This paper is devoted to the function introduced by M. P. Appell in connection with decomposition of elliptic functions of the third kind into simple elements. We show that this function (which appeared as a Fukaya triple product in math.AG/9803017) is related to global sections of rank-2 vector bundles on elliptic curves. We derive analogues of theta-identities for this function and prove the divisibility property for the action of modular group which should be considered as a replacement for functional equation.
We study moduli of odd-framed $\mathcal{N}=2$ elliptic curves subject to certain conditions, and show that the fermionic part of the moduli problem is essentially controlled by the Appell-Lerch sum, familiar from the theory of mock modular forms.
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
We study the splitting properties of the Verlinde bundles over elliptic curves. Our methods rely on the explicit description of the moduli space of semistable vector bundles on elliptic curves, and on the analysis of the symmetric powers of the Schrodinger representation of the Theta group.
M. Nori proved that on a projective smooth variety, a bundle is finite, (that is the ring it generates has dimension 0), if and only if it trivializes on a finite cover. In this note, we consider bundles of degree 0 on an elliptic curve. We prove that the dimension of the ring generated by such a bundle is the same as the dimension of a torsor trivializing the bundle (and is 1).
Let $E$ be a smooth elliptic curve over $\mathbb{C}$. For $E$ embedded into $\mathbb{P}^2$ as Hesse cubic and $V$ an Ulrich bundle on $E$ we derive a explicit presentation of $V$ using Moore matrices and theta functions.
The aim of this paper is to establish an equivalence of certain categories of Higgs bundles on a non-isotrivial elliptic surface $π: X \rightarrow C$ with $χ(X) > 0$ and certain categories of Parabolic Higgs bundles on $C$
We study vector bundles on the moduli stack of elliptic curves over a local ring R. If R is a field or a discrete valuation ring of (residue) characteristic not 2 or 3, all these vector bundles are sums of line bundles. For R the 3-local integers, we construct higher rank indecomposable vector bundles and give a classification of vector bundles that are iterated extensions of line bundles. For R the 2-local integers, we show that there are even indecomposable vector bundles of arbitrary high rank.
In this paper, we give a numerical criterion of Reider-type for the $d$-very ampleness of the adjoint line bundles on quasi-elliptic surfaces, and meanwhile we give a new proof of the vanishing theorem on quasi-elliptic surfaces emailed from Langer and show that it is the optimal version.
In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincaré relations on a Jacobian variety.
We study topologically trivial $G$-Higgs bundles over an elliptic curve $X$ when the structure group $G$ is a connected real form of a complex semisimple Lie group $G^{\mathbb{C}}$. We achieve a description of their (reduced) moduli space, the associated Hitchin fibration and the finite morphism to the moduli space of $G^{\mathbb{C}}$-Higgs bundles.
In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.
We show that fiberwise stable vector bundles are preserved by relative Fourier-Mukai transforms between elliptic threefolds with relative Picard number one. Using these bundles we define new invariants of elliptic fibrations, and we relate the invariants of a space with those of a relative moduli space of stable sheaves on it. As a byproduct, we calculate the intersection form of a certain new example of an elliptic Calabi-Yau threefold.
Let $X$ be a rational elliptic surface with elliptic fibration $π:X\to\Bbb{P}^1$ over an algebraically closed field $k$ of any characteristic. Given a conic bundle $φ:X\to\Bbb{P}^1$ we use numerical arguments to classify all possible fibers of $φ$ and study the interplay between singular fibers of $π$ and $φ$.
We give normal forms of determinantal representations of a smooth projective plane cubic in terms of Moore matrices. Building on this, we exhibit matrix factorizations for all indecomposable vector bundles of rank 2 and degree 0 without nonzero sections, also called Ulrich bundles, on such curves.
It is well-known that del Pezzo surfaces of degree $9-n$ one-to-one correspond to flat $E_n$ bundles over an elliptic curve. In this paper, we construct $ADE$ bundles over a broader class of rational surfaces which we call $ADE$ surfaces, and extend the above correspondence to all flat $G$ bundles over an elliptic curve, where $G$ is any simply laced, simple, compact and simply-connected Lie group. In the sequel, we will construct $G$ bundles for non-simply laced Lie group $G$ over these rational surfaces, and extend the above correspondence to non-simply laced cases.
To count bundles on curves, we study zetas of elliptic curves and their zeros. There are two types, i.e., the pure non-abelian zetas defined using moduli spaces of semi-stable bundles, and the group zetas defined for special linear groups. In lower ranks, we show that these two types of zetas coincide and satisfy the Riemann Hypothesis. For general cases, exposed is an intrinsic relation on automorphism groups of semi-stable bundles over elliptic curves, the so-called counting miracle. All this, together with Harder-Narasimhan, Desale-Ramanan and Zagier's result, gives an effective way to coun…
We study deformation theory of elliptic fibre bundles over curves in positive characteristics. As applications, we give examples of non-liftable elliptic surfaces in charactertic two and three, which answers a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of F. Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.
We present a systematic study of involutions on the moduli space of $G$-Higgs bundles over an elliptic curve $X$, where $G$ is complex reductive affine algebraic group. The fixed point loci in the moduli space of $G$-Higgs bundles on $X$, and in the moduli space of representations of the fundamental group of $X$ into $G$, are described. This leads to an explicit description of the moduli spaces of pseudo-real $G$-Higgs bundles over $X$.
In this paper we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the…
We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalization of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.
Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.
We show that the moduli space of semistable G-bundles on an elliptic curve for a reductive group G is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This generalises a result of Laszlo to arbitrary connected components and recovers the global description of the moduli space due to Friedman--Morgan--Witten and Schweigert. The proof is entirely in the realm of algebraic geometry and works in arbitrary characteristic.
Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to anti-canonical divisor $-K_M$) at least 6, and we give a complete classification for elliptic curves of degree $6$. Moreover, if $g>4$, we show that any elliptic curve passing through the generic point of $M$ has degree at least $12$. We also formulate a conjecture for higher rank.
Resonance of rank-two vector bundles over elliptic curves
University of Bucharest, Faculty of Mathematics and Informatics（布加勒斯特大学数学与信息学院）；”Simion Stoilow” Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛瓦数学研究所）
Let $M$ be the moduli space of rank 3 stable bundles with fixed determinant of degree 1 on a smooth projective curve of genus $g\geq 2$. When $C$ is generic, we show that any essential elliptic curve on $M$ has degree (respect to anti-canonical divisor $-K_{M}$) at least 6, and we give a complete classification for elliptic curves of degree 6, which is not in conformity with Sun's Conjecture. Moreover, if $g>12$, we show that any elliptic curve passing through the generic point of $M$ has degree at least 18.
The goal of this paper is the study of simple rank 2 parabolic vector bundles over a $2$-punctured elliptic curve $C$. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$. We also showcase a special curve $Γ$ isomorphic to $C$ embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over $\mathbb{P}^1$ via a modular map which turns out to be the 2:1 cover ramified in $Γ$. We recover the geometry of del Pezzo surfaces of deg…
For any non-simply laced Lie group $G$ and elliptic curve $Σ$, we show that the moduli space of flat $G$ bundles over $Σ$ can be identified with the moduli space of rational surfaces with $G$-configurations which contain $Σ$ as an anti-canonical curve. We also construct $Lie(G)$-bundles over these surfaces. The corresponding results for simply laced groups were obtained by the authors in another paper. Thus we have established a natural identification for these two kinds of moduli spaces for any Lie group $G$.
Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.
We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.
We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are $\CC P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block-Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invaria…
Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we determine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and degree d on the curve. When r and d are coprime, we describe the topology of the real locus and give a modular interpretation of its points. We also study, for arbitrary rank and degree, the moduli space of indecomposable vector bundles of rank r and degree d, and determine its isomorphism class as a real algebraic variety.
In a recent paper Ben-Zvi and Nadler proved that the induction map from $B$-bundles of degree 0 to semistable $G$-bundles of degree 0 over an elliptic curve is a small map with Galois group isomorphic to the Weyl group of $G$. We generalize their result to all connected components of $\Bun_G$ for an arbitrary reductive group $G$. We prove that for every degree (i.e. topological type) there exists a unique parabolic subgroup such that any semistable $G$\nobreakdash-bundle of this degree has a reduction to it and moreover the induction map is small with Galois group the relative Weyl group of th…
Let $X$ be any smooth prime Fano threefold of degree $2g-2$ in $\mathbb{P}^{g+1}$, with $g \in \{3,\ldots,10,12\}$. We prove that for any integer $d$ satisfying $\left\lfloor \frac{g+3}{2} \right\rfloor \leq d \leq g+3$ the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree $d$ in $X$ is nonempty and has a component of dimension $d$, which is furthermore reduced except for the case when $(g,d)=(4,3)$ and $X$ is contained in a singular quadric. Consequently, we deduce that the moduli space of rank--two slope--stable $ACM$ bundles $\mathcal{F}_d$ on $X$ such that $\det(\ma…
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is proven that the moduli space of stable rank-2 vector bundles with Chern classes c_1=1,c_2=5 on a generic X is isomorphic to the curve of genus 7 obtained by taking an orthogonal linear section of the spinor tenfold. This is an inverse of Mukai's result on the isomorphism of a non-abelian Brill--Noether locus on a curve of genus 7 to a Fano threefold of genus 7. An explicit geometric construction of both isomorphisms and a similar result f…
Let G be a simple and simply connected complex linear algebraic group. In this paper, we discuss the generalization of the parabolic construction of holomorphic principal G-bundles over a smooth elliptic curve to the case of a singular curve of arithmetic genus one and to a fibration of Weierstrass cubics over a base B. Except for G of type E_8, the method gives a family of weighted projective spaces associated to a sum of line bundles over B. Working with the universal family of Weierstrass curves over affine two-space and its natural C^*-action, we determine the line bundles that arise in th…
We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.
We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over complex projective line with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible parabolic structure such that the parabolic degree of the Higgs bundle is 0. Our method relies on elliptic fibrations of the rational elliptic surface, an equivalence of categories between irregular Higgs bundles and some sheaves on a ruled surface, and an analysis of stability conditions.
In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study birational morphisms between projective spaces appearing as such moduli spaces.
The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is étale. As auxiliary results, the irreducibil…
In this paper, the first of a series of three, we classify holomorphic principal G-bundles over an elliptic curve, where G is a reductive group. We also study the local and global properties of the moduli space of semistable G-bundles. We identify canonical representatives for each S-equivalence class of semistable G-bundles, and study their automorphism groups.
In this note we show the existence of a family of elliptic conic bundles in P^4 of degree 8. This family has been overlooked and in fact falsely ruled out in a series of classification papers. Our surfaces provide a counterexample to a conjecture of Ellingsrud and Peskine. According to this conjecture there should be no irregular m-ruled surface in P^4 for m at least 2.
In this paper, we study the moduli spaces $\mathcal{M}_{δ,c_2}$ of stable rank-2 vector bundles on non-K\" ahler elliptic surfaces, thus giving a classification these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system.
This paper, the last in a series of three, studies vector bundles on an elliptic surface whose determinant has odd intersection number with a general fiber and uses this study to calculate certain coefficients of Donaldson polynomials.
New local and global non-abelian zeta functions for elliptic curves are studied using certain refined Brill-Noether loci in moduli spaces of semi-stable bundles. Examples of these zeta functions and a justification of using only semi-stable bundles are given too. We end this paper with an appendix on the so-called Weierstrass Groups for general curves, which is motivated by a construction of Euler systems from torsion points (of elliptic curves).
In this paper, we study holomorphic rank-2 vector bundles on non-K\" ahler elliptic surfaces. Our main tool for analysing these bundles is of course the spectral cover. However, given the non-Kähler condition, the elliptic surfaces we are considering do not have sections and gerbes naturally arise in this context. The spectral construction presented in this paper is a modification of the Fourier-Mukai transform for elliptic fibrations without a section. After examining some of the properties of this Fourier-Mukai transform, we give a complete classification of vector bundles on these surfaces.
In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our approach, based on the representation theory of boxes, also yields an explicit description of the corresponding universal families of simple vector bundles.
For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that $X$ has a Kähler metric if and only both Chern classes (with real coefficients) of $π$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry locally conformally Kähler (LCK, for short) metrics. We show that in the case when the Chern classes are linearly independent, $X$ carries no LCK metric.
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves…
We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. V.2: Main theorem is corrected for small finite fields.
Let $G$ be a simply connected simple algebraic group. The classical multiplicative and additive Grothendieck-Springer resolutions are simultaneous resolutions of singularities for the maps from $G$ and its Lie algebra to their invariant theory quotients by the conjugation action of $G$. In this paper, we construct an elliptic version of the Grothendieck-Springer resolution, which is a simultaneous log resolution of a family whose total space is the stack of principal $G$-bundles on an elliptic curve. Our construction extends a well-known simultaneous resolution of the coarse moduli space map f…
