We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.
We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they can be recovered from the attached logarithmic vector bundles.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.
We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to prove a Torelli type theorem in some cases.
We characterize Kaehler manifolds with trivial logarithmic tangent bundle (with respect to a divisor D) as a class of certain compatifications of complex semi-tori.
In previous papers it was shown that the left and right O-module structure of the jet bundles on the projective line differed. In this paper we show that similar statements hold for jet bundles on projective space in any dimension. We also consider some classes associated to jet bundles in the algebraic K-theory of projective space.
Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tauto…
Let X be a smooth complete complex toric variety such that the boundary is a simple normal crossing divisor, and let E be a holomorphic vector bundle on X. We prove that E admits an equivariant structure if and only if E admits a logarithmic connection singular over D. More precisely, we show that an equivariant vector bundle on X has a tautological integrable logarithmic connection singular over D. This is used in computing the Chern classes of the equivariant vector bundles on X. We also prove a version of the result for holomorphic vector bundles on log parallelizable G-pairs (X,D), where G…
In this article we introduce a notion of logarithmic co-Higgs sheaves associated to a simple normal crossing divisor on a projective manifold, and show their existence with nilpotent co-Higgs fields for fixed ranks and second Chern classes. Then we deal with various moduli problems with logarithmic co-Higgs sheaves involved, such as coherent systems and holomorphic triples, specially over algebraic curves of low genus.
In this paper, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety, under suitable conditions. In particular when the Picard number is one, we obtain results for any log Del Pezzo surface, log Fano threefolds, and for log Fano $n$-folds of dimension $n\leq 6$.
We consider deformations of the Weyl arrangement of type $A_2$, which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.
We study k-jet ampleness of line bundles on hyperelliptic surfaces using vanishing theorems. Our main result states that on a hyperelliptic surface of an arbitrary type a line bundle of type (m,m) with m\geq k+2 is k-jet ample.
The aim of this paper is to open the problem of construction of a nonlinear connection $Γ=(M^{(i)}_{(α)β}, N^{(i)}_{(α)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(α)(β)}_{(i)(j)}=h^{αβ}g_{ij}$, possibly provided by multi-time dependent quadratic Lagrangians coming from various branches of theoretical physics: bosonic strings theory, electrodynamics or elasticity.
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.
Let $X$ be a connected smooth complex projective variety of dimension $n \geq 1$. Let $D$ be a simple normal crossing divisor on $X$. Let $G$ be a connected complex Lie group, and $E_G$ a holomorphic principal $G$-bundle on $X$. In this article, we give criterion for existence of a logarithmic connection on $E_G$ singular along $D$.
Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection si…
We propose an approach to study logarithmic sheaves T(-log A) associated with a hyperplane arrangements A on the projective space, based on projective duality, direct image functors and vector bundles methods. We focus on freeness of line arrangements having a point with high multiplicity.
We investigate the relative logarithmic connections on a holomorphic vector bundle over a complex analytic family. We give a sufficient condition for the existence of a relative logarithmic connection on a holomorphic vector bundle singular over a relative simple normal crossing divisor. We define the relative residue of relative logarithmic connection and express relative Chern classes of a holomorphic vector bundle in terms of relative residues.
We consider irreducible logarithmic connections $(E,\,δ)$ over compact Riemann surfaces $X$ of genus at least two. The underlying vector bundle $E$ inherits a natural parabolic structure over the singular locus of the connection $δ$; the parabolic structure is given by the residues of $δ$. We prove that for the universal isomonodromic deformation of the triple $(X,\,E,\,δ)$, the parabolic vector bundle corresponding to a generic parameter in the Teichmüller space is parabolically stable. In the case of parabolic vector bundles of rank two, the general parabolic vector bundle is even parabolica…
Spectral and Logarithmic Atiyah Classes for Higgs Bundles
Shiv Nadar University（希夫纳达尔大学）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；International Center for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究院国际理论科学中心）；Indian Institute of Technology Patna（巴特那印度理工学院）
Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $…
Winkelmann considered compact complex manifolds $X$ equipped with a reduced effective normal crossing divisor $D\, \subset\, X$ such that the logarithmic tangent bundle $TX(-\log D)$ is holomorphically trivial. He characterized them as pairs $(X,\, D)$ admitting a holomorphic action of a complex Lie group $\mathbb G$ satisfying certain conditions \cite{Wi1}, \cite{Wi2}; this $\mathbb G$ is the connected component, containing the identity element, of the group of holomorphic automorphisms of $X$ that preserve $D$. We characterize the homogeneous holomorphic principal $H$--bundles over $X$, wher…
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of Noguchi and Winkelmann by replacing the semi-abelian condition by a differential one.
Let $X$ be a smooth hypersurface of degree $d$ in $\CP^3$. By totally algebraic calculations, we prove that on the third Demailly-Semple jet bundle $X_3$ of $X$, the bundle $\ocal_{X_3}(1)$ is big for $d\geq 11$, and that on the fourth Demailly-Semple jet bundle $X_4$ of $X$, the bundle $\ocal_{X_4}(1)$ is big for $d\geq 10$, improving a recent result of Diverio.
A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle $K_X$ of a Kobayashi hyperbolic compact complex manifold $X$ is ample. In this note we prove that $K_X$ is ample if $X$ is projective and satisfies a stronger condition of nondegenerate negative total $k$-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on $X$.
Let $ \mathcal{D} = \{D_{1}, ..., D_{\ell}\} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \mathbf{P}^{n} $, with degrees $ d_{1}, ..., d_{\ell} $; let $ Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $ \mathcal{D} $ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$ d_{i} $ hypersurfaces of $ Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $. Then, when $ n = 2 $, by describing the moduli spaces containing $ Ω_{\mathbf{P}^{2}…
We characterize the triples (X,L,H), consisting of holomorphic line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J_k(L), is isomorphic to a direct sum H+...+H. Given the geometrical constrains imposed by a projectivized line bundle being a product of the base and a projective space it is natural to expect that this would happen only under very rare circumstances. It is shown, in fact, that X is either an Abelian variety or projective space. In the former case L\cong H is any line bundle of Chern class zero. In …
Chern number formulas for holomorphic jet bundles are computed for projective curves and for projective surfaces. These formulas are used to show that certain minimal surfaces of general type (generic hypersurfaces of degree at least 5 in projective 3-spaces in particular) are hyperbolic in the sense of complex hyperbolic geometry.
We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao's Conjecture for arrangements having a line with at most 4 intersection points.
Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let Ω^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that Ω^{1}_{P^n}(log D) admits a resolution of lenght 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when all the D_{i}'s have the same degree d and l >= {{n+d} \choose {d}}+3: indeed, we recover the components of D as unstable smooth hypersurfaces of Ω^{1}_{P^n}(log D). Finally we analyze the cases of one q…
Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D \subset X$, and for each $x\in D$ fix $w_x \in \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic …
The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods and generalized Verma modules. We calculate the $P$-module of the dual jet bundle $J^l(L)^*$ and prove it equals the $l$'th piece of the canonical filtration for $H^0(X,L)^*$. We use the results obtained to prove the discriminant of any linear system on any grassmannian is irreducible.
In this paper we state and prove some results on the structure of the jetbundles as left and right module over the structure sheaf on the projective line and projective space using elementary techniques involving diagonalization of matrices, multilinear algebra and sheaf cohomology.
Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schon assumptions. As the second application, we confirm an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, whi…
We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and Kühn, in their extension of the arithmetic intersection theory of Gillet and Soulé, aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of Bost-Gillet-Soulé, as well as the properties established by Faltings for heights of points attached to hermitian line bundles whose metrics have logarit…
We prove the logarithmic divergence of equivariant analytic torsion for one-parameter degenerations of projective algebraic manifolds, when the coefficient vector bundle is given by a Nakano semi-positive vector bundle twisted by the relative canonical bundle.
We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.
In the last twenty years a number of papers appeared aiming to construct locally free replacements of the sheaf of principal parts for families of Gorenstein curves. The main goal of this survey is to present to the widest possible audience of mathematical readers a catalogue of such constructions, discussing the related literature and reporting on a few applications to classical problems in Enumerative Algebraic Geometry.
For a reduced curve $C:f=0$ in the complex projective plane $\mathbb{P}^2$, we study the set of jumping lines for the rank two vector bundle $T\langle C \rangle $ on $\mathbb{P}^2$, whose sections are the logarithmic vector fields along $C$. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of $f$ and with the Bourbaki ideal of the module of Jacobian syzygies of $f$. In particular, when the vector bundle $T\langle C \rangle $ is unstable, a line is a jumping line if and only if it meets the 0-dimensional subscheme defined by this Bourba…
Let $f:\Cal C\to S$ be a flat family of curves over a smooth curve $S$ such that $f$ is smooth over $S_0=S\ssm\{s_0\}$ and $f^{-1}(s_0)=\Cal C_0$ is irreducible with one node. We have an associated family $\Cal M_{S_0}\to S_0$ of moduli spaces of semistable vector bundles and the relative theta line bundle $Θ_{S_0}$. We are interested in the problem: to find suitable degeneration $\Cal M_S$ of moduli spaces and extension $Θ_S$ of theta line bundles such that the direct image of $Θ_S$ is a vector bundle on $S$ with a logarithmic projective connection. In this paper, we figured out the condition…
We study various aspects on nontrivial logarithmic co-Higgs structure associated to unstable bundles on algebraic curves. We check several criteria for (non-)existence of nontrivial logarithmic co-Higgs structures and describe their parameter spaces. We also investigate the Segre invariants of these structures and see their non-simplicity. In the end we also study the higher dimensional case, specially when the tangent bundle is not semistable.
Motivated by the Green-Griffiths conjecture, we study maximal rank holomorphic maps from $\C^p$ into complex manifolds. When $p>1$ such maps should in principle be more tractable than entire curves. We extend to this setting the jet-bundles techniques introduced by Semple, Green-Griffiths and Demailly. Our main application is the non-existence of maximal rank holomorphic maps from $\C^2$ into the very general degree $d$ hypersurface in $\bP^4$, as soon as $d\geq 93.$
In this article, we study projective log smooth pairs with numerically flat normalized logarithmic tangent bundle. Generalizing works of Jahnke-Radloff and Greb-Kebekus-Peternell, we show that, passing to an appropriate finite cover and up to isomorphism, these are the projective spaces or the log smooth pairs with numerically flat logarithmic tangent bundles blown-up at finitely many points away from the boundary. On the other hand, the structure of log smooth pairs with numerically flat logarithmic tangent bundle is well understood: they are toric fiber bundles over finite étale quotients of…
The consequences of the splitting of higher jet sequences of vector bundles on a compact complex (Kaehler) manifold are studied, proving an infinitesimal convers to a result of I.Biswas.
We prove that the complement of a very generic curve of degree $d$ at least equal to 15 in the projective plane is hyperbolic in the sens of Kobayashi (here, the terminology ``very generic'' refers to complements of countable unions of proper algebraic subsets of the parameter space). We first consider the Dethloff and Lu's generalisation to the logarithmic situation of Demailly's jet bundles. We study their base loci for surfaces of log-general type in the same way as it was done in the compact case by Demailly and El Goul. With some condition on log-Chern classes, any entire holomorphic map …
We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S\, =\, \sum_{i=1}^d x_i$, where $x_i \,\in\, X$ are distinct points. After defining branched ${\rm SO}(3,{\mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${\rm SO}(3,{\mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)\otimes {\mathcal O}_X(S))…
We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.
We study logarithmic jet schemes of a log scheme and generalize a theorem of M. Mustata from the case of ordinary jet schemes to the logarithmic case. If X is a normal local complete intersection log variety, then X has canonical singularities if and only if the log jet schemes of X are irreducible.
Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of length $ 1 $ depending on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then, we pro…
