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3 results on '"Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586))"'

1. Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

Author

Henri Guenancia, Mihai Păun, Junyan Cao, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), University of Illinois [Chicago] (UIC), University of Illinois System, Universität Bayreuth, Stony Brook University [SUNY] (SBU), State University of New York (SUNY), Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, Mathematics::Complex Variables, Mathematics - Complex Variables, Applied Mathematics, General Mathematics, 010102 general mathematics, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], 01 natural sciences, Mathematics - Algebraic Geometry, symbols.namesake, Variation (linguistics), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 0103 physical sciences, symbols, Kodaira dimension, Mathematics::Differential Geometry, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, 0101 mathematics, Einstein, Mathematics::Symplectic Geometry, Mathematics
Abstract

Given a K\"ahler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular K\"ahler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted K\"ahler-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 fiber-wise Song-Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension)., Comment: v2: Title changed as the initial text is now split into two separate articles (authored by the same persons). The main results we obtain here are less general than their respective counterparts in the previous version, due to a gap in our arguments

Published
2021
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2. Pro unitality and pro excision in algebraic K-theory and cyclic homology

Author

Matthew Morrow, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Noetherian, Pure mathematics, 19D55 (primary), 16E40, 13D03 (secondary), General Mathematics, Cyclic homology, Mathematics::Algebraic Topology, 01 natural sciences, Mathematics::K-Theory and Homology, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Algebraic number, Excision theorem, Commutative property, ComputingMilieux_MISCELLANEOUS, Mathematics, Mathematics::Commutative Algebra, Hochschild homology, Applied Mathematics, Unital, 010102 general mathematics, K-Theory and Homology (math.KT), Algebraic K-theory, Mathematics - K-Theory and Homology, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], 010307 mathematical physics
Abstract

We study pro excision in algebraic K-theory, following Suslin--Wodzicki, Cuntz--Quillen, Corti\~nas, and Geisser--Hesselholt, as well as Artin--Rees and continuity properties of Andr\'e--Quillen, Hochschild, and cyclic homology. Our key tool is to first establish the equivalence of various pro Tor vanishing conditions which appear in the literature. Using this we prove that all ideals of commutative, Noetherian rings are pro unital in a certain sense, and show that such ideals satisfy pro excision in $K$-theory as well as in cyclic and topological cyclic homology. In addition, our techniques yield a strong form of the pro Hochschild--Kostant--Rosenberg theorem, an extension to general base rings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and a generalisation of the Fe\u{\i}gin--Tsygan theorem., Comment: Version 2: The paper now includes generalisations of the Cuntz--Quillen and Fe\u{i}gin--Tsygan theorems on periodic cyclic homology to general base rings of characteristic zero. Various minor improvements and corrections. To appear in Journal fur die reine und angewandte Mathematik

Published
2015
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3. Hyper-Kähler fourfolds and Grassmann geometry

Author

Olivier Debarre, Claire Voisin, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Applied Mathematics, General Mathematics, Geometry, Construct (python library), 14J70, K3 surface, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Hilbert scheme, Genus (mathematics), 14J35, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Projective test, 14M15, Mathematics::Symplectic Geometry, Mathematics
Abstract

We construct a new 20-dimensional family of algebraic hyper-Kaehler fourfolds and prove that they are deformation-equivalent to the second punctual Hilbert scheme of a K3 surface of degree 22., Comment: Typos corrected. Connection made with the work of Gritsenko, Hulek, and Sankaran on the Kodaira dimension of moduli spaces of hyper-Kaehler manifolds (arXiv:0802.2078)

Published
2010
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