On the Kähler-Ricci Flow on Projective Manifolds of General Type

2006 ◽  
Vol 27 (2) ◽  
pp. 179-192 ◽  
Author(s):  
Gang Tian* ◽  
Zhou Zhang
Keyword(s):  
Ricci Flow ◽  
General Type ◽  
Projective Manifolds

scholarly journals Maximal time existence of unnormalized conical Kähler–Ricci flow

2020 ◽  
Vol 2020 (760) ◽  
pp. 169-193
Author(s):  
Liangming Shen
Keyword(s):  
Ricci Flow ◽  
General Type ◽  
Normal Crossing ◽  
Log Canonical ◽  
Flow Through ◽  
Projective Manifolds ◽  
Time Existence ◽  
Simple Normal Crossing ◽  
Maximal Time ◽  
Invent Math

AbstractWe generalize the maximal time existence of Kähler–Ricci flow in [G. Tian and Z. Zhang, On the Kähler–Ricci flow on projective manifolds of general type, Chin. Ann. Math. Ser. B 27 (2006), no. 2, 179–192] and [J. Song and G. Tian, The Kähler–Ricci flow through singularities, Invent. Math. 207 (2017), no. 2, 519–595] to the conical case. Furthermore, if the log canonical bundle {K_{M}+(1-\beta)[D]} is big or big and nef, we can examine the limit behaviors of such conical Kähler–Ricci flow. Moreover, these results still hold when D is a simple normal crossing divisor.

scholarly journals A note on Lang's conjecture for quotients of bounded domains

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Sébastien Boucksom ◽  
Simone Diverio
Keyword(s):  
Plurisubharmonic Function ◽  
General Type ◽  
Universal Cover ◽  
Projective Manifold ◽  
Bounded Domains ◽  
Final Version ◽  
Lang's Conjecture ◽  
Projective Manifolds ◽  
Lang’S Conjecture ◽  
Complex Projective

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains. Comment: 10 pages, no figures, comments are welcome. v3: following suggestions made by the referee, the exposition has been improved all along the paper, we added a variant of Theorem A which includes manifolds whose universal cover admits a bounded psh function which is strictly psh just at one point, and we added a section of examples. Final version, to appear on \'Epijournal G\'eom. Alg\'ebrique

scholarly journals Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

2014 ◽  
Vol 151 (2) ◽  
pp. 351-376 ◽  
Author(s):  
Fréderic Campana ◽  
Benoît Claudon ◽  
Philippe Eyssidieux
Keyword(s):  
General Type ◽  
Linear Representation ◽  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Shafarevich Conjecture ◽  
Classical Work ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Holomorphic Convexity

AbstractWe extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

scholarly journals Convergence of the Weak Kähler–Ricci Flow on Manifolds of General Type

2019 ◽  
Author(s):  
Tat Dat Tô
Keyword(s):  
Ricci Flow ◽  
General Type ◽  
Einstein Metric ◽  
Canonical Class ◽  
Open Set ◽  
Long Time Existence ◽  
Long Time ◽  
Viscosity Sense ◽  
Monge Ampere ◽  
Time Existence

Abstract We study the Kähler–Ricci flow on compact Kähler manifolds whose canonical bundle is big. We show that the normalized Kähler–Ricci flow has long-time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular Kähler–Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge–Ampère flows in big classes that we develop, extending and refining the approach of Eyssidieux–Guedj–Zeriahi.

scholarly journals Ricci flow on quasi-projective manifolds

2011 ◽  
Vol 156 (1) ◽  
pp. 87-123 ◽  
Author(s):  
John Lott ◽  
Zhou Zhang
Keyword(s):  
Ricci Flow ◽  
Projective Manifolds
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