Vector Bundles on Certain Non-algebraic Surfaces

It is well-known that every holomorphic vector bundle is filtrable on a projective algebraic     

η-Invariant and Flat Vector Bundles

We present an alternate definition of the mod Z component of the Atiyah-Patodi-Singer η invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer.     

η-Invariant and Flat Vector Bundles

Abstract We present an alternate definition of the mod Z component of the Atiyah-Patodi-Singer η invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer. * Project supported by the Cheung-Kong Scholarship of the Ministry of Education of China and the 973 Project of the Ministry of Science and Technology of China. (Dedicated to the memory of Shiing-Shen Chern)     

Some Results on Holomorphic Vector Bundles Over General Hopf Manifolds

1 Introduction In recent years, holomorphic vector bundles on non-algebraic manifolds received increased attention. The main problem is to determine which continuous complex vector bundle carries a holomorphic structure or furthermore a holomorphic filtrable structure. D. Mall studied vector bundles on the primary Hopf manifolds。     

Some Results of Holomorphic Vector Bundles on Non-primary Hopf Manifolds

In recent years,there are a lot of work on holomorphic vector bundles on non-algebraic     

Pseudo-Effective Vector Bundles with Vanishing First Chern Class on Astheno-K¨ahler Manifolds*

Let E be a holomophic vector bundle over a compact Astheno-K¨ahler manifold (M, ω). The authors would prove that E is a numerically flat vector bundle if E is pseudoeffective and the first Chern class c^BC₁ (E) is zero.     

The Structure of Vector Bundles on Non-primary Hopf Manifolds∗

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X, with trivial pull-back to Cⁿ ? {0}. The authors show that there exists a line bundle L over X such that E ? L has a nowhere vanishing section. It is proved that in case dim(X) ≥ 3, π^?(E) is trivial if and only if E is filtrable by vector bundles. With the structure theorem, the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E.     

Action of μq(g) on Its Positive Part μq+(g)

In this paper,two kinds of skew derivations of a type of Nichols algebras are introduced,and then the relationship between them is investigated.In particular they satisfy the quantum Serre-relations.Therefore,the algebra generated by these derivations and corresponding automorphisms is a homomorphic image of the Drinfeld-Jimbo quantum enveloping algebra μq(g),which proves the Nichols algebra becomes aμq(g)-module algebra.But the Nichols algebra considered here is exactly μq+(g),namely,the positive part of the Drinfeld-Jimbo quantum enveloping algebra μq(g),it turns out that μq+(g)is a μq(g)-module algebra.     

Submanifolds with Parallel Mean Curvature Vector

In a previous paper[1],S.T.Yau proved the following theorem:Theorem A Let M~n be an n-dimensional compact submanifold with parallel meancurvature in S~n p with p>1.If(3 n~(1/2)-1/p-1)S≤n,then M~n lies in a totally geodesicS~n 1.Lemma1[2]If a given set of n 1(n≥2)real numbers a_1,…,a_n and k satisfy thein(?)ality     

(U(p, q), U(p − 1, q)) is a generalized Gelfand pair

Denote by G = U(p, q) the orthogonal group of the sesqui-linear quadratic form on and let H ₁ = U(p − 1, q) be the stabilizer of the first unit vector e ₁. Let H ₀ = U(1) and set H = H ₀ × H ₁. Define the character χ _l of H by where . Define the anti-involution σ on G by . In this note we show that any distribution T on G satisfying T(h ₁ gh ₂) = χ _l (h ₁ h ₂) T(g) (g ∈ G; h ₁, h ₂ ∈ H) is invariant under the anti-involution σ. This result implies that (G,  H ₁) is a generalized Gelfand pair.      

Erratum to: Paracontact Tangent Bundles with Cheeger–Gromoll Metric

Mediterranean Journal of Mathematics -     

Vectors bundles with theta divisors I—Bundles on Castelnuovo curves

In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g ≥ 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g ≥ 2 which extend to semistable vector bundles on any flat Castelnuovo degeneration of the general curve admit a theta divisor. Received: 8 August 2008     

Chen’s Theorem with Small Primes

Let N be a sufficiently large even integer. Let p denote a prime and P2 denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N - p+ P2 (p ≤ AT0.945) is solvable.     

K-Convergent Sequences in Topological Vector Spaces with a Basis

Ｋ┐ＣｏｎｖｅｒｇｅｎｔＳｅｑｕｅｎｃｅｓｉｎＴｏｐｏｌｏｇｉｃａｌＶｅｃｔｏｒＳｐａｃｅｓｗｉｔｈａＢａｓｉｓ＊）ＷｕＪｕｎｄｅ（武俊德）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＤａｑｉｎｇＰｅｔｒｏｌｅｕｍＩｎｓｔｉｔｕｔｅ，Ａｎｄａ，...     

Lp-Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.     

On Surfaces of General Type with p g = q = 1 Isogenous to a Product of Curves

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F)/G. In this article, we classify the surfaces of general type with p_g = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces 𝔐_I, 𝔐_II, 𝔐_IV are irreducible, whereas 𝔐_III is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples.