scholarly journals The Hodge theory of flat vector bundles on a complex torus

1982 ◽  
Vol 271 (1) ◽  
pp. 117-117
Author(s):  
Jerome William Hoffman
Keyword(s):  
Vector Bundles ◽  
Hodge Theory ◽  
Complex Torus

scholarly journals A theorem of Matsushima

1974 ◽  
Vol 54 ◽  
pp. 123-134 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Complex Torus ◽  
Frobenius Endomorphism

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

scholarly journals A Note on Holomorphic Vector Bundles Over Complex Tori

1971 ◽  
Vol 41 ◽  
pp. 101-106 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Quotient Group ◽  
Vector Bundles ◽  
Additive Group ◽  
Complex Vector ◽  
Complex Torus ◽  
Holomorphic Vector Bundles ◽  
Complex Tori

Let (ω: Z2r→Cr be an isomorphism of the free additive group of rank 2r into the complex vector n-space such that the quotient group T = Crω/(Z2r) is compact, i.e., Tω is a complex torus.

scholarly journals Heisenberg groups and holomorphic vector bundles over a complex torus

1976 ◽  
Vol 61 ◽  
pp. 161-195 ◽  
Author(s):  
Yozo Matsushima
Keyword(s):  
Vector Space ◽  
Vector Bundles ◽  
Hermitian Form ◽  
Complex Vector Space ◽  
Heisenberg Groups ◽  
Complex Vector ◽  
Complex Torus ◽  
Holomorphic Vector Bundles

Let V be a complex vector space of dimension n, L a lattice of V and E = V/L a complex torus. Let H be a Hermitian form on V. We introduce a multiplication in L × C* by

scholarly journals Positivity of vector bundles and Hodge theory

2021 ◽  
pp. 2140008
Author(s):  
Mark Green ◽  
Phillip Griffiths
Keyword(s):  
Differential Geometry ◽  
Vector Bundles ◽  
Special Property ◽  
Hodge Theory ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Curvature Invariants ◽  
Norm Property ◽  
Expository Paper

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are [Formula: see text] order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property.

scholarly journals Asymptotic Hodge theory of vector bundles

2015 ◽  
Vol 23 (3) ◽  
pp. 559-609
Author(s):  
Benoit Charbonneau ◽  
Mark Stern
Keyword(s):  
Vector Bundles ◽  
Hodge Theory

scholarly journals Toric vector bundles: GAGA and Hodge theory

2020 ◽  
Author(s):  
Jonas Stelzig
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Hodge Theory ◽  
Algebraic Construction ◽  
Smooth Base

Abstract We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques. Both results make use of the Rees-bundle construction.

scholarly journals A geometrical characterization of a class of holomorphic vector bundles over a complex torus

1976 ◽  
Vol 61 ◽  
pp. 197-202 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Vector Bundle ◽  
Heisenberg Group ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Projective Bundle ◽  
Geometrical Characterization ◽  
Complex Torus ◽  
Holomorphic Vector Bundles ◽  
Holomorphic Representation

This note is to be a supplement of the preceeding paper in the journal by Matsushima, settling a question raised by him. In his paper he associates a holomorphic vector bundle over a complex torus to a holomorphic representation of what he calls Heisenberg group. We shall show that a simple holomorphic vector bundle is determined in this manner if and only if the associated projective bundle admits an integrable holomorphic connection. A theorem by Morikawa ([3], Theorem 1) is the motivation of this problem and is somewhat strengthened by our result.

scholarly journals Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles

2019 ◽  
Vol 155 (2) ◽  
pp. 289-323 ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Thomas Peternell ◽  
Behrouz Taji
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Hodge Theory ◽  
Resolution Of Singularities ◽  
Independent Interest ◽  
Restriction Theorem ◽  
Projective Varieties ◽  
To Come ◽  
Smooth Locus

We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety$X$and any resolution of singularities, any vector bundle on the resolution that appears to come from$X$numerically, does indeed come from $X$. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.

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