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KRF-CY TERMINATED

The Kaehler-Ricci flow and Singular Calabi-Yau manifolds

Total Cost €

EC-Contrib. €

Partnership

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Outcomes and
results

KRF-CY project word cloud

Explore the words cloud of the KRF-CY project. It provides you a very rough idea of what is the project "KRF-CY" about.

techniques initial flow isolated eyssidieux zero la yau understand analogies metrics g2 behavior lu myself physicists theoretical admits starting data flat physics existence happens smoothed strategy singularities guessed de attempt candelas seminal regularity paper interesting metric folds kaehler outside maximal potentials positive immediately variety showing bet last calabi establishes incomplete holonomy singular lelong guedj regularizing zeriahi time effect run ricci look setting smoothing singularitiy points near asymptotic few first ideas fano simplest conifold degenerate einstein admit analytic decades generally shown ossa manifolds prove relate

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 title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
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 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 title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	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

List of publications.
year	authors and title	journal	last update
2017	E. Di Nezza, V. Guedj Geometry and topology of the space of KÃ¤hler 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-AmpÃ¨re equations with prescribed singularity published pages: , ISSN: , DOI:		2019-06-18
2018	TamÃ¡s Darvas, Eleonora Di Nezza, Chinh H. Lu On the singularity type of full mass currents in bigÂ 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 KÃ¤hlerâ€“Ricci flow published pages: 953-993, ISSN: 0001-8708, DOI:	Advances in Mathematics	2018-01-25

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