In this paper, we compute the dimensions of the 1st and 2nd cohomology groups of all the pluricanonical bundles for Hirzebruch surfaces, and the dimensions of the 1st and 2nd cohomology groups of the second pluricanonical bundles for a blow-up of a projective plane along finite many distinct points. Both of them give examples showing that the higher cohomology of the pluricanonical bundle is not deformation invariant.
We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf.
In this note, we generalize slightly Popa--Schnell's theorem regarding to direct images of pluricanonical bundles to the case when the ample line bundle is not globally generated. We also treat the case of positive characteristic.
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.
We discuss the local freeness and the numerical semipositivity of direct images of relative pluricanonical bundles for surjective morphisms between smooth projective varieties with connected fibers. We give a desirable semipositivity theorem under the assumption that the geometric generic fiber has a good minimal model.
We show that techniques inspired by Kollár and Viehweg's study of weak positivity, combined with vanishing theorems for log-canonical pairs, lead to new consequences regarding generation and vanishing properties for direct images of pluricanonical bundles. We formulate the strongest such results as Fujita conjecture-type statements, which are then shown to govern a range of fundamental properties of direct images of pluricanonical and pluriadjoint line bundles, like effective vanishing theorems, weak positivity, or generic vanishing.
We extend the so called Chen-Jiang decomposition for pushforward of pluricanocanical bundles to abelian varieties to the setting of klt pairs. We also provide a geometric application of this decomposition.
We give effective bounds on the generation of pushforwards of log-pluricanonical bundles twisted by ample line bundles. This gives a partial answer to a conjecture proposed by Popa and Schnell. We prove two types of statements: first, more in the spirit of the general conjecture, we show generic global generation with predicted bound when the dimesnion of the variety if less than 4 and more generally, with a quadratic Angehrn-Siu type bound. Secondly, assuming that the relative canonical bundle is relatively semi-ample, we make a very precise statement. In particular, when the morphism is smoo…
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.
Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* ω_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.
In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is $F$-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases.
We study pseudonorms on pluricanonical bundles over Stein manifolds. We prove that the pseudonorms determine holomorphic structures of Stein manifolds under certain assumptions. This theorem is based on and a generalization of the result obtained by Deng, Wang, Zhang and Zhou \cite{DWZZ} for bounded domains in $\mathbb{C}^n$. We also investigate Stein morphisms and the pseudonorms on direct images of pluricanonical bundles. Our main goal in this paper is to show that the pseudonorms also determine holomorphic structures of Stein morphisms. One important technique is an $L^{2/m}$-variant of the…
Given a fibration $f$ between two projective manifolds $X$ and $Y$, we discuss the effective generation of the higher direct images $R^{i}f_{\ast}(K^{m}_{X})$, where $K^{m}_{X}$ is the $m$-th tensor power of the canonical bundle of $X$. In particular, we answer two questions posed by Popa--Schnell in [PS14].
Given a fibration $f$ between two projective manifolds $X$ and $Y$, we provide a sufficient condition such that the direct images $f_{\ast}(K_{X/Y}\otimes L\otimes\mathscr{I}(f,\|L\|))$ is nef, where $L$ is a holomorphic line bundle with non-negative relative Iitaka dimension and $\mathscr{I}(f,\|L\|)$ is the relative asymptotic multiplier ideal sheaf.
Generalizing the recent result of Berndtsson, we prove the Nakano semipositivity of the direct image of relative pluricanonical systems and the direct image of relative adjoint (singular) hermitian line bundle with semipositive curvature. This gives a new proof of Kawamata's theorem on the semipositivity of the direct image of relative pluricanonical systems.
We prove an effective vanishing theorem for direct images of log pluricanonical bundles of projective semi-log canonical pairs. As an application, we obtain a semipositivity theorem for direct images of relative log pluricanonical bundles of projective semi-log canonical pairs over curves, which implies the projectivity of the moduli spaces of stable varieties. It is worth mentioning that we do not use the theory of variation of (mixed) Hodge structure.
In this paper, we show an extension type theorem for twisted pluricanonical sections on a family of smooth projective manifolds (the twisting line bundle being pseudo-effective and having a prescribed multiplier ideal on the central fiber).
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
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…
We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector bundles from their cohomology.
We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.
Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.
We study the vector bundles without intermediate cohomology on Fano threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a complete characterization in the case of rank-two vector bundles. For arbitrary rank, we give all possible Chern classes of such vector bundles, under some general condition.
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.
In an earlier paper we conjectured a relation between the quantum $\mathcal D$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete intersection in a toric variety. We also use the conjecture to show that the relations of the small quantum cohomology ring of $X$ that come from differential operators lift to the projective bundle. The basic cohomology relation of the projective bundle deforms to a relation in the small quantum cohomology.
Étale Nori finite vector bundles are those bundles defined by representations of a finite étale group scheme in the usual way. In this note we show that in many cases the dimensions of the Hodge cohomology groups of such a vector bundle and of a twist of it by an automorphism of the ground field are the same. This generalizes to the higher rank case a result of Pink-Roessler.
We find an algorithm to compute the cohomology groups of spherical vector bundles on complex projective K3 surfaces, in terms of their Mukai vectors. In many good cases, we give significant simplifications of the algorithm. As an application, when the Picard rank is one, we show a numerical condition that is equivalent to weak Brill-Noether for a spherical vector bundle.
In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that generic flat bundles (in its moduli space) can be expected to have nontrivial classes, which is in contrast with some Gauss-Manin systems.
In this paper, we attempt to determine the quantum cohomology of projective bundles over the projective space P^n. In contrast to the previous examples, the relevant moduli spaces in our case frequently do not have expected dimensions. It makes the calculation more difficult. We overcome this difficulty by using excessive intersection theory.
Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful.
We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the genus $0$ Gromov-Witten theory of a toric stack bundle.
We show that a normalized rank two vector bundle, E, on P2 splits if and only if h1(E(-1)) = 0. Using this fact we give another proof of a theorem of Chiantini and Valabrega. Finally we describe the normalized bundles with h1(E(-1)) <= 4.
We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivisation of the bundle from a small number of low-degree Gromov--Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch programme.
The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.
Two decades ago, as part of their work of generic vanishing theorems, Green-Lazarsfeld showed that over a compact Kahler manifold $X$, the cohomology jump loci in the $Pic^τ(X)$ are all translates of subtori. In this paper, we generalize this result to higher dimensional vector bundles. It is showed by Nadel that locally the moduli space of vector bundles with vanishing chern classes is canonically isomorphic to a quadratic cone in the Zariski tangent space of a point. We prove that under the isomorphism, the cohomology jump loci are defined by linear equations.
This note corrects a certain construction in (the old version of) the paper [GL]. The construction in question starts from a Brylinski-Deligne datum, which is an extension of a group G by K_2, and produces a factorization line bundle on the affine Grassmannian of the group G.
It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not true, as already remarked by M. Saito. We explain how one has to modify the latter, show that the Picard group can be expressed by absolute Hodge cohomology, too, and introduce an intermediate object between Picard group and usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne cohomology one can relate line bundles with a regular connec…
For $C$ a stable curve of arithmetic genus $g\ge 2$, and $\mathcal{D}$ the determinant of cohomology line bundle on $\operatorname{Bun}_{\operatorname{SL}(r)}(C)$, we show the section ring for the pair $(\operatorname{Bun}_{\operatorname{SL}(r)}(C), \mathcal{D})$ is finitely generated. Three applications are given.
In this paper, we show that the motive of the quaternionic Grassmannian $HP^n$ (as defined by I. Panin and C. Walter) splits in the category of effective MW-motives (as defined by B. Calmès, F. Déglise and J. Fasel). Moreover, we extend this result to an arbitrary symplectic bundle, obtaining the so-called quaternionic projective bundle theorem. Finally, we give the Gysin triangle in MW-motivic cohomology.
In their paper "Quantum cohomology of projective bundles over $P^n$" (Trans. Am. Math. Soc. (1998)350:9 3615-3638) Z.Qin and Y.Ruan introduce interesting techniques for the computation of the quantum ring of manifolds which are projectivized bundles over projective spaces; in particular, in the case of splitting bundles they prove under some restrictions the formula of Batyrev about the quantum ring of toric manifolds. Here we prove the formula of Batyrev on the quantum Chern-Leray equation for a large class of splitting bundles.
We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Δ^+-Δ^-$ of polyhedra in the character space $M_{\mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Δ^- \setminus Δ^+$. Here, we provide a short and direct proof of this formula.
We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of relative sections of line bundles (proving that the equivariant, divisorial multi-variable Hilbert series is topological), a combinatorial description of divisors of analytic function-germs, and an expression for the multiplicity of the singularity from its resolution graph. Additional, we establish a new formula for the Seiberg-Witten invariants of any rati…
We give sharp bounds on the vanishing of the cohomology of a tensor product of vector bundles on the n-dimensional projective space in terms of the vanishing of the cohomology of the factors. For this purpose we introduce regularity indices generalizing the Castelnuovo-Mumford regularity. As an application we give a sufficient condition for a vector bundle to have an unobstructed deformation theory that depends only on the cohomology table of the bundle. We construct complete families of bundles with such cohomology tables.
A vector bundle on a projective variety has a natural cohomology if for every twist its cohomology is concentrated in a single degree. Eisenbud and Schreyer conjectured there should be vector bundles on $\mathbb{P}^1 \times \mathbb{P}^1$ with natural cohomology with respect to bundles $\mathcal{O}(1,0),\mathcal{O}(0,1)$ with prescribed Hilbert polynomial. We prove this conjecture.
We study the moduli space of Higgs bundles on a compact Riemann surface. It was shown by Thaddeus and Hausel (in rank 2) and Markman (in general rank) that the rational cohomology ring of this space is generated by universal classes. In this paper, we use their results to compute the Hodge structure on the cohomology of this moduli space. We show that the natural Hodge structure on H^k is pure of weight k.
This paper is a companion of the paper "Weil's conjecture for function fields" by J. Lurie and the author. We present a different exposition of essentially the same algebro-geometric proof of the Atiyah-Bott for the cohomology of Bun(G), which subsequently leads to the proof of the Tamagawa number formula for the volume of the automorphic space for function fields.
We compute some Hodge and Betti numbers of the moduli space of stable rank $r$ degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles on a curve. Our methods also yield a formula for the Poincare polynomial of the moduli stack that is valid over any ground field.
Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $α$. Let $P \mathcal{M}_ξ$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $ξ$ over $X$ with weight system $α$, where $r$ is a prime number and $ξ$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_ξ$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.
