Space Of Holomorphic Sections Research Articles

Energy of sections of the Deligne\u2013Hitchin twistor space

We study a natural functional on the space of holomorphic sections of the Deligne–Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating S^1-action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy functional is basically given by the Willmore energy of corresponding equivariant conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne–Hitchin moduli space from the space of twistor lines.

Universality results for zeros of random holomorphic sections

In this work we prove a universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space X X . Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections. In the case when X X is a compact Kähler manifold, we also prove an off-diagonal exponential decay estimate for the Bergman kernels of a sequence of positive line bundles on X X .

Noncommutative complex geometry of the quantum projective space

We define holomorphic structures on canonical line bundles of the quantum projective space C P q ℓ and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of C P q ℓ is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for C P q 1 and C P q 2 .

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C ∗ -algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach–Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.

Quantum superalgebra representations on cohomology groups of non-commutative bundles

Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic sub-supergroup to the category of locally finite modules of the quantum general linear supergroup. The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles. We explicitly compute the cohomology groups, which are given in terms of well understood modules over the quantized universal enveloping algebra of the general linear superalgebra.

A Hitchin-Kobayashi Correspondence for Coherent Systems on Riemann Surfaces

A coherent system (E, V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (that is, the Hermitian–Einstein equation with an additional zeroth order term), and the other is an orthonormality condition on a frame for the subspace V⊂H0(E).

Involutions on moduli spaces and refinements of the Verlinde formula

The moduli space M of semi-stable rank 2 bundles with trivial deter- minant over a complex curvecarries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle L. We ob- tain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in M, which we also determine completely using the order 4 Weil pairing. The lifted involutions act on the spaces of holomorphic sections of powers of L, whose dimensions are given by the Verlinde formula. We compute the characters of these vector spaces as representations of the group generated by our lifts, and we obtain an explicit isomorphism (as group representations) with the combinatorial-topological TQFT-vector spaces of (BHMV). As an application, we describe a 'brick decomposition', with explicit dimension formulas, of the Verlinde vector spaces. We also obtain similar results in the twisted (i.e., degree one) case.

Explicit realization of irreducible representations of classical compact lie groups in the spaces of sections of line bundles

We use the Borel-Weil scheme for the construction of irreducible representations of compact Lie groups in the spaces of holomorphic sections of line bundles over homogeneous manifolds. We find the explicit form of the space of sections and construct an invariant scalar product. We show that the space of holomorphic sections locally satisfies the Zhelobenko indicator system.

Quantum line bundles on S2 and method of orbits for SUq(2)

The quantum sphere is covered by two quantum complex planes. Quantum line bundle on S2 is introduced by specifying a transition function. The structure of the quantum complex planes enables one to construct representations of SUq(2) acting in spaces of holomorphic sections. In the quantum case, dressing transformation replaces coadjoint action.

The Pfaffian line bundle

We analyze the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold. Using the infinite dimensional relative Pfaffian, we produce a Fock space structure on the space of holomorphic sections of the dual of this bundle. On this Fock space, an explicit and rigorous construction of the spin representations of the loop groupsLOn is given. We also discuss and prove some facts about the connection between the Pfaffian line bundle over the Grassmannian and the Pfaffian line bundle of a Dirac operator.

The singularities of H-space

The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.

On Higher Order Weierstrass Points

Let S be a compact Riemann surface of genus g > 2, let X be a holomorphic line bundle of positive Chern class, and suppose that X admits holomorphic sections which have no common zero. Let F(S, x) = the space of holomorphic sections of x, and let ry(X) = the complex dimension of F(S, x). Let %n = ... X A) X* n times. Define W.(X) to be the set of points p of S for which there exists an element a of F(S, \n) such that the order of c at p is not less than y(Xn) Let W(X) be the union of the W,(X). Our main result was conjectured by Lipman Bers for the case X = a, the canonical bundle: