On Bott–Chern forms and their applications
نویسندگان
چکیده
We use Chern–Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every ∂̄∂-exact real form of the type (k, k) on an n-dimensional complexmanifold X arises as a difference of the Chern character forms of trivial Hermitian vector bundles with canonical connections, and that modulo Im ∂ + Im ∂̄ every real form of type (k, k), k < n, arises as a Bott– Chern form for two Hermitian metrics on some trivial vector bundle over X . The latter result is a complex manifold analogue of Proposition 2.6 in the paper by Simons and Sullivan (Am Math Soc 11:579–599, 2010). As an application, we obtain an explicit formula for the Bott–Chern form of a short exact sequence of holomorphic vector bundles considered by Bott and Chern (Acta Math 114:71–112, 1965), for the case when the first term is a line bundle. Mathematics Subject Classification (2010) 32L05 · 32Q · 58A10 The work of L. T. was partially supported by the NSF grants DMS-0705263 and DMS-1005769. V. P. Pingali Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, Baltimore, MD 21218, USA e-mail: [email protected] L. A. Takhtajan (B) Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA e-mail: [email protected] L. A. Takhtajan Euler Mathematical Institute, Pesochnaya nab. 10, St. Petersburg 197022, Russia 123 Author's personal copy
منابع مشابه
On the arithmetic Chern character
of the underlying vector bundles on X , (i.e. in which we ignore the hermitian metrics). Then the difference ĉh(E0) + ĉh(E2)− ĉh(E1), is represented by a secondary characteristic class c̃h first introduced by Bott and Chern [1] and defined in general in [2]. These Bott-Chern forms measure the defect in additivity of the Chern forms associated by Chern-Weil theory to the hermitian bundles in the ...
متن کاملBott-Chern Forms and Arithmetic Intersections
Let E : 0 → S → E → Q → 0 be a short exact sequence of hermitian vector bundles with metrics on S and Q induced from that on E. We compute the Bott-Chern form φ̃(E ) corresponding to any characteristic class φ, assuming E is projectively flat. The result is used to obtain a new presentation of the Arakelov Chow ring of the arithmetic Grassmannian.
متن کاملTransgression on Hyperkähler Manifolds and Generalized Higher Torsion Forms
Transgression of the characteristic classes taking values in the differential forms is a reach source of the interesting algebraic objects. The examples include Chern-Simons and Bott-Chern forms which are given by the transgression of the Chern character form. Chern-Simons forms are defined for a vector bundle over an arbitrary real manifold and are connected with the representation of combinat...
متن کاملAnalytic Cycles, Bott-chern Forms, and Singular Sets for the Yang-mills Flow on Kähler Manifolds
It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kähler manifolds is completely determined by the Harder-NarasimhanSeshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves. We derive a ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014