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KRF-CY TERMINATED

The Kaehler-Ricci flow and Singular Calabi-Yau manifolds

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EC-Contrib. €

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Partnership

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Project "KRF-CY" data sheet

The following table provides information about the project.

Coordinator
IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE 

Organization address
address: SOUTH KENSINGTON CAMPUS EXHIBITION ROAD
city: LONDON
postcode: SW7 2AZ
website: http://www.imperial.ac.uk/

contact info
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surname: n.a.
function: n.a.
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 Coordinator Country United Kingdom [UK]
 Project website https://www.imperial.ac.uk/people/e.di-nezza
 Total cost 183˙454 €
 EC max contribution 183˙454 € (100%)
 Programme 1. H2020-EU.1.3.2. (Nurturing excellence by means of cross-border and cross-sector mobility)
 Code Call H2020-MSCA-IF-2014
 Funding Scheme MSCA-IF-EF-ST
 Starting year 2015
 Duration (year-month-day) from 2015-07-01   to  2017-11-08

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE UK (LONDON) coordinator 183˙454.00

Map

 Project objective

Smoothing properties of the Kaehler-Ricci flow have been known and used for a long time. Attempt to run the Kaehler-Ricci flow from a degenerate initial data has been of great interest in the last decades. The bet result so far was recently obtained by Guedj and Zeriahi that were able to define the maximal flow for any initial current with zero Lelong number. This initial current will be smoothed out immediately. One example was also given showing that there might be no regularity at all in the case of Fano manifolds when starting from a current with positive Lelong number. However it is expected that the regularizing effect happens outside analytic sets. The first goal of this proposal is to prove such a regularity result.

In the last few years, Eyssidieux, Guedj and Zeriahi have shown that every Calabi-Yau variety admits a unique singular Kaehler-Ricci flat metric. Their work establishes the existence of such singular Kaehler-Ricci flat metric but it does not establish the expected asymptotic behavior near the singular points. The main goal of my proposal is to study the asymptotic behavior and the regularity properties of these metrics/potentials near singularities. More generally, given a Kaehler-Einstein metric on a singular variety, it would be interesting to understand how we can relate the asymptotic behavior of such a metric near to the singularities of the variety. Such a result would be of great interest also in theoretical physics. Indeed, since the seminal paper of Candelas and de la Ossa in the 90's, physicists have guessed that Calabi-Yau 3-folds with the simplest isolated singularities should admit incomplete Kaehler-Ricci flat metrics which near each singularitiy look like the conifold metric.

A related goal would be to go after the analogies by these singular Calabi-Yau problems in the singular G2 holonomy setting.

A possible strategy would be to try to develop the techniques and the ideas recently used by Lu and myself.

 Publications

year authors and title journal last update
List of publications.
2017 E. Di Nezza, V. Guedj
Geometry and topology of the space of Kähler metrics on singular varieties
published pages: , ISSN: 0010-437X, DOI:
Compositio Mathematica 2019-06-18
2017 T. Darvas, E. Di Nezza, C. Lu
Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity
published pages: , ISSN: , DOI:
2019-06-18
2018 Tamás Darvas, Eleonora Di Nezza, Chinh H. Lu
On the singularity type of full mass currents in big cohomology classes
published pages: 380-409, ISSN: 0010-437X, DOI: 10.1112/s0010437x1700759x
Compositio Mathematica 154/02 2019-06-18
2017 E. Di Nezza, C. Lu
Uniqueness and short time regularity of the weak Kähler–Ricci flow
published pages: 953-993, ISSN: 0001-8708, DOI:
Advances in Mathematics 2018-01-25

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