Canonical Nonlinear Connections on Jet Bundles of First Order
Mircea Neagu
备注：
7 pages, Versions of this paper were presented at: -Workshop on Differential Geometry, Global Analysis and Lie Algebras, Aristotle University of Thessaloniki, Greece, Aug. 28-Sept. 2, 2000; -Seminar "Gh. Vranceanu", University of Bucharest, Romania, November, 2000
发表信息：
"d-Metrics and nonlinear connections on 1-jet fibre bundles", Seminarul de Mecanica 84 (2003), 1-12
The aim of this paper is to open the problem of construction of a nonlinear connection $Γ=(M^{(i)}_{(α)β}, N^{(i)}_{(α)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(α)(β)}_{(i)(j)}=h^{αβ}g_{ij}$, possibly provided by multi-time dependent quadratic Lagrangians coming from various branches of theoretical physics: bosonic strings theory, electrodynamics or elasticity.
This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general Kähler classes. We give some examples of explicit families, in particular, a formula for the two dimensional family of asymptotically conical metrics on the canonical bundle of $F_{1,2}$.
In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by primary and secondary structures belong to the same atlas. We prove that double jet bundles can be considered as a quotient of second order jet bundle. We show that there exists a natural involution that interchanges between primary and secondary vector bundle structures on double…
The affine bundle theorem in synthetic differential geometry of jet bundles
在我们的[Higher-order preconnections in synthetic differential geometry of jet bundles，Beiträge zur Algebra und Geometrie，45（2004），677-696]中，我们已经建立了在无穷小空间D的条件下对jet bundle进行合成的仿射束定理。在我们的成功[合成微分几何高阶全微分，出现在拓扑学手册等几何Différentielle Catégoriques]我们介绍了另一种合成方法，以喷射束方面的无穷小空间D_{n}的，并比较了它与前一种方法都在一般微线性设置和在有限维设置。然而，我们在有限维条件下的比较是不完全的，因为我们使用维数计数技术已经默认了用D和D_{n}表示的喷流丛是向量丛，除非给定的丛已经是向量丛，否则情况不是这样。本文的主要目的是建立仿射丛定理在我们的第二个综合方法喷丛，然后比较不仅基于这两种方法的喷丛，但仿射丛作为一个整体构建在一般的微线性设置和有限维设置。这就完成了我们在有限维环境中两种方法之间的比较。
In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
We identify and study a matrix algebra consisting of Pascal-type matrices. The generator of the matrix algebra is shown to well define a canonical bundle map, called the Pascal map on jet bundles, and we use it to give an intrinsic definition of point-wise contact between Hermitian vector bundles in terms of unitary equivalence of the Pascal maps.
We discuss the Morse estimates for the curvature of several metrics on Semple weighted projective bundle over a projective variety. Following Demailly works on holomorphic Morse inequalities we show an analogue of his results along the Green-Griffiths conjecture for invariant jets.
For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between holomorphic and smooth gerbes. For example, we introduce an Atiyah class for gerbes and prove a Koszul-Malgrange type theorem.
Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic…
In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).
Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumer…
We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an explicit formula in terms of the Levi-Civita connection is achieved. Our methods include the use of complex integral forms and the recently developed theory of superholonomy.
A family of holomorphic vector bundles is constructed on a complex manifold $X$. The space of the holomorphic sections of these bundles are calculated in certain cases. As an application, if $X$ is an $N$-dimensional compact Kähler manifold with holonomy group $SU(N)$, the space of holomorphic vector fields on its jet scheme $J_m(X)$ is calculated. We also prove that the space of the global sections of the chiral de Rham complex of a K3 surface is the simple $N=4$ superconformal vertex algebra with central charge $6$.
We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on $(m,0)$ forms, i.e., sections of the canonical bundle of Kähler manifolds, where $m$ is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of $(m, 0)$ forms, which involves only the Ricci curvature and the gradient of the scalar curvature.
Mathematics Program and Center for Quantum and Topological Systems, New York University Abu Dhabi（数学项目与量子拓扑系统中心，阿布扎比纽约大学）；Institute of Mathematics of the Czech Academy of Sciences, Prague（捷克科学院数学研究所）
This paper introduces $\infty$- and $n$-fold vector bundles as special functors from the $\infty$- and $n$-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of $n$-fold vector bundles and we prove that any $n$-fold vector bundle admits a non-canonical isomorphism to a decomposed $n$-fold vector bundle. A colimit argument then shows that $\infty$-fold vector bundles admit as well non-canonical decompositions. For the convenience of the reader, the case of triple vector bundles is discussed in detail.
