Holomorphic Triples on Elliptic Curves

Here we study the α-stability for holomorphic triples over curves of genus g = 1. We provide necessary and sufficient conditions for the moduli space of α-stable triples to be non-empy and, in these cases, we show that it is smooth and irreducible. Received: January 10, 2007.     

Variations of hodge structures of rank three k-Higgs bundles and moduli spaces of holomorphic triples

Geometriae Dedicata - There is an isomorphism between the moduli spaces of $$\sigma $$ -stable holomorphic triples and some of the critical submanifolds of the moduli space of k-Higgs bundles of...     

Moduli Problems of Sheaves Associated with Oriented Trees

To every oriented tree we associate vector bundle problems. We define semistability concepts for these vector bundle problems and establish the existence of moduli spaces. As an important application, we obtain an algebraic construction of the moduli space of holomorphic triples.     

Holomorphic Line Bundles over a Tower of Coverings

We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furthermore, we obtain a variance estimate for those random zero currents, which yields the almost sure convergence under some geometric condition.     

Hodge polynomials of the moduli spaces of rank 3 pairs

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple on X consists of two holomorphic vector bundles E ₁ and E ₂ over X and a holomorphic map . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E ₁) = 3, rk(E ₂) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.      

On the range of the Berezin transform

We study the range of the Berezin transform B. More precisely, we characterize all triples (f,g,u) where f and g are non-constant holomorphic functions on the unit disc D in the complex plane and u is integrable on D such that . It turns out that there are very ‘few’ such triples. This problem arose in the study of Bergman space Toeplitz operators and its solution has application to the theory of such operators.     

Anelli Henseliani topologici - II

Summary In the present work we study some problems about henselian triples and henselization which have been introduced in our article [18]. Mainly we prove that henselian triples coincide with strong henselian triples and give a new formulation of the Hensel lemma, stronger than that we gave in [18]. Then we investigate some properties of henselian triples (changement of ideal or of topology, ecc.) and prove commutativity with quotient.

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Lavoro eseguito nell'ambito dei contratti di ricerca del Comitato Nazionale per la Matematica del C.N.R.

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Entrata in Redazione il 27 maggio 1971.     

Montel定则的进一步推广

本文研究了全纯函数族的正规性问题.利用Zalcman引理,证明了全纯函数族的几个正规定则,推广了Montel正规定则.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

Non-Hermitian Yang-Mills connections

We study Yang-Mills connections on holomorphic bundles over complex K?hler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk?hler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk?hler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K?hler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk?hler manifold and the space of rational curves in the twistor space of the ‘Mukai’ dual hyperk?hler manifold.     

On the axiomatic definition of real JB*–triples

In the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J*B–triple. These J*B–triples include real C*–algebras and complex JB*–triples. However, concerning J*B–triples, an important problem was left open. Indeed, the question was whether the complexification of a J*B–triple is a complex JB*–triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J*B–triples. In this paper we characterize those J*B–triples with a unitary element whose complexifications are complex JB*–triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J*B–triples with a unitary element which are J*B–algebras in the sense of [1] or real JB*–triples in the sense of [4].     

Some results on the moduli spaces of quiver bundles

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.      

Teichmüller spaces and holomorphic motions

The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmüller space of a closed subset of the Reimann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the “Harmonic γ-Lemma” of Bers and Royden. We also study some other applications.     

Vanishing cohomology theorems and stability of complex analytic foliations

In this paper we deal with a complex analytic foliation of a compact complex manifold endowed with a bundle-like metric and give a transversally holomorphic rigidity theorem (Theorem 9.1) for these foliations, depending on curvature conditions. We give some examples for which we study holomorphic rigidity. The classical vanishing theorems of Nakano, Griffiths and Le Potier are the main tools we use to prove our results.     

关于商高数的Jemanowicz猜想

本文研究了商高数的Jemanowicz猜想的整数解问题.利用初等数论方法,获得了该猜想的两个新结果并给出证明,推广了文献[4–8]的结果.     

Twisted holomorphic chains and vortex equations over non-compact Kähler manifolds

In this paper, we study twisted holomorphic chains and related gauge equations over non-compact Kähler manifolds. We use the heat flow method to solve the Dirichlet boundary problem for the related gauge equations, and prove a Hitchin-Kobayashi type correspondence for twisted holomorphic chain over some non-compact Kähler manifolds.     

Holomorphic transforms with application to affine processes

In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multi-dimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

Modified Bers operators and the duality of multiplicative holomorphic automorphic forms

The integral Bers operator, related to a reflection in some quasicircle, plays an important role in the theory of single-valued automorphic forms [1, 2]. Study was started in [3] of the normed spaces of measurable and multiplicative holomorphic automorphic forms for a Fuchsian group. In the present article we introduce some basic multiplicative modifications of the Bers operator and the corresponding bilinear pairing in connection with duality in the spaces of multiplicative holomorphic automorphic forms. Under study we obtain a universal norm estimate and establish selfadjointness for all operators.     

Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in \(\mathbb {P}^2\).