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Several Complex Variables
Mihai Paun is interested in problems arising from global complex geometry. The main techniques used to treat these questions have their origin in several complex variables and partial differential equations. In this spirit, he has obtained (jointly with JeanPierre Demailly) a numerical characterization of the Kaehler cone of a compact manifold; this result parallels the celebrated NakaiMoishezon criteria in algebraic geometry.
His current research concerns the extension properties of twisted pluricanonical forms (with applications to birational geometry) and also the analysis of complex Kaehler manifolds endowed with conic singularities metrics.



Publication at kias
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AUTHOR  Pun, Mihai 
TITLE  Kodaira dimension of algebraic fiber spaces over abelian varieties 
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JOURNAL  INVENTIONES MATHEMATICAE, 2017 
ABSTRACT  In this article we provide a proof of the Iitaka conjecture for algebraic fiber spaces over tori. 

Publication at kias
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AUTHOR  Paun, Mihai,Campana, Frederic 
TITLE  Positivity properties of the bundle of logarithmic tensors on compact Kahler manifolds 
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JOURNAL  COMPOSITIO MATHEMATICA, 2016 
ABSTRACT  Let X be a compact Kahler manifold, endowed with an effective reduced divisor B = Sigma Yk having simple normal crossing support. We consider a closed form of (1,1)type alpha on X whose corresponding class {alpha} is nef, such that the class c(1) (KX + B)+ {alpha} is an element of H1,H1(X, R) is pseudoeffective. A particular case of the first result we establish in this short note states the following. Let in be a positive integer, and let L be a line bundle on X, such that there exists a generically injective morphism L > circle times Tm(X)star < B >, where we denote by TX(star) < B > the logarithmic cotangent bundle associated to the pair (X, B). Then for any Kahler class {omega} on X, we have the inequality integral(x)c1(L)boolean AND{omega}(n1) <= m integral(x) (c1 (KX + B) + {alpha}) boolean AND {omega}(n1) If X is projective, then this result gives a generalization of a criterion due to Y. Miyaoka, concerning the generic semi positivity: under the hypothesis above, let Q be the quotient of circle times(m) TX star < B > by L. Then its degree on a generic complete intersection curve C subset of X is bounded from below by As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Paun, Orbifold generic semipositivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835861]. 

Publication at kias
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AUTHOR  Paun, Mihai 
TITLE  CONIC SINGULARITIES METRICS WITH PRESCRIBED RICCI CURVATURE: GENERAL CONE ANGLES ALONG NORMAL CROSSING DIVISORS 
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JOURNAL  JOURNAL OF DIFFERENTIAL GEOMETRY, 2016 
ABSTRACT  Let X be a nonsingular compact Kahler manifold, endowed with an effective divisor D = Sigma(1  beta(kappa))Ykappa having simple normal crossing support, and satisfying beta(kappa) is an element of (0, 1). The natural objects one has to consider in order to explore the differential geometric properties of the pair (X, D) are the socalled metrics with conic singularities. In this article, we complete our earlier work [CGP13] concerning the Mange Ampere equations on (X, D) by establishing Laplacian and L2,Lalpha,Lbeta estimates for the solution of these equations regardless of the size of the coefficients 0 < beta(kappa) < 1. In particular, we obtain a general theorem concerning the existence and regularity of KahlerEinstein metrics with conic singularities along a normal crossing divisor. 

Publication at kias
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AUTHOR  Paun, Mihai 
TITLE  ORBIFOLD GENERIC SEMIPOSITIVITY: AN APPLICATION TO FAMILIES OF CANONICALLY POLARIZED MANIFOLDS 
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JOURNAL  ANNALES DE L INSTITUT FOURIER, 2015 
ABSTRACT  Let X be a normal projective manifold, equipped with an effective orbifold divisor Delta, such that the pair (X, Delta) is logcanonical. We first define the notion of orbifold cotangent bundle Omega(perpendicular to)(X, Delta), living on any suitable ramified cover of X. We are then in position to formulate and prove (in a completely different way) an orbifold version of Y. Miyaokas generic semipositivity theorem: Omega(1)(X, Delta) is generically semipositive if KX + Delta is pseudoeffective. Using the deep results of the LMMP, we immediately get a statement conjectured by E. Viehweg: if X is smooth, and if A is a reduced divisor with simple normal crossings on X such that some tensor power of Omega(1)(X, Delta) = Omega(1)(X)(Log(Delta)) contains the injective image of a big line bundle, then KX + Delta is big. This implies, by fundamental results of ViehwegZuo, the ShafarevichViehweg hyperbolicity conjecture: if an algebraic family of canonically polarized manifolds parametrised by a quasiprojective manifold B has maximal variation, then B is of loggeneral type. 

Publication at kias
NUMBER  
AUTHOR  Paun, Mihai 
TITLE  TECHNIQUES OF CONSTRUCTON OF HOMOMORPHIC AND HYPERBOLICITY DIFFERENTIALS (according to J.P Demailly, S.Diverio, J.Merker, E.Rousseau, Y.T Siu...) 
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JOURNAL  ASTERISQUE, 2014 
ABSTRACT  

Publication at kias
NUMBER  
AUTHOR  Paun, Mihai 
TITLE  METRICS WITH CONE SINGULARITIES ALONG NORMAL CROSSING DIVISORS AND HOLOMORPHIC TENSOR FIELDS 
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JOURNAL  ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE, 2013 
ABSTRACT  We prove the existence of nonpositively curved KahlerEinstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kahler manifold, under a technical condition on the cone angles, and we also discuss the case of positivelycurved KahlerEinstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields. 
 8/2012 present Professor, Korea Institute for Advanced Study, Korea
 9/2005 7/2012 Professor, Université de Nancy, France
 9/19988/2005 Assistant Professor, Université de Strasbourg, France
 1998: PhD (advisor JeanPierre Demailly), Université de Grenoble, France
 Prize
 Invited address at the International Congress of Mathematicians for a 45' talk, India 2010.
 Junior fellow of the
 KIAS scholar, 20092012.
 Prize
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 School of Mathematics, Korea Institute for Advanced Study
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