Natural Hermitian Research Articles

Perturbations of principal submodules in the Drury–Arveson space

We study the geometry in the perturbations of principal submodules in the Drury–Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy.

$L^2$-cohomology for affine spaces and an application to monads

Given a locally free sheaf ℱ on a projective space ℙn, a natural question is to compute the L2-cohomology of its restriction to an affine subspace ℂn=ℙn∖ℙn−1 equipped with a flat Kähler form ωℂn. This problem arises naturally when studying the geometry of moduli of monads. We identify the known construction of a natural hermitian structure on the moduli space of the associated vector bundles with the Weil–Petersson approach to a Kähler metric based on harmonic theory for Kodaira–Spencer tensors. We show that for a holomorphic family of monads on ℙ2 or on a modification ℙ ˜2 of ℙ2, there is a Kodaira–Spencer map providing the base with a Kähler structure.

A Natural Hermitian Line Bundle on the Moduli Space of Semistable Representations of a Quiver

This paper describes the construction of a natural Hermitian holomorphic line bundle on the stratified moduli space of complex representations of a finite quiver, which are semistable with respect to a fixed rational weight and have a fixed type. It is shown that the curvature of this Hermitian line bundle on each stratum of the moduli space is essentially the Kahler form of that stratum.

Kähler Forms for Families of Calabi–Yau Manifolds

Kähler–Einstein metrics for polarized families (f\colon \mathcal X \to S, \lambda_{\mathcal X/S}) of Calabi–Yau manifolds define a natural hermitian metric on the relative canonical bundle K_{\mathcal X/S} . The computation of the curvature form being equal to the pullback of the Weil–Petersson form up to a numerical constant is used for the construction of a Kähler–Einstein form on \mathcal X , whose restriction to the fibers is Ricci flat.

A new kind of Hermitian matrices for digraphs

In an earlier work, the author together with Guo [7] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory.

Twisted Hodge filtration: Curvature of the determinant

AbstractGiven a holomorphic family of compact complex manifolds and a relatively ample line bundle , the higher direct images carry a natural hermitian metric. An explicit formula for the curvature tensor of these direct images is given in [8]. We prove that the determinant of the twisted Hodge filtration is (semi‐)positive on the base S if L itself is (semi‐)positive on .

Hermitian Metric and the Infinite Dihedral Group

For a tuple A = (A1,A2,…, An) of elements in a unital Banach algebra B, the associated multiparameter pencil is A(z) = z1A1 + z2A2 + … + znAn. The projective spectrum P(A) is the collection of z ∈ ℂn such that A(z) is not invertible. Using the fundamental form ΩA = −ω * ∧ ωA, where ωA(z) = A−1(z) dA(z) is the Maurer–Cartan form, R. Douglas and the second author defined and studied a natural Hermitian metric on the resolvent set Pc(A) = ℂn \ P(A). This paper examines that metric in the case of the infinite dihedral group, D∞ = , with respect to the left regular representation λ. For the non-homogeneous pencil R(z) = I + z1λ(a) + z2λ(t), we explicitly compute the metric on Pc(R) and show that the completion of Pc(R) under the metric is ℂ2 \ {(±1, 0), (0, ±1)}, which rediscovers the classical spectra σ(λ(a)) = σ(λ(t)) = {± 1}. This paper is a follow-up of the papers by R. G. Douglas and R. Yang (2018) and R. Grigorchuk and R. Yang (2017).

Curvature of the Determinant Line Bundle for the Noncommutative Two Torus

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation $\delta_{w}\delta_{\bar{w}}\log\det(\Delta)$.

Hawking radiation and classical tunneling: A ray phase space approach

Acoustic waves in fluids undergoing the transition from sub- to supersonic flow satisfy governing equations similar to those for light waves in the immediate vicinity of a black hole event horizon. This acoustic analogy has been used by Unruh and others as a conceptual model for “Hawking radiation.” Here, we use variational methods, originally introduced by Brizard for the study of linearized MHD, and ray phase space methods, to analyze linearized acoustics in the presence of background flows. The variational formulation endows the evolution equations with natural Hermitian and symplectic structures that prove useful for later analysis. We derive a 2 × 2 normal form governing the wave evolution in the vicinity of the “event horizon.” This shows that the acoustic model can be reduced locally (in ray phase space) to a standard (scalar) tunneling process weakly coupled to a unidirectional non-dispersive wave (the “incoming wave”). Given the normal form, the Hawking “thermal spectrum” can be derived by invoking standard tunneling theory, but only by ignoring the coupling to the incoming wave. Deriving the normal form requires a novel extension of the modular ray-based theory used previously to study tunneling and mode conversion in plasmas. We also discuss how ray phase space methods can be used to change representation, which brings the problem into a form where the wave functions are less singular than in the usual formulation, a fact that might prove useful in numerical studies.

Hessian of the natural Hermitian form on twistor spaces

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperk\"ahler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic K\"ahler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperk\"ahler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic K\"ahler manifold of non-positive scalar curvature. We check that no non-K\"ahler strong KT manifold occurs as such a twistor space.

A natural hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class

Let π:X→S be a local universal family of compact Kähler manifolds with zero first Chern class over a smooth base S. The complexified Kähler cones of each manifold of the family form a holomorphic fiber bundle K→S. We show that there exists a natural hermitian metric ω on the fiber product X×SK. We then discuss the example of elliptic curves in some detail.

Limits of families of Brieskorn lattices and compactified classifying spaces

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space.

Moment Maps and Equivariant Szegö Kernels

Let M be a connected n-dimensional complex projective manifold and consider an Hermitian ample holomorphic line bundle (L; hL) on M. Suppose that the unique compatible covariant derivative ▽L on L has curvature -2πiΩ where Ω is a Kahler form. Let G be a compact connected Lie group and μ: G x M → M a holomorphic Hamiltonian action on (M; Ω ). Let \frac g be the Lie algebra of G, and denote by Φ : M → g* the moment map. Let us also assume that the action of G on M linearizes to a holomorphic action on L; given that the action is Hamiltonian, the obstruction for this is of topological nature [GS1]. We may then also assume that the Hermitian structure hL of L, and consequently the connection as well, are G-invariant. Therefore for every k ∈ N there is an induced linear representation of G on the space H0(M;L⊗k) of global holomorphic sections of L⊗k. This representation is unitary with respect to the natural Hermitian structure of H0(M;L⊗k) (associated to Ω and hL in the standard manner). We may thus decompose H0(M;L⊗k) equivariantly according to the irreducible representations of G. The subject of this paper is the local and global asymptotic behaviour of certain linear series defined in terms this decomposition. Namely, we shall first consider the asymptotic behaviour as k →+ ∞ of the linear subseries of H0(M;L⊗k) associated to a single irreducible representation, and then of the linear subseries associated to a whole ladder of irreducible representations. To this end, we shall estimate the asymptoptic growth, in an appropriate local sense, of these linear series on some loci in M defined in terms of the moment map Φ.

Natural quark mass patterns.

We incorporate the idea of natural mass matrices into the construction of phenomenologically viable quark mass matrix patterns. The general texture pattern for natural Hermitian mass matrices is obtained and several applications of this result are made.