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Canonical complex extensions of Kähler manifolds
... which is an affine bundle over M. In [14], we showed that the holomorphic convexity of Z M yields numerous restrictions on the geometry of M. In particular, we have the following conjecture [10], see also [14, Conj.1.1]: Conjecture 1.1 Let M be a projective manifold such that a canonical extension Z M is affine for some Kähler class. ...
... where Q is a line bundle. Since by (12) one has e * C ω * (10,2), we obtain that Q O C×l (7, −1). Letẽ C : U C → S ⊂ P(T M ) be the map determined by the quotient e * C T M → Q. Thenẽ C factors e C and by the universal property of the tautological bundle one hasẽ * C ζ M O C×l (7, −1). ...
... We are therefore reduced to the case ρ(M) = 1. Since Z M is affine, the tangent bundle T M is big [10,Prop.4.2]. Hence, we know by [13, Cor.1.2] ...
Let M be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that M is rational homogeneous.
... Thus by GAGA (or by the second part of Proposition 2.10) the functions f j are algebraic and the embedding is actually an algebraic embedding X \Y → A N . Hence X \Y is affine and N Y /X is big by [22,Prop. 4.2]. Thus we have reached a contradiction. ...
... be the canonical extension of T A given by some Kähler class (see Sect. 4). Set X = P(V ) and Y = P(T A ). Then by [22,Prop. 2.13] the complement X \Y is isomorphic to (C * ) 2n , hence Stein. Yet the normal bundle N Y /X = ζ P(T A ) is globally generated, but not big. ...
... Assume now that Z M is even affine. Then the tangent bundle T M is big by [22,Prop. 4.20], so M is obviously not an étale quotient of a torus. By the first part of the statement we are left to exclude the case of ruled surfaces f : M P(E) → B over a curve of genus at least one. ...
Given a projective or compact Kähler manifold X and a (smooth) hypersurface Y, we study conditions under which X\Y could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle TM of a projective manifold M with Y being the projectivization of TM itself.
... which is an affine bundle over M . In this context, we have the following conjecture [GW20], see also [HP21, Conj.1.1]: ...
... Fano threefolds are of course rather special manifolds, but they are a natural testing ground for Conjecture 1.1: since Z M is affine, the tangent bundle T M is big [GW20,Prop.4.2], a very restrictive property for manifolds that are not rational homogeneous [HLS22]. Thus it seems likely that potential counterexamples to Conjecture 1.1 share at least some properties of rational homogeneous spaces, e.g. ...
... We are therefore reduced to the case ρ(M ) = 1. Since Z M is affine, the tangent bundle T M is big [GW20,Prop.4.2]. Hence we know by [HL21, Cor.1.2] ...
Let M be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that M is rational homogeneous.
... In particular, we see from this that Z a is indeed a manifold and we also see that we may view the affine space structure on the fibres Z a | x as arising from the embedding Z a | x V a | x . This is the definition of Z a used in [GW20]. ...
... A more precise version of this statement may be found in [ The construction of extensions is clearly functorial: Proposition 1.6. (see also [GW20,Lemma 1.16(b)]) Let f : X → T be a holomorphic map between complex manifolds. Let E be a holomorphic vector bundle on T and fix any cohomology class a ∈ H 1 (T, E). ...
... The following conjecture arose out of the work of [GW20] and [HP21] on canonical extensions: Conjecture 1.8. (Greb-Wong, Höring-Peternell) Let X be a compact Kähler manifold. ...
To any compact K\"ahler manifold one may associate a bundle of affine spaces called a of X. In this paper we prove that (assuming a well-known conjecture of Campana-Peternell to hold true) if the tangent bundle of X is nef, then the total space is a Stein manifold. This partially answers a question raised by Greb-Wong of whether these two properties are actually equivalent. We also complement some known results for surfaces in the converse direction.
... We then consider the complement Z M := Z α M := P(V α ) \ P(T M ). Following Greb and Wong [GW20] we call Z M a canonical extension of M . They showed that Z M does contain any compact subvarieties and it is Stein if M is a torus or has a Kähler metric of non-negative holomorphic bisectional curvature. ...
... They showed that Z M does contain any compact subvarieties and it is Stein if M is a torus or has a Kähler metric of non-negative holomorphic bisectional curvature. In view of their results one is tempted to make the following conjecture; already formulated as a question in [GW20], 1.1. Conjecture. ...
... For the other implication observe first that if Y := P(T M ) ⊂ P(V α ) =: X is the embedding defined above, the normal bundle N Y /X identifies to the tautological bundle O P(TM ) (1). Let us recall that Greb and Wong [GW20,Prop.4.2] used Goodman's theorem [Goo69] to show that if X is projective and X \ Y is affine, the normal bundle N Y /X is big. Thus the other implication would essentially reduce to a conjecture of Goodman: 1.3. ...
Given a projective or compact K\"ahler manifold X and a (smooth) hypersurface Y, we study conditions under which could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle of a projective manifold M with Y being the projectivization of itself.
... We shall now apply Theorem 4.12 to Kähler quotients. [17], Cor. 4.4 states that this property is sufficient for T N to be big (see also the remark below). ...
... Remark 4.14: The results in [17] are formulated for varieties called canonical extensions. These varieties are discussed in Appendix to this paper under the name twisted cotangent bundles. ...
... These varieties are discussed in Appendix to this paper under the name twisted cotangent bundles. We believe that this name is more preferable in the context of hyperkähler geometry than the name used in [17]. In particular, Corollary 5.6 states that canonical extensions/twisted cotangent bundles are isomorphic to Y J (the variety Y with "twisted" complex structure). ...
In this paper we study the geometry of the total space Y of a cotangent bundle to a K\"ahler manifold N where N is obtained as a K\"ahler reduction from . Using the hyperk\"ahler reduction we construct a hyperk\"ahler metric on Y and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion of the space Y is equipped with a structure of a stratified hyperk\"ahler space. We give a necessary condition for the Feix-Kaledin metric to be complete using an observation of R.Bielawski. Pick a complex structure J on induced from quaternions. Suppose that where I is the complex structure whose restriction to is induced by the complex structure on N. We prove that the space admits an algebraic structure and is an affine variety.
... Hence by [CP11,Theorem 0.1], the bigness of the tangent bundle T X implies the uniruledness of X. This fact also appears in [GW20,Proposition 4.6]. Futhermore, Hsiao [Hsi15, Corollary 1.3] gave a criterion for the bigness of the tangent bundle T X in terms of its section ring and as a corollary, he showed that T X is big if X is a smooth projective toric variety or a partial flag variety(i.e. a rational homogeneous space G/P where G is of type A). ...
... He also asked a question whether Fano manifolds with nef tangent bundles have big tangent bundles. Greb and Wong [GW20,Corollary 4.4] showed that the affineness of canonical extensions implies the bigness of the tangent bundles and as a corollary they proved that rational homogeneous spaces have big tangent bundles (see Proposition 6.2 for another proof). Hence Hsiao's question has a positive answer if the Campana-Peternell conjecture holds. ...
Let X be an irreducible Hermitian symmetric space of compact type (IHSS for short). In this paper, we give the irreducible decomposition of Sym r T X. As a by-product, we give a cohomological characterization of the rank of X. Moreover, we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.
... Note that the notion of complexification of manifolds is not uniquely defined (see e.g. [29,30] and references therein). On a real analytic manifold X one can always define a complex structure on the tangent bundle T X ∼ = X (on a neighbourhood of zero section) and this is a minimal complexification [29,30]. ...
... [29,30] and references therein). On a real analytic manifold X one can always define a complex structure on the tangent bundle T X ∼ = X (on a neighbourhood of zero section) and this is a minimal complexification [29,30]. For the minimal complexification the inclusion X → X is a homotopy equivalence. ...
We consider the ambitwistor description of =4 supersymmetric extension of U(N) Yang-Mills theory on Minkowski space . It is shown that solutions of super-Yang-Mills equations are encoded in real-analytic U(N)-valued functions on a domain in superambitwistor space of real dimension . This leads to a procedure for generating solutions of super-Yang-Mills equations on via solving a Riemann-Hilbert-type factorization problem on two-spheres in .
... Hence by [CP11,Theorem 0.1], the bigness of the tangent bundle T X implies the uniruledness of X. This fact also appears in [GW20,Proposition 4.6]. Futhermore, Hsiao [Hsi15, Corollary 1.3] gave a criterion for the bigness of the tangent bundle T X in terms of its section ring and as a corollary, he showed that T X is big if X is a smooth projective toric variety or a partial flag variety(i.e. a rational homogeneous space G/P where G is of type A). ...
... He also asked a question whether Fano manifolds with nef tangent bundles have big tangent bundles. Greb and Wong [GW20,Corollary 4.4] showed that the affineness of canonical extensions implies the bigness of the tangent bundles and as a corollary they proved that rational homogeneous spaces have big tangent bundles (see Proposition 6.2 for another proof). Hence Hsiao's question has a positive answer if the Campana-Peternell conjecture holds. ...
Let X be an irreducible Hermitian symmetric space of compact type (IHSS for short). In this paper, we give the irreducible decomposition of . As a by-product, we give a cohomological characterization of the rank of X. Moreover, we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.
... e canonical extension is a reflexive sheaf E X on X that appears in the middle of the short exact sequence 0 → O X → E X → T X → 0 whose extension class is given by the first Chern class of the Q-Cartier divisor −K X . For first Chern classes of line bundles over manifolds the construction is classical, see [Ati57], and generalisations of it have appeared in many other problems of Kähler geometry, see for instance [Tia92,Sem92,Don02,GW20]. Section 4 discusses the construction of the canonical extension in the singular case. ...
... e following construction is classical and well-known in the smooth setup, see [Ati57], and has been studied for various questions in Kähler geometry, for example in [Tia92,Don02,GW20]. ...
We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by -Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.
... We expect that bigness of the tangent bundle is a rather restrictive property, so it would be very interesting to have a more geometric characterisation. Greb and Wong [GW19] show that the affineness of a canonical complex extension implies that the tangent bundle is big [GW19,Cor.4.4]. Our results indicate that many Fano manifolds do not admit such affine extensions. ...
... We expect that bigness of the tangent bundle is a rather restrictive property, so it would be very interesting to have a more geometric characterisation. Greb and Wong [GW19] show that the affineness of a canonical complex extension implies that the tangent bundle is big [GW19,Cor.4.4]. Our results indicate that many Fano manifolds do not admit such affine extensions. ...
Let X be a Fano manifold. While the properties of the anticanonical divisor and its multiples have been studied by many authors, the positivity of the tangent bundle is much more elusive. We give a complete characterisation of the pseudoeffectivity of for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.
... Proof of Proposition 1. 9 Suppose to the contrary that X admits a non-isomorphic surjective endomorphism. By [42,Proposition 2.7], such X satisfies Bott vanishing and then X is locally rigid. ...
Let be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle is big. We show that f is an isomorphism unless Y is a projective space. As applications, we explore the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.
... This is known as the twisted cotangent bundle, and appears, for example, in Donaldson's reformulation of Kähler geometry [18, §2], as also explained in [9, §2]. The complex structure induced by Ω ζ is not the original one, but is biholomorphic to the affine bundle modeled on T * X defined by the Kähler class [ω] ∈ H 1 (X, T * X) (see [1,Appendix] and [21]). ...
The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem.
In this paper, we discuss the ℂ-algebra H0(X, S•TX) for a smooth complex projective variety X. We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.
Our workshop focused on recent results in our main field (complex geometry) and its connection with other branches of mathematics. The main theme of an important proportion of the talks was Hodge theory, combined with differential-geometric methods in the study of singular spaces. One special lecture was a very comprehensive introduction in Scholze-Clausen’s theory of condensed mathematics.
Let f : X → Y be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle T X is big. We show that f is an isomorphism unless Y is a projective space. As applications, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.
We consider slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class c_1(T_X) on Picard-rank-1 Fano varieties. In cases where the index divides the dimension or the dimension plus one, we show that stability of the tangent bundle implies (semi)stability of the canonical extension. One consequence of our result is that the canonical extensions on moduli spaces of stable vector bundles with a fixed determinant on a curve are at least semistable, and stable in some cases.
We consider the ambitwistor description of N=4 supersymmetric extension of U(N) Yang-Mills theory on Minkowski space R3,1. It is shown that solutions of super-Yang-Mills equations are encoded in real analytic U(N)-valued functions on a domain in superambitwistor space LR5|6 of real dimension (5|6). This leads to a procedure for generating solutions of super-Yang-Mills equations on R3,1 via solving a Riemann-Hilbert-type factorization problem on two-spheres in LR5|6.
We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by Q \mathbb {Q} -Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.
The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.
In this paper, we study the geometry of the total space Y of a cotangent bundle to a Kähler manifold N where N is obtained as a Kähler reduction from . Using the hyperkähler reduction, we construct a hyperkähler metric on Y and prove that it coincides with the canonical Feix–Kaledin metric. This metric is in general non-complete. We show that the metric completion of the space Y is equipped with a structure of a stratified hyperkähler space (in the sense of [ 31]). We give a necessary condition for the Feix–Kaledin metric to be complete using an observation of R. Bielawski. Pick a complex structure J on induced from the quaternions. Suppose that where I is the complex structure whose restriction to is induced by the complex structure on N. We prove that the space admits an algebraic structure and is an affine variety.
We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such that P(T∗X) has nef tangent bundle, then X≅Pn.
Let A be a complex abelian variety. The moduli space of rank one algebraic connections on A is a principal bundle over the dual abelian variety A∨ = Pic⁰(A) for the group . Take any line bundle L on A∨; let be the algebraic principal -bundle over A∨ given by the sheaf of connections on L. The line bundle L produces a homomorphism . We prove that is isomorphic to the principal -bundle obtained by extending the structure group of the principal -bundle using this homomorphism given by L. We compute the ring of algebraic functions on . As an application of the above result, we show that does not admit any nonconstant algebraic function, despite the fact that it is biholomorphic to (ℂ*)2 dim A implying that it has many nonconstant holomorphic functions.
In this paper we prove a generalization of a theorem of Schneider, which gives a criterion for a projective surface over the complex numbers to have an ample cotangent bundle. After reviewing different notions of positivity, we introduce a slightly weaker notion of ampleness, which we call quasi-ample, and then are able to extend Schneider's result to higher dimensions. Comment: 15 pages. v2. Minor revisions. To appear in Michigan Math. J
Contemporary research in algebraic geometry is the focus of this collection, which presents articles on modern aspects of the subject. The list of topics covered is a roll-call of some of the most important and active themes in this thriving area of mathematics: the reader will find articles on birational geometry, vanishing theorems, complex geometry and Hodge theory, free resolutions and syzygies, derived categories, invariant theory, moduli spaces, and related topics, all written by leading experts. The articles, which have an expository flavour, present an overall picture of current research in algebraic geometry, making this book essential for researchers and graduate students. This volume is the outcome of the conference Recent Advances in Algebraic Geometry, held in Ann Arbor, Michigan, to honour Rob Lazarsfeld's many contributions to the subject on the occasion of his 60th birthday.
This two-volume book on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Most of the material in the present Volume II has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. Both volumes are also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete."
Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions.
The second edition contains five new chapters which present some of the most important recent result on the subject. Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture.
This book is based on courses given at Columbia University on vector bun dles (1988) and on the theory of algebraic surfaces (1992), as well as lectures in the Park City lIAS Mathematics Institute on 4-manifolds and Donald son invariants. The goal of these lectures was to acquaint researchers in 4-manifold topology with the classification of algebraic surfaces and with methods for describing moduli spaces of holomorphic bundles on algebraic surfaces with a view toward computing Donaldson invariants. Since that time, the focus of 4-manifold topology has shifted dramatically, at first be cause topological methods have largely superseded algebro-geometric meth ods in computing Donaldson invariants, and more importantly because of and Witten, which have greatly sim the new invariants defined by Seiberg plified the theory and led to proofs of the basic conjectures concerning the 4-manifold topology of algebraic surfaces. However, the study of algebraic surfaces and the moduli spaces of bundles on them remains a fundamen tal problem in algebraic geometry, and I hope that this book will make this subject more accessible. Moreover, the recent applications of Seiberg Witten theory to symplectic 4-manifolds suggest that there is room for yet another treatment of the classification of algebraic surfaces. In particular, despite the number of excellent books concerning algebraic surfaces, I hope that the half of this book devoted to them will serve as an introduction to the subject.
Given a holomorphic principal bundle , the universal space of holomorphic connections is a torsor for such that the pullback of Q to has a tautological holomorphic connection. When , where P is a parabolic subgroup of a complex simple group G, and Q is the frame bundle of an ample line bundle, we show that may be identified with G/L, where is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-K\"ahler metric on , recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.
The space of K\"ahler potentials in a compact K\"ahler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of K\"ahler potentials, and prove existence and uniqueness for such isometries.
Let X be a projective manifold such that the anticanonical bundle is nef. We prove that the Albanese map is locally isotrivial. In particular, p is a submersion.
By studying the Higgs bundle equations with the gauge group replaced by the
group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of
a folded hyperkaehler 4-manifold and conjecture the existence of a family of
such metrics parametrised by an infinite-dimensional analogue of Teichmueller
space.
I Lie Groups and Lie Algebras.- II Elementary Representation Theory.- III Representative Functions.- IV The Maximal Torus of a Compact Lie Group.- V Root Systems.- VI Irreducible Characters and Weights.- Symbol Index.
Ce n’est que depuis peu de temps que les ensembles analytiques dans le domaine réel ont fait l’objet d’études approfondies. La théorie des variétés de Stein vient d’être transportée au cas des ensembles analytiques-réels « cohérents » par H. Cartan [6], qui a montré que, dans R
n
, un ensemble lieu des zéros d’un faisceau cohérent d’idéaux est « C-analytique », c’est-à-dire est la partie réelle d’un ensemble analytique-complexe, et est globalement définissable par l’annulation d’un nombre fini de fonctions analytiques-réelles. Par ailleurs, F. Bruhat et H. Cartan ont étudié dans [2] et [3] le cas général des ensembles analytiques-réels qui ne sont pas C-analytiques et ont montré que, si ces ensembles ont toujours de bonnes propriétés «locales» (i. e. sur un compact), ils peuvent avoir un comportement global très pathologique: en particulier, il n’existe pas toujours de «bonne» décomposition en composantes irréductibles.
For any real-analytic Kahler manifold we shall prove the existence of a hyperkähler metric in a neighbourhood of the zero section of its cotangent bundle using the twistor method. This hyperkähler metric is compatible with the canonical holomorphic-symplectic structure of the cotangent bundle, extends the given Kähler metric, and the circle action given by scalar multiplication in the fibres is isometric. By deriving necessary conditions we show that many of these hyperkähler metrics will be incomplete.
The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.
Soit (X, h) une variete de Kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit (X~, h~) son espace de recouvrement universel. Alors il existe des entiers non negatifs k, N 1 ,…, N l et des espaces symetriques hermitiens compacts ineductibles M 1 ,…, M k de rang ≥2, tels que (X~, h~) est isometriquement bi-holomorphe a (C k , g 0 )×(P Nu 1, θ 1 )×…×(P Nu l, θ l )×(M 1 , g 1 )×…×(M p , g p ), ou g 0 est la metrique euclidienne sur C k , g 1 ,…, g p sont des metriques canoniques sur M 1 , …, M p , et θ i , 1≤i≤l, est une metrique de Kahler sur P Nu i portant une courbure bisectionnelle holomorphe non negative
We show that a complex projective manifold X which satisfies Gromov's h-principle is special in the sense of the first author's paper "orbifolds, Special Varieties, and Classification Theory" (Annals of the Institut Fourier, 2004), and raise some questions about the reverse implication, the extension to the quasi-Kähler case, and the relationships of these properties to the Oka property. The guiding principle is that the existence of many Stein manifolds with degenerate Kobayashi pseudometric gives strong obstructions to the complex hyperbolicity of projective manifolds satisfying the h-principle.
Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X,which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville–Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series |kL| as k→∞.
In this paper we study a version of the Dirichlet problem for the homogeneous complex Monge-Amperè equation.
- Campana F.
Complex analytic and differential geometry’ version of
- J.-P Demailly
Complex analytic connections in fibre bundles
- Atiyah M. F.
Approximation theorems on differentiable submanifolds of a complex manifold
- Nirenberg R.
Review of the cohomology of compact Lie groups
- Y Zhang