Narrow Results

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve $M$, we construct explicitly a bundle ${{𝒲}}_M$ with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.

Generalized Calabi–Gray manifolds are non-Kähler complex manifolds with very explicit geometry yet not being homogeneous. In this note, we demonstrate how generalized Calabi–Gray manifolds can be used to answer some questions in non-Kähler geometry.

Developing ideas of [B. L. Feigin, Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, in Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vols. 1 and 2 (Mathematical Society of Japan, Tokyo, 1991), pp. 71–85], we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold $M$. Graded differential cohomology of a sheaf of Lie algebras ${𝒢}$ via the cosimplicial cohomology of ${𝒢}$-formal series for any covering by Stein spaces on $M$ is computed. A relation between cosimplicial cohomology (on a special set of open domains of $M$) of formal series of an infinite-dimensional Lie algebra ${𝒢}$ and singular cohomology of auxiliary manifold associated to a ${𝒢}$-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.

An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u : M → [0, π/4] which are plurisubharmonic in M and such that u(p) = 0 for each p ∈ V. If admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E_{V, M} on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge–Ampère models introduced by Lempert and Szöke in [18].

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are $1^{\mathrm{\text{st}}}$ order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property.

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds.

The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space , but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a complex structure on . When the latter is a complex-analytic manifold admitting just one complex structure, there is a unique quantization whose classical limit is . Then the notion of coherence is the same for all observers. However, when admits two or more nonbiholomorphic complex structures, there is one different quantization per different complex structure on . The lack of analyticity in transforming between nonbiholomorphic complex structures can be interpreted as the loss of quantum-mechanical coherence under the corresponding transformation. Observers using one complex structure perceive as coherent the states that other observers, using a different complex structure, do not perceive as such. This is the notion of a quantum-mechanical duality transformation: the relativity of the notion of a quantum.

In this paper, we construct the fuzzy (finite-dimensional) analogs of the conifold Y⁶ and its base X⁵. We show that fuzzy X⁵ is (the analog of) a principal U(1) bundle over fuzzy spheres and explicitly construct the associated monopole bundles. In particular, our construction provides an explicit discretization of the spaces T^{κ, κ} and T^{κ, 0}.

The classical phase of the matrix model of 11-dimensional M-theory is complex, infinite-dimensional Hilbert space. As a complex manifold, the latter admits a continuum of nonequivalent, complex-differentiable structures that can be placed in 1-to-1 correspondence with families of coherent states in the Hilbert space of quantum states. The moduli space of nonbiholomorphic complex structures on classical phase space turns out to be an infinite-dimensional symmetric space. We argue that each choice of a complex differentiable structure gives rise to a physically different notion of an elementary quantum.