scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

scholarly journals A Survey on Seifert Fiber Space Theorem

ISRN Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jean-Philippe Préaux
Keyword(s):  
Fiber Space ◽  
Seifert Fiber Space ◽  
History Of

We review the history of the proof of the Seifert fiber space theorem, as well as its motivations in 3-manifold topology and its generalizations.

scholarly journals RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450006 ◽  
Author(s):  
GAUTAM BHARALI ◽  
INDRANIL BISWAS
Keyword(s):  
Riemann Surface ◽  
Special Form ◽  
General Class ◽  
Specific Information ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Fiber Space ◽  
Rigidity Result ◽  
Holomorphic Map ◽  
Genus 2

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.

scholarly journals ON THE CONCEPT OF FIBER SPACE

1955 ◽  
Vol 41 (11) ◽  
pp. 956-961 ◽  
Author(s):  
W. Hurewicz
Keyword(s):  
Fiber Space

scholarly journals A pathological fiber space

1968 ◽  
Vol 12 (4) ◽  
pp. 623-625
Author(s):  
Gerald S. Ungar
Keyword(s):  
Fiber Space

scholarly journals Proof of the radical conjecture for homogeneous Kähler manifolds

1989 ◽  
Vol 114 ◽  
pp. 77-122 ◽  
Author(s):  
Josef Dorfmeister
Keyword(s):  
Lie Algebra ◽  
Kähler Manifold ◽  
Transitive Group ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fiber Space ◽  
Kahler Manifold ◽  
Solvable Ideal ◽  
Simply Connected ◽  
Bounded Homogeneous Domain

In 1967 Gindikin and Vinberg stated the Fundamental Conjecture for homogeneous Kähler manifolds. It (roughly) states that every homogeneous Kähler manifold is a fiber space over a bounded homogeneous domain for which the fibers are a product of a flat with a simply connected compact homogeneous Kähler manifold. This conjecture has been proven in a number of cases (see [6] for a recent survey). In particular, it holds if the homogeneous Kähler manifold admits a reductive or an arbitrary solvable transitive group of automorphisms [5]. It is thus tempting to think about the general case. It is natural to expect that lack of knowledge about the radical of a transitive group G of automorphisms of a homogeneous Kähler manifold M is the main obstruction to a proof of the Fundamental Conjecture for M. Thus it is of importance to consider the Kähler algebra generated by the radical of the Lie algebra of G. Computations in this context suggest that one rather considers Kähler algebras generated by an arbitrary solvable ideal.

Sign in / Sign up

Export Citation Format

Share Document