Natural Line Bundle Research Articles

Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds

Let X be a compact Calabi–Yau 3-fold, and write M,M‾ for the moduli stacks of objects in coh(X),Dbcoh(X). There are natural line bundles KM→M, KM‾→M‾, analogues of canonical bundles. Orientation data on M,M‾ is an isomorphism class of square root line bundles KM1/2,KM‾1/2, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman [35, §5] in their theory of motivic Donaldson–Thomas invariants, and is also important in categorifying Donaldson–Thomas theory using perverse sheaves.We show that natural orientation data can be constructed for all compact Calabi–Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi–Yau 3-folds X that admit a spin smooth projective compactification X↪Y. This proves a long-standing conjecture in Donaldson–Thomas theory.These are special cases of a more general result. Let X be a spin smooth projective 3-fold. Using the spin structure we construct line bundles KM→M, KM‾→M‾. We define spin structures on M,M‾ to be isomorphism classes of square roots KM1/2,KM‾1/2. We prove that natural spin structures exist on M,M‾. They are equivalent to orientation data when X is a Calabi–Yau 3-fold with the trivial spin structure.We prove this using our previous paper [33], which constructs ‘spin structures’ (square roots of a certain complex line bundle KPE•→BP) on differential-geometric moduli stacks BP of connections on a principal U(m)-bundle P→X over a compact spin 6-manifold X.

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants includes: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle.The first part contains the definition of ‘generic structure’ based on the work of Laufer [14]. The second technical ingredient is the Abel map developed in [21].The results can be compared with certain parallel statements from the Brill–Noether theory and from the theory of Abel map associated with projective smooth curves (see e.g. [1] and [6]), though the tools and machineries are very different.

Rigidity properties of holomorphic Legendrian singularities

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold.

Relations in the cohomology ring of the moduli space of flat SO(2n + 1)-connections on a Riemann surface.

We consider the moduli space of flat SO(2n + 1)-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalizing a conjecture of Newstead.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

On a relation between spectral theory of lens spaces and Ehrhart theory

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO(n) is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.

Extension groups of tautological sheaves on Hilbert schemes

We give formulas for the extension groups between tautological sheaves and more generally between tautological objects twisted by natural line bundles on the Hilbert scheme of points on a smooth quasi-projective surface. As a consequence we observe that a tautological object can never be a spherical or P n \mathbb {P}^n -object. We also provide a description of the Yoneda products.

Teichmüller curves, triangle groups, and Lyapunov exponents

We construct a Teichmuller curve uniformized by a Fuchsian triangle group commensurable to �(m, n, ∞) for every m, n ≤ ∞. In most cases, for example when m 6 n and m or n is odd, the uniformizing group is equal to the triangle group �(m, n, ∞). Our construction includes the Teichmuller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find billiard tables that generate these Teich- muller curves. We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmuller curve. We determine the Lyapunov exponents for the Teichmuller curves we construct.

The Volume of a Differentiable Stack

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the groupoid. Sections of a square root of this line bundle constitute an "intrinsic Hilbert space" of the stack.

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kahler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.

Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle

Let $\pi \colon X \to B$ be a holomorphic submersion between compact K\ahler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature K\ahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature K\ahler metric on $X$. The condition involves the $\CM$-line bundle---a certain natural line bundle on $B$---which is proved to be nef. Knowing this, the condition is then implied by $c_1(B) <0$. This provides infinitely many K\ahler manifolds of constant scalar curvature in every dimension, each with K\ahler class arbitrarily far from the canonical class.

$A_{\infty}$-structures, Brill–Noether loci and the Fourier–Mukai transform

We use the -formalism to study variation in cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of some singular points in twisted Brill–Noether loci in the moduli spaces of vector bundles on a curve. Another application is a computation of the Fourier–Mukai transform of some natural line bundles on symmetric powers of a curve.

Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems

AbstractThis paper treats the moduli spaceg,1(Λ) of representations of the fundamental group of a Riemann surface of genusgwith one boundary component which send the loop around the boundary to an element conjugate to exp Λ, where Λ is in the fundamental alcove of a Lie algebra. We construct natural line bundles overg,1(Λ) and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rankkand degreed.

Conformal blocks and generalized theta functions

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks, which is defined in terms of representations of the Lie algebra sl(r, C((z))).

On the natural line bundle on the moduli space of stable parabolic bundles

We construct the natural holomorphic line bundle on the moduli space of stable parabolic bundles on a compact marked Riemann surface, which is the prequantum line bundle for the Chern-Simons gauge theory. The fusion rule in the Chern-Simons gauge theory can be viewed as the existence condition of this line bundle.

Half density quantization of the moduli space of flat connections and witten's semiclassical manifold invariants

LET PpB denote the moduli space of flat connections on (the trivial SU(2) bundle on) a compact, oriented two-manifold P. The space s”, is a stratified space containing an open dense set Yg which is a symplectic manifold of (real) dimension 6g - 6. There exists a natural line bundle 9 -+ ,Fb, with hermitian metric ( , ), whose restriction to Y9 has a connection V with curvature given by the symplectic form o on Y9. If we equip the two-manifold Xg with a metric, the resulting Hodge star operator turns ,Yg into a Klhler manifold, with w as the Kahler form. Setting aside for the moment concern with the singularities of Pg,, we arrive at the usual setting for geometric quantization: we are given a symplectic manifold (qg, o), a line bundle 3’ + 9, with connection of curvature CO, and a polarization of the space of sections of 3. The quantization CYY~(C~) = H”(Pg, Yk) of this system has been of importance in attempts to construct a topological field theory. A topological field theory would assign to every two-manifold Cg the quantization XDk(Cg) of the system (Pg, CO, 9, V), which must be proved independent of the choice of Kahler structure on -Fg, coming from a choice of metric on X9. The space Xk(Xg) can be given the structure of a Hilbert space with the metric given by ((s1,s~)) = j(si>s~)~~~-~. A topological field theory would also assign an element sN of the quantization Xk(Xg) to every three-manifold N3 whose boundary is given by 8N3 = Cg. If this assignment satisfied the axioms of topological field theory, the space Zk(Eg) would be equipped with a representation of the group Diff(Xg) of diffeomorphisms of Eg. Furthermore, we would obtain invariants of closed three-manifolds as follows. Let N be a closed three-manifold, and let N = H be a