Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on \mathbbP²{\mathbb{P}^2} . The top non-vanishing equivariant Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators.     

Generators of the unoriented bordism ring which are fibered over products of projective spacesRP(2 r −1)

For each positive integerk≦∞ we construct a family {M _k ⁿ } of generators of the unoriented bordims ring. The manifoldsM _k ⁿ are total spaces of fiber bundles whose base spaces are high-dimensional products of projective spaces wherer _i≦k. The fibers are themselves iterated projective bundles with maximal fiber dimension two. In the special casek=3 we obtain generatorsM ₃ ⁿ which admit approximately 7/8·n pointwise linearly independent vector fields. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

First steps in tropical intersection theory

We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in \mathbbRⁿ{\mathbb{R}^{n}} and then for “abstract” cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in \mathbbRⁿ{\mathbb{R}^{n}} .     

Vector bundles and regulous maps

Let $X$ be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on $X$ becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic $\mathbb{R }$ -vector bundle on $X$ are algebraic. We also derive that the Chern classes of any pre-algebraic $\mathbb{C }$ -vector bundles and the Pontryagin classes of any pre-algebraic $\mathbb{R }$ -vector bundle are blow- $\mathbb{C }$ -algebraic. We also provide several results on line bundles on $X$ .     

Some results on special stable vector bundles of rank 3 on algebraic curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 2 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3^2,d and w3 respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3^2,4 and a complete discussion on the structure of ω3^3,2/3g＋2.     

The incidence structure of subspaces with well-scaled frames

In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S - F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular.Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±2⁰, ±2¹, ±2², …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.     

A relation between the parabolic Chern characters of the de Rham bundles

In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism for . We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.     

The Thom isomorphism for nonorientable bundles

The classical theory of Thom isomorphisms is extended to nonorientable vector bundles. The properties of orientation sheaves of bundles and of the Thom and Euler classes τ and e with respect to projections, fiber maps, Cartesian products, and Whitney sums of bundles are studied. The validity of standard constructions used in the applications of the classes τ and e is confirmed. It is shown that the Thom isomorphisms, together with their form, are consequences of the Poincaré duality. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 55–103, 2003.     

Inflectional loci of scrolls

Let be a scroll over a smooth curve C and let denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.      

Free abelian covers, short loops, stable length, and systolic inequalities

We explore the geometry of the Abel–Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus . Applying M. Gromov’s filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, , classified by f. We show that the finite-dimensionality of the rational homology of is a sufficient condition for the homological non-triviality of the fiber. When applied to nilmanifolds, our “fiberwise” inequality typically gives stronger information than the filling inequality for X itself. In dimension 3, we present a sufficient non-vanishing condition in terms of Massey products. This condition holds for certain manifolds that do not fiber over their Jacobi torus, such as 0-framed surgeries on suitable links. Our systolic inequality applies to surface bundles over the circle (provided the algebraic monodromy has 1-dimensional coinvariants), even though the Massey product invariant vanishes for some of these bundles. A. I. Suciu was supported by the National Science Foundation (grant DMS-0105342).     

On mappings in the Orlicz–Sobolev classes on Riemannian manifolds

We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz–Sobolev classes W_loc^1,j W_{loc}^{1,\varphi } under a condition of the Calderon type for the function φ and, in particular, in the Sobolev classes W_loc^1,p W_{loc}^{1,p} for p > n − 1.     

Teichmüller Spaces and Negatively Curved Fiber Bundles

In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres ${\mathbb{S}^k}In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres \mathbbS^k{\mathbb{S}^k}, the forgetful map F_\mathbbS^k{F_{\mathbb{S}^k}} is not one-to-one. This result follows from Theorem A, which proves that the quotient map MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} is the space of negatively curved metrics on M and T  ^{sec < 0} (M) = MET  ^{sec < 0} (M)/ DIFF₀(M){\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)} is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M. In particular we conclude that T  ^{sec < 0} (M){\mathcal{T}^{\,\,sec <0 }(M)} is, in general, not connected. Two remarks: (1) the nontrivial elements in p_kMET  ^{sec < 0} (M){\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)} constructed in [FO3] have trivial image by the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; (2) the nonzero classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} constructed in [FO2] are not in the image of the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; the nontrivial classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} given here, besides coming from MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} and being harder to construct, have a different nature and genesis: the former classes – given in [FO2] – come from the existence of exotic spheres, while the latter classes – given here – arise from the non-triviality and structure of certain homotopy groups of the space of pseudo-isotopies of the circle \mathbbS¹{\mathbb{S}^1}. The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in Theorem B.     

Simply connected symplectic 4-manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript><Superscript>+</Superscript>=1 and <Emphasis Type="Italic">c</Emphasis><Subscript>1</Subscript><Superscript>2</Superscript>=2

In this article we construct a new simply connected symplectic 4-manifold with b₂⁺=1 and c₁²=2 which is homeomorphic, but not diffeomorphic, to a rational surface by using rational blow-down technique. As a corollary, we conclude that a rational surface admits an exotic smooth structure. Mathematics Subject Classification (2000) 53D05, 14J26, 57R55, 57R57     

Chern classes of tropical vector bundles

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in Torchiani, C., Line Bundles on Tropical Varieties, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern, 2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.     

Bicomplex hyperfunctions

In this paper, we consider bicomplex holomorphic functions of several variables in _boxclose C^n{{\mathbb B}{\mathbb C}^n} .We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space \mathbb Rⁿ{{\mathbb R}^n} within the bicomplex space \mathbb B\mathbb Cⁿ{{\mathbb B}{\mathbb C}^n}, and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1.     

Natural connections on the bundle of Riemannian metrics

Let be the bundles of linear frames and Riemannian metrics of a manifold M, respectively. The existence of a unique Diff M-invariant connection form on , which is Riemannian with respect to the universal metric on , is proved. Applications to the construction of universal Pontryagin and Euler forms, are given. Authors’ addresses: R. Ferreiro Pérez, Departamento de Economía Financiera y Contabilidad I, UCM, Campus de Somosaguas, 28223 Madrid, Spain; J. Mu?oz Masqué, Insituto de Física Aplicada, CSIC, C/Serrano 144, 28006 Madrid, Spain     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.     

The PENROSE Transform for Bundles Non-Trivial on the General Line

Let L₀ be a fixed projective line in CP ³ and let M ? C ⁴ be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP ³ with L ∩ L ₀ ?? Ø. Let D ? M , D ′ ? CP ³/ L ₀ be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.     

On Some Rational Triangles

We prove that every set of n ≥ 3 points in \mathbbR²{\mathbb{R}^2} can be slightly perturbed to a set of n points in \mathbbQ²{\mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that A^￠￠, B^￠￠, C^￠￠ ? \mathbbQ²{A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε.