Logarithmic Connection Research Articles

Localization formulas of cohomology intersection numbers

We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of the canonical form.

Parabolic opers and differential operators

Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.

Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation

The basic filters in mathematical morphology are dilation and erosion. They are defined by a structuring element that is usually shifted pixel-wise over an image, together with a comparison process that takes place within the corresponding mask. This comparison is made in the grey value case by means of maximum or minimum formation. Hence, there is easy access to max-plus algebra and, by means of an algebra change, also to the theory of linear algebra. We show that an approximation of the maximum function forms a commutative semifield (with respect to multiplication) and corresponds to the maximum again in the limit case. In this way, we demonstrate a novel access to the logarithmic connection between the Fourier transform and the slope transformation. In addition, we prove that the dilation by means of a fast Fourier transform depends only on the size of the structuring element used. Moreover, we derive a bound above which the Fourier approximation yields results that are exact in terms of grey value quantisation.

On the relative logarithmic connections and relative residue formula

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.

Branched SL(r, ℂ)-Opers

Abstract Branched projective structures were introduced by Mandelbaum [22, 23], and opers were introduced by Beilinson and Drinfeld [2, 3]. We define the branched analog of ${\textrm SL}(r, {\mathbb C})$-opers and investigate their properties. For the usual ${\textrm SL}(r, {\mathbb C})$-opers, the underlying holomorphic vector bundle is actually determined uniquely up to tensoring with a holomorphic line bundle of order $r$. For the branched ${\textrm SL}(r, {\mathbb C})$-opers, the underlying holomorphic vector bundle depends more intricately on the oper. While the holomorphic connection for a branched ${\textrm SL}(r, {\mathbb C})$-oper is nonsingular, given a branched ${\textrm SL}(r, {\mathbb C})$-oper, we associate to it a certain holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle in question supporting a logarithmic connection does not depend on the branched oper. We characterize the branched ${\textrm SL}(r, {\mathbb C})$-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.

Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

Moduli spaces of stable parabolic bundles of parabolic degree 0 over the Riemann sphere are stratified according to the Harder–Narasimhan filtration of underlying vector bundles. Over a Zariski open subset mathscr {N}_{0} of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function mathscr {S} is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of mathscr {S} depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that -mathscr {S} is a primitive for a (1,0)-form vartheta on mathscr {N}_{0} associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that -mathscr {S} is a Kähler potential for (Omega -Omega _{mathrm {T}})|_{mathscr {N}_{0}}, where Omega is the Narasimhan–Atiyah–Bott Kähler form in mathscr {N} and Omega _{mathrm {T}} is a certain linear combination of tautological (1, 1)-forms associated with the marked points. These results provide an explicit relation between the cohomology class [Omega ] and tautological classes, which holds globally over certain open chambers of parabolic weights where mathscr {N}_{0} = mathscr {N}.

A criterion for the existence of logarithmic connections on curves over a perfect field

Let k be a perfect field, and X an irreducible smooth projective curve over k. We give a criterion for a vector bundle over X to admit a logarithmic connection singular over a finite subset of X with given residues, where residues are assumed to be rigid.

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.

A Map Between Moduli Spaces of Connections

We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve C with elliptic quotient, and the logarithmic connection, we may pullback the connection to the elliptic curve to obtain a new connection on C. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve C with two simple poles and trivial determinant.

Criterion for existence of a logarithmic connection on a principal bundle over a smooth complex projective variety

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 $D$.

On a Possible Logarithmic Connection between Einstein’s Constant and the Fine-Structure Constant, in Relation to a Zero-energy Hypothesis

This paper brings into attention a possible logarithmic connection between Einstein’s constant and the fine-structure constant, based on a hypothetical electro-gravitational resistivity of vacuum: we also propose a zero-energy hypothesis (ZEH) which is essentially a conservation principle applied on zero-energy that mainly states a general quadratic equation having a pair of conjugate mass solutions for each set of coefficients, thus predicting a new type of mass “symmetry” called here “mass conjugation” between elementary particles (EPs) which predicts the zero/non-zero rest masses of all known/unknown EPs to be conjugated in boson-fermion pairs; ZEH proposes a general formula for all the rest masses of all EPs from Standard model, also indicating a possible bijective connection between the three types of neutrinos and the massless bosons (photon, gluon and the hypothetical graviton), between the electron/positron and the W boson and predicting two distinct types of neutral massless fermions (modelled as conjugates of the Higgs boson and Z boson respectively) which are plausible candidates for dark energy and dark matter. ZEH also offers a new interpretation of Planck length as the approximate length threshold above which the rest masses of all known elementary particles have real number values (with mass units) instead of complex/imaginary number values (as predicted by the unique quadratic equation proposed by ZEH).

ALGEBRAIC ISOMONODROMIC DEFORMATIONS AND THE MAPPING CLASS GROUP

AbstractThe germ of the universal isomonodromic deformation of a logarithmic connection on a stable $n$-pointed genus $g$ curve always exists in the analytic category. The first part of this article investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this work studies the dynamics of this action in the particular case of reducible rank 2 representations and genus $g>0$, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.

Jumps, folds and hypercomplex structures

We investigate the geometry of the Kodaira moduli space M of sections of $$\pi :Z\rightarrow {\mathbb {P}}^1$$ , the normal bundle of which is allowed to jump from $${\mathcal {O}}(1)^{n}$$ to $${\mathcal {O}}(1)^{n-2m}\oplus {\mathcal {O}}(2)^{m}\oplus {\mathcal {O}}^{m}$$ . In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of M extends to a logarithmic connection on M.

On the direct images of parabolic vector bundles and parabolic connections

Let φ:Y⟶X be a finite surjective morphism between smooth complex projective curves, where X is irreducible but Y need not be so. Let V∗ be a parabolic vector bundle on Y. We construct a parabolic structure on the direct image φ∗V on X, where V is the vector bundle underlying V∗. The parabolic vector bundle φ∗V∗ on X obtained this way has a ramified torus sub-bundle; it is a torus bundle of Ad(φ∗V) outside the parabolic divisor for φ∗V∗ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle E∗ on X, and a ramified torus sub-bundle T for it, we construct a ramified covering Z of X and a parabolic vector bundle W∗ on Z, such that the parabolic bundle E∗ is the direct image of W∗. A connection on V∗ produces a connection on φ∗V∗. The ramified torus sub-bundle for φ∗V∗ is preserved by the logarithmic connection on End(φ∗V) induced by this connection on φ∗V∗. If the parabolic vector bundle E∗ on X is equipped with a connection D such that the connection on the endomorphism bundle induced by it preserves the ramified torus sub-bundle T, then we prove that the corresponding parabolic vector bundle W∗ on Z has a connection that produces the connection D on the direct image E∗.

Logarithmic connections on principal bundles over a Riemann surface

Let [Formula: see text] be a holomorphic principal [Formula: see text]-bundle on a compact connected Riemann surface [Formula: see text], where [Formula: see text] is a connected reductive complex affine algebraic group. Fix a finite subset [Formula: see text], and for each [Formula: see text] fix [Formula: see text]. Let [Formula: see text] be a maximal torus in the group of all holomorphic automorphisms of [Formula: see text]. We give a necessary and sufficient condition for the existence of a [Formula: see text]-invariant logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text]. We also give a necessary and sufficient condition for the existence of a logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text], under the assumption that each [Formula: see text] is [Formula: see text]-rigid.

Criterion for logarithmic connections with prescribed residues

A theorem of Weil and Atiyah says that a holomorphic vector bundle E on a compact Riemann surface X admits a holomorphic connection if and only if the degree of every direct summand of E is zero. Fix a finite subset S of X, and fix an endomorphism \(A(x) \in \text {End}(E_x)\) for every \(x \in S\). It is natural to ask when there is a logarithmic connection on E singular over S with residue A(x) at every \(x \in S\). We give a necessary and sufficient condition for it under the assumption that the residues A(x) are rigid.

Two-dimensional Riemann problem for rigid representations on an elliptic curve

We consider a generalization of Riemann–Hilbert problem on elliptic curves. For a given elliptic curve and irreducible representation of free group with two generators we construct explicitly a semistable vector bundle of degree zero obeying a logarithmic connection such that its monodromy over fundamental parallelogram is equivalent to given free group representation, monodromy along a -cycle is trivial and monodromy along b -cycle belong to certain orbit.

Equivariant principal bundles and logarithmic connections on toric varieties

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 singular over $D$.

Equivariant principal bundles and logarithmic connections on toric varieties

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 singular over $D$.

Isomonodromic deformations of logarithmic connections and stability

Let \(X_0\) be a compact connected Riemann surface of genus g with \(D_0 \subset X_0\) an ordered subset of cardinality n, and let \(E_G\) be a holomorphic principal G-bundle on \(X_0\), where G is a reductive affine algebraic group defined over \(\mathbb C\), that is equipped with a logarithmic connection \(\nabla _0\) with polar divisor \(D_0\). Let \((\mathcal {E}_G , \nabla )\) be the universal isomonodromic deformation of \((E_G ,\nabla _0)\) over the universal Teichmuller curve \((\mathcal {X}, \mathcal {D})\,{\longrightarrow }\, \text {Teich}_{g,n}\), where \(\text {Teich}_{g,n}\) is the Teichmuller space for genus g Riemann surfaces with n–marked points. We prove the following (see Sect. 5): (1) Assume that \(g \ge 2\) and \(n= 0\). Then there is a closed complex analytic subset \(\mathcal {Y} \subset \text {Teich}_{g,n}\), of codimension at least g, such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is semistable, where \({\mathcal X}_t\) is the compact Riemann surface over t. (2) Assume that \(g\ge 1\), and if \(g= 1\), then \(n > 0\). Also, assume that the monodromy representation for \(\nabla _0\) does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \(\mathcal {Y}' \subset \text {Teich}_{g,n}\), of codimension at least g, such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is semistable. (3) Assume that \(g\ge 2\). Assume that the monodromy representation for \(\nabla _0\) does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \(\mathcal {Y}'' \subset \text {Teich}_{g,n}\), of codimension at least \(g-1\), such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is stable. In [12], the second-named author proved the above results for \(G= \text {GL}(2,{\mathbb C})\).