative. When $E$ is of higher rank, we show that the answer is affirmative under some conditions on $E$. We then study the case of $X \,=\, G/P$, where $G$ is a reductive complex affine algebraic group, and $P$ is a parabolic subgroup of $G$. In this case, we show that th…
We survey results concerning behavior of positivity of line bundles and possible vanishing theorems in positive characteristic. We also try to describe variation of positivity in mixed characteristic. These problems are very much related to behavior of strong semistability of vector bundles, which is another main topic of the paper.
Positivity for Higgs vector bundles: criteria and applications
SISSA (International School for Advanced Studies)（国际高等研究院）；INFN (Istituto Nazionale di Fisica Nucleare), Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；IGAP (Institute for Geometry and Physics)（几何与物理研究所）
The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stabi…
We study the positivity of the first Chern class of a rank r Ulrich vector bundle E on a smooth n-dimensional variety $X \subseteq \mathbb P^N$. We prove that $c_1(E)$ is very positive on every subvariety not contained in the union of lines in X. In particular if X is not covered by lines, then E is big and $c_1(E)^n \ge r^n$. Moreover we classify rank r Ulrich vector bundles E with $c_1(E)^2=0$ on surfaces and with $c_1(E)^2=0$ or $c_1(E)^3=0$ on threefolds (with some exceptions).
In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,Δ)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing s…
Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2.
