Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham complex and the Dolbeault complex, and discuss spinors. In Chapter 19, we discuss complex geometry and establish the Kodaira Embedding Theorem.

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Esteban Calvino-Louzao is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He received his Ph.D. in 2011 from the University of Santiago under the direction of E. Garcia-Rio and R. Vazquez-Lorenzo. His research specialty is Riemannian and pseudo-Riemannian geometry. He has published more than 20 research articles and booksEduardo Garcia-Rio is a Professor of Mathematics at the University of Santiago de Compostela (Spain). He is a member of the editorial board of Differential Geometry and its Applications and The Journal of Geometric Analysis and leads the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He received his Ph.D. in 1992 from the University of Santiago under the direction of A. Bonome and L. Hervella. His research specialty is Differential Geometry. He has published more than 120 research articles and books.Peter B. Gilkey is a Professor of Mathematics and a member of the Institute of Theoretical Science at the University of Oregon. He is a fellow of the American Mathematical Society and is a member of the editorial board of Results in Mathematics, Differential Geometry and its Applications, and The Journal of Geometric Analysis. He received his Ph.D. in 1972 from Harvard University under the direction of L. Nirenberg. His research specialties are Differential Geometry, Elliptic Partial Differential Equations, and Algebraic topology. He has published more than 275 research articles and books.JeongHyeong Park is a Professor of Mathematics at Sungkyunkwan University and is an associate member of the KIAS (Korea). She received her Ph.D. in 1990 from Kanazawa University in Japan under the direction of H. Kitahara. Her research specialties are spectral geometry of Riemannian submersion and geometric structures on manifolds like eta-Einstein manifoldsand H-contact manifolds. She organized the geometry section of AMC 2013 (The Asian Mathematical Conference 2013), the ICM 2014 satellite conference on Geometric analysis, and geometric structures on manifolds (2016). She has published more than 90 research papers and books.Ramon Vazquez-Lorenzo is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He is a member of the Spanish Research Network on Relativity and Gravitation. He received his Ph.D. in 1997 from the University of Santiago de Compostela under the direction of E. Garcia-Rio and R. Castro. His research focuses mainly on Differential Geometry, with special emphasis on the study of the curvature and the algebraic properties of curvature operators in the Lorentzian and in the higher signature settings. He has published more than 60 research articles and books.