Metric Contact Manifolds and Their Dirac Operators Christoph Stadtm¨Uller
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Kähler Manifolds and Holonomy
K¨ahlermanifolds, lecture 3 M. Verbitsky K¨ahler manifolds and holonomy lecture 3 Misha Verbitsky Tel-Aviv University December 21, 2010, 1 K¨ahlermanifolds, lecture 3 M. Verbitsky K¨ahler manifolds DEFINITION: An Riemannian metric g on an almost complex manifiold M is called Hermitian if g(Ix; Iy) = g(x; y). In this case, g(x; Iy) = g(Ix; I2y) = −g(y; Ix), hence !(x; y) := g(x; Iy) is skew-symmetric. DEFINITION: The differential form ! 2 Λ1;1(M) is called the Hermitian form of (M; I; g). REMARK: It is U(1)-invariant, hence of Hodge type (1,1). DEFINITION: A complex Hermitian manifold (M; I; !) is called K¨ahler if d! = 0. The cohomology class [!] 2 H2(M) of a form ! is called the K¨ahler class of M, and ! the K¨ahlerform. 2 K¨ahlermanifolds, lecture 3 M. Verbitsky Levi-Civita connection and K¨ahlergeometry DEFINITION: Let (M; g) be a Riemannian manifold. A connection r is called orthogonal if r(g) = 0. It is called Levi-Civita if it is torsion-free. THEOREM: (\the main theorem of differential geometry") For any Riemannian manifold, the Levi-Civita connection exists, and it is unique. THEOREM: Let (M; I; g) be an almost complex Hermitian manifold. Then the following conditions are equivalent. (i)( M; I; g) is K¨ahler (ii) One has r(I) = 0, where r is the Levi-Civita connection. 3 K¨ahlermanifolds, lecture 3 M. Verbitsky Holonomy group DEFINITION: (Cartan, 1923) Let (B; r) be a vector bundle with connec- tion over M. For each loop γ based in x 2 M, let Vγ;r : Bjx −! Bjx be the corresponding parallel transport along the connection. -
Hermitian Geometry
1 Hermitian Geometry Simon M. Salamon Introduction These notes are based on graduate courses given by the author in 1998 and 1999. The main idea was to introduce a number of aspects of the theory of complex and symplectic structures that depend on the existence of a compatible Riemannian metric. The present notes deal mostly with the complex case, and are designed to introduce a selection of topics and examples in differential geometry, accessible to anyone with an acquaintance of the definitions of smooth manifolds and vector bundles. One basic problem is, given a Riemannian manifold (M; g), to determine whether there exists an orthogonal complex structure J on M , and to classify all such J . A second problem is, given (M; g), to determine whether there exists an orthogonal almost-complex structure for which the corresponding 2-form ! is closed. The first problem is more tractible in the sense that there is an easily identifiable curvature obstruction to the existence of J , and this obstruction can be interpreted in terms of an auxiliary almost-complex structure on the twistor space of M . This leads to a reasonably complete resolution of the problem in the first non-trivial case, that of four real dimensions. The theory has both local and global aspects that are illustrated in Pontecorvo's classification [89] of bihermitian anti-self-dual 4-manifolds. Much less in known in higher dimensions, and some of the basic classification questions concerning orthogonal complex structures on Riemannian 6-manifolds remain unanswered. The second problem encompasses the so-called Goldberg conjecture. -
Spinc GEOMETRY of K¨AHLER MANIFOLDS and the HODGE
SPINc GEOMETRY OF KAHLER¨ MANIFOLDS AND THE HODGE LAPLACIAN ON MINIMAL LAGRANGIAN SUBMANIFOLDS O. HIJAZI, S. MONTIEL, AND F. URBANO Abstract. From the existence of parallel spinor fields on Calabi- Yau, hyper-K¨ahleror complex flat manifolds, we deduce the ex- istence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the sub- manifolds are compact, we obtain sharp estimates on their Betti numbers. When the ambient manifold is K¨ahler-Einstein with pos- itive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of K¨ahlerian Killing spinor fields for some particular spinc struc- tures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. Applications on the Morse index of minimal Lagrangian submanifolds are obtained. 1. Introduction Recently, connections between the spectrum of the classical Dirac operator on submanifolds of a spin Riemannian manifold and its ge- ometry were investigated. Even when the submanifold is spin, many problems appear. In fact, it is known that the restriction of the spin bundle of a spin manifold M to a spin submanifold is a Hermitian bun- dle given by the tensorial product of the intrinsic spin bundle of the submanifold and certain bundle associated with the normal bundle of the immersion ([2, 3, 6]). In general, it is not easy to have a control on such a Hermitian bundle. Some results have been obtained ([2, 24, 25]) when the normal bundle of the submanifold is trivial, for instance for hypersurfaces. -
The Chirka-Lindelof and Fatou Theorems for D-Bar Subsolutions
The Chirka - Lindel¨of and Fatou theorems for ∂J-subsolutions Alexandre Sukhov* * Universit´ede Lille, Laboratoire Paul Painlev´e, U.F.R. de Math´e-matique, 59655 Villeneuve d’Ascq, Cedex, France, [email protected] The author is partially suported by Labex CEMPI. Institut of Mathematics with Computing Centre - Subdivision of the Ufa Research Centre of Russian Academy of Sciences, 45008, Chernyshevsky Str. 112, Ufa, Russia. Abstract. This paper studies boundary properties of bounded functions with bounded ∂J differential on strictly pseudoconvex domains in an almost complex manifold. MSC: 32H02, 53C15. Key words: almost complex manifold, ∂-operator, strictly pseudoconvex domain, the Fatou theorem. Contents 1 Introduction 2 2 Almost complex manifolds and almost holomorphic functions 3 arXiv:1808.04266v1 [math.CV] 10 Aug 2018 2.1 Almostcomplexmanifolds............................ 3 2.2 Pseudoholomorphicdiscs............................. 4 2.3 The ∂J -operator on an almost complex manifold (M,J)............ 6 2.4 Plurisubharmonic functions on almost complex manifolds: the background . 8 2.5 Boundary properties of subsolutions of the ∂-operator in the unit disc . 9 3 The Chirka-Lindel¨of principle for strictly pseudoconvex domains 10 3.1 Localapproximationbyhomogeneousmodels . .. 12 3.2 Caseofmodelstructures . .. .. .. 13 3.3 DeformationargumentandproofofTheorem3.4 . ... 16 4 The Fatou theorem 17 1 1 Introduction The first fundamental results on analytic properties of almost complex structures (in sev- eral variables) are due to Newlander - Nirenberg [9] and Nijenhuis - Woolf [10]. After the seminal work by M.Gromov [7] the theory of pseudoholomorphic curves in almost complex manifolds became one of the most powerful tools of the symplectic geometry and now is rapidly increasing. -
Arxiv:1208.0648V1 [Math.DG] 3 Aug 2012 4.6
CALCULUS AND INVARIANTS ON ALMOST COMPLEX MANIFOLDS, INCLUDING PROJECTIVE AND CONFORMAL GEOMETRY A. ROD GOVER AND PAWEŁ NUROWSKI Abstract. We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and effi- cient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants. Contents 1. Introduction 2 1.1. Conventions 4 2. Calculus on an almost complex affine manifold 5 2.1. Almost complex affine connections 5 2.2. Torsion and Integrability 6 2.3. Compatible affine connections 7 3. Almost Hermitian geometry 8 4. Conformal almost Hermitian manifolds 12 4.1. A canonical torsion free connection 12 4.2. Canonical conformal almost complex connections 14 4.3. Compatibility in the conformal setting 15 4.4. Characterising the distinguished connection 16 4.5. The Nearly Kähler Weyl condition 18 arXiv:1208.0648v1 [math.DG] 3 Aug 2012 4.6. The locally conformally almost Kähler condition 18 4.7. The Gray-Hervella types 20 4.8. Higher conformal invariants 20 4.9. Summary 25 5. Projective Geometry 25 5.1. -
IMMERSIONS of SURFACES in Spinc -MANIFOLDS with a GENERIC POSITIVE SPINOR
appeared in Annals of Global Analysis and Geometry 26 (2004), 175{199 and 319 IMMERSIONS OF SURFACES IN SPINc -MANIFOLDS WITH A GENERIC POSITIVE SPINOR Andrzej Derdzinski and Tadeusz Januszkiewicz Abstract: We define and discuss totally real and pseudoholo- morphic immersions of real surfaces in a 4-manifold which, instead of an almost complex structure, carries only a \framed spinc-structure," that is, a spinc-structure with a fixed generic section of its positive half-spinor bundle. In particular, we describe all pseudoholomorphic immersions of closed surfaces in the 4-sphere with a standard framed spin structure. Mathematics Subject Classification (2000): primary 53C27, 53C42; secondary 53C15. Key words: spinc-structure, totally real immersion, pseudoholo- morphic immersion 1. Introduction Almost complex structures on real manifolds of dimension 2n are well- known to be, essentially, a special case of spinc-structures. This amounts to a specific Lie-group embedding U(n) Spinc(2n) (see [6, p. 392] and Remark 7.1 below). The present paper! deals with the case n = 2. The relation just mentioned then can also be couched in the homotopy theorists' language: for a compact four-manifold M with a fixed CW-decomposition, a spinc-structure over M is nothing else than an almost complex structure on the 2-skeleton of M, admitting an extension to its 3-skeleton; the ex- tension itself is not a part of the data. (Kirby [5] attributes the italicized comment to Brown.) Our approach is explicitly geometric and proceeds as follows. Given an almost complex structure J on a 4-manifold M, one can always choose a Riemannian metric g compatible with J, thus replacing J by an almost Hermitian structure (J; g) on M. -
1. Complex Vector Bundles and Complex Manifolds Definition 1.1. A
1. Complex Vector bundles and complex manifolds n Definition 1.1. A complex vector bundle is a fiber bundle with fiber C and structure group GL(n; C). Equivalently, it is a real vector bundle E together with a bundle automor- phism J : E −! E satisfying J 2 = −id. (this is because any vector space with a linear map 2 n J satisfying J = −id has an real basis identifying it with C and J with multiplication by i). The map J is called a complex structure on E. An almost complex structure on a manifold M is a complex structure on E. An almost complex manifold is a manifold together with an almost complex structure. n Definition 1.2. A complex manifold is a manifold with charts τi : Ui −! C which are n −1 homeomorphisms onto open subsets of C and chart changing maps τi ◦ τj : τj(Ui \ Uj) −! τi(Ui \ Uj) equal to biholomorphisms. Holomorphic maps between complex manifolds are defined so that their restriction to each chart is holomorphic. Note that a complex manifold is an almost complex manifold. We have a partial converse to this theorem: Theorem 1.3. (Newlander-Nirenberg)(we wont prove this). An almost complex manifold (M; J) is a complex manifold if: [J(v);J(w)] = J([v; Jw]) + J[J(v); w] + [v; w] for all smooth vector fields v; w. Definition 1.4. A holomorphic vector bundle π : E −! B is a complex manifold E together with a complex base B so that the transition data: Φij : Ui \ Uj −! GL(n; C) are holomorphic maps (here GL(n; C) is a complex vector space). -
RICCI CURVATURES on HERMITIAN MANIFOLDS Contents 1. Introduction 1 2. Background Materials 9 3. Geometry of the Levi-Civita Ricc
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0 RICCI CURVATURES ON HERMITIAN MANIFOLDS KEFENG LIU AND XIAOKUI YANG Abstract. In this paper, we introduce the first Aeppli-Chern class for com- plex manifolds and show that the (1, 1)- component of the curvature 2-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K¨ahler Calabi-Yau manifolds by using the first Aeppli- Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct 2n−1 1 explicit Levi-Civita Ricci-flat metrics on Hopf manifolds S × S . We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Rie- mannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian man- ifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature. Contents 1. Introduction 1 2. Background materials 9 3. Geometry of the Levi-Civita Ricci curvature 12 4. Curvature relations on Hermitian manifolds 22 5. Special metrics on Hermitian manifolds 26 6. Levi-Civita Ricci-flat and constant negative scalar curvature metrics on Hopf manifolds 29 7. Appendix: the Riemannian Ricci curvature and ∗-Ricci curvature 35 References 37 1. -
NOTES on INTEGRABILITY of COMPLEX STRUCTURES Recall
NOTES ON INTEGRABILITY OF COMPLEX STRUCTURES NILAY KUMAR Recall that every complex manifold is by definition a smooth manifold. This raises the following question: when can a smooth manifold be given the structure of a complex manifold, i.e. a complex structure? We begin by reviewing some definitions. Definition 1. An almost complex structure J on a 2n-dimensional smooth mani- fold M is a (real) vector bundle endomorphism J : TM ! TM satisfying J 2 = − id. Exercise 2. Show that if M has an almost complex structure then it must be even-dimensional. Suppose (M; J) is an almost complex manifold of dimension 2n. Define the complexified tangent bundle TCM = TM ⊗ C to be the tensor product of the real tangent bundle with the trivial complex line bundle. Notice that TCM has fibers that are 4n dimensional. Complexifying the almost complex structure J, we obtain J = J ⊗ . Notice that J 2 = − id and hence J has eigenvalues ±i. Thus we can C C C C decompose 1;0 0;1 TCM = T M ⊕ T M; 1 the ±i-eigenbundles of JC. These are the holomorphic and antiholomorphic tan- gent bundles of (M; J). Proposition 3. There is a natural almost complex structure J on any complex manifold X. Proof. Let fUα; φαg be an atlas for X. Each Uα is biholomorphic to an open subset of Cn, and hence has real coordinates (x1; : : : ; xn; y1; : : : ; yn) comprising the i i i coordinates z = x + iy . Consider, in this chart, the endomorphism J of TUα i i i i sending @=@x 7! @=@y and @=@y 7! −@=@x at every point of Uα. -
Results Concerning Almost Complex Structures on the Six-Sphere
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 034, 21 pages Results Concerning Almost Complex Structures on the Six-Sphere Scott O. WILSON Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd., Queens, NY 11367, USA E-mail: [email protected] URL: http://qcpages.qc.cuny.edu/~swilson/ Received November 20, 2017, in final form April 09, 2018; Published online April 17, 2018 https://doi.org/10.3842/SIGMA.2018.034 Abstract. For the standard metric on the six-dimensional sphere, with Levi-Civita connec- tion r, we show there is no almost complex structure J such that rX J and rJX J commute for every X, nor is there any integrable J such that rJX J = JrX J for every X. The latter statement generalizes a previously known result on the non-existence of integrable ortho- gonal almost complex structures on the six-sphere. Both statements have refined versions, expressed as intrinsic first order differential inequalities depending only on J and the met- ric. The new techniques employed include an almost-complex analogue of the Gauss map, defined for any almost complex manifold in Euclidean space. Key words: six-sphere; almost complex; integrable 2010 Mathematics Subject Classification: 53C15; 32Q60; 53A07 1 Introduction One knows from topology that the only spheres which admit almost complex structures are those in dimensions two and six, [3]. On the two-sphere, every almost complex structure is integrable, i.e., induced by a complex structure, and furthermore this is unique up to biholomorphism. The situation for the six-sphere S6 is not well understood: while there are many almost complex structures, Hopf's original question from 1926, asking whether the six-sphere admits a complex structure, remains unsolved [5]. -
Almost Complex Manifolds an Almost-Complex Manifold Is A
Almost complex manifolds An almost-complex manifold is a smooth real manifold 2 M equipped with a smooth endomorphism field J : T M T M satisfying J = Ix for → x − all x M. The linear algebra introduced above may be applied pointwise to the tangent bundle∈ of M. The complexified tangent bundle is TCM := T M C, where C is regarded as a trivial 1,0 0,1 ⊗ vector bundle over M. Since TC,xM = Tx M Tx M for any point x M as above, the tensor field J splits the complexified tangent bundle⊕ into bundles of eigen∈spaces 1,0 0,1 TCM = T M T M, ⊕ and the smooth complex vector bundles (TM,J) and (T 1,0M, i) are C-linearly isomorphic. We notice that a real smooth manifold carrying an almost complex structure may not be a complex manifold. We’ll discuss later that a real smooth manifold carrying an integral almost complex structure is a complex manifold. [Example] (Almost complex structure on spheres) It is natural to ask which spheres S2n admit almost-complex structures or complex structure (or other ones such as K¨ahler structure). 2 CP1 p m R R S = is the Riemann sphere, a complex manifold. It is known HDR(S , ) = • for p =0or p = m; 0 otherwsie. On the other hand, if X is a compact K¨ahler manifold, it is known that the Betti number b (X) 1. Thus S2 is the only sphere that admits 2 ≥ a K¨ahler structure. S2 has only one almost-complex structure up to equivalence. -
From Smooth to Almost Complex
FROM SMOOTH TO ALMOST COMPLEX WEIYI ZHANG Abstract. This article mainly aims to overview the recent efforts on developing algebraic geometry for an arbitrary compact almost complex manifold. We review the results obtained by the guiding philosophy that a statement for smooth maps between smooth manifolds in terms of Ren´e Thom’s transversality should also have its counterpart in pseudoholo- morphic setting without requiring genericity of the almost complex struc- tures. These include intersection of compact almost complex subman- ifolds, structure of pseudoholomorphic maps, zero locus of certain har- monic forms, and eigenvalues of Laplacian. In addition to reviewing the compact manifolds situation, we also extend these results to orbifolds and non-compact manifolds. Motivations, methodologies, applications, and further directions are discussed. The structural results on the pseudoholomorphic maps lead to a no- tion of birational morphism between almost complex manifolds. This motivates the study of various birational invariants, including Kodaira dimensions and plurigenera, in this setting. Some other aspects of almost complex algebraic geometry in dimension 4 are also reviewed. These include cones of (co)homology classes and subvarieties in spheri- cal classes. Contents 1. Introduction 2 2. Intersection theory of almost complex submanifolds 4 2.1. Further discussions 9 arXiv:2010.03858v1 [math.DG] 8 Oct 2020 2.2. Orbifolds 10 3. Maps between almost complex manifolds and birational morphisms 11 3.1. Some further discussions 14 4. From harmonic forms to J-anti-invariant forms 16 4.1. Relation with Taubes’ SW=Gr 18 5. Non-compact manifolds 19 6. Birational invariants for almost complex manifolds 22 7.