Global analysis describes diverse yet interrelated research areas in analysis and algebraic geometry, particularly those in which Kunihiko Kodaira made his most outstanding contributions to mathematics. The eminent contributors to this volume, from Japan, the United States, and Europe, have prepared original research papers that illustrate the progress and direction of current research in complex variables and algebraic and differential geometry. The authors investigate, among other topics, complex manifolds, vector bundles, curved 4-dimensional space, and holomorphic mappings. Bibliographies facilitate further reading in the development of the various studies.Originally published in 1970.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Global Analysis: Papers in Honor of K. Kodaira (Pms-29)

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2\pi \int_{\theta\in S^1}\chi(\Sigma_{\theta})\geq \frac{1}{2}\int_{\theta\in S^1}\int_{\Sigma_{\theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $\Sigma_{\theta}=u^{-1}\{\theta\}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;\mathbb{Z})$ in terms of $\|R_M^-\|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(\min R_M)sys_2(M)\leq 8\pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.

Scalar curvature and harmonic maps to $S^1$

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. 
In the first part of this paper we lay the foundation of the theory of multicomplexes. We construct the singular multicomplex K(X) associated to a topological space X, and we prove that K(X) is homotopy equivalent to $X$ for every CW complex X. Following Gromov, we introduce the notion of completeness, which translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes. 
In the second part we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. We provide complete proofs of Gromov's Mapping Theorem (which implies that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. 
The third part is devoted to the study of locally finite chains on non-compact spaces. We expand some ideas of Gromov to provide complete proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the l^1-invisibility of closed manifolds in terms of amenable covers. As an application, we give the first complete proof of the vanishing of the simplicial volume of the product of three open manifolds.

Gromov's theory of multicomplexes with applications to bounded cohomology and simplicial volume

We use the theory of singular foliations to study $$ \mathcal{N}=1 $$
 compactifications of eleven-dimensional supergravity on eight-manifolds M down to AdS3 spaces, allowing for the possibility that the internal part ξ of the supersymmetry generator is chiral on some locus $$ \mathcal{W} $$
 which does not coincide with M . We show that the complement M \ $$ \mathcal{W} $$
 must be a dense open subset of M and that M admits a singular foliation $$ \overline{\mathrm{\mathcal{F}}} $$
 endowed with a longitudinal G
 2 structure and defined by a closed one-form ω, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet $$ \mathcal{W} $$
 . When ω is a Morse form, the chiral locus is a finite set of points, consisting of isolated zero-dimensional leaves and of conical singularities of seven-dimensional leaves. In that case, we describe the topology of $$ \overline{\mathrm{\mathcal{F}}} $$
 using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the Spin(7)± structures which exist on the complement of $$ \mathcal{W} $$
 .

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Singular foliations for M-theory compactification

The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of rk ω and Sing ω. The set of the ranks of all forms defining a given foliation without minimal com- ponents is described. It is shown that if ω has more centers than conic singularities then b1(M) = 0 and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree.

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On the structure of a Morse form foliation

Morse theory of harmonic forms

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theorem \ref{th3.5}). The definitions of boundary generic and traversally generic vector fields $v$ are inspired by some classical notions from the singularity theory of smooth Bordman maps \cite{Bo}. Like in that theory (cf. \cite{Morin}), we establish local versal algebraic models for the way a sheaf of $v$-trajectories interacts with the boundary $\d X$. For fields from the space $\mathcal V^\ddagger(X)$, the finite list of such models depends only on $\dim(X)$; as a result, it is universal for all equidimensional manifolds. In specially adjusted coordinates, the boundary and the $v$-flow acquire descriptions in terms of universal deformations of real polynomials whose degrees do not exceed $2\cdot \dim(X)$.

Traversally Generic & Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary

As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary by compact submanifolds , where . Our main observation is that this stratification re-flects the stratified convexity/concavity of the boundary  with respect to the  v-flow. We study the behavior of this stratification under deformations of the vector field v. We also investigate the restrictions that the existence of a convex/concave traversing  v-flow imposes on the topology of X. Let be the orthogonal projection of on the tangent bundle of . We link the dynamics of theon the boundary with the property of in X being convex/concave. This linkage is an instance of more general phenomenon that we call “holography of traversing fields”—a subject of a different paper to follow.

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Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

For a non-vanishing gradient-like vector field on a compact manifold $$X^{n+1}$$
 with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity n, which is the maximum possible. (Among them are trajectories that are tangent to $$\partial X$$
 exactly n times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of X by adapting methods of Gromov, in particular his “amenable reduction lemma”. We apply these bounds to vector fields on hyperbolic manifolds.

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Gabriel Katz

Papers

Morse theory of harmonic forms

Traversally Generic & Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary

Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

Using simplicial volume to count multi-tangent trajectories of traversing vector fields

The stratified spaces of real polynomials and trajectory spaces of traversing flows