Text references of the type (4.6) refer to the 6th equation (or theorem, lemma, etc.) in Sect. In the nearly four decades since the first edition of this book appeared, many of the topics covered have evolved in a variety of interesting ways.
Elliptic Operator Theory 108 1. Sobolev Spaces 108
Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217
Manifolds
If D is an open subset of Kn, we will deal with the following function spaces on D:. So in what follows we will use three categories – differentiable (S= E), real-analytic (S = A) and holomorphic (S=O) categories – and the note above states that each is a subcategory of the first.
Vector Bundles
In the same way as in Example 2.4, we can make the union of these tangent spaces into a holomorphic vector bundle over X, i.e. This clearly defines an equivalence relation on the S-bundles over an S-manifold,X. The statement that a bundle is locally trivial now becomes the following:.
Presheaves and Sheaves
Then a stack of modules over the structure OX of X is called an analytic stack. Rp, thus defined, is clearly a stack of R-modules and is called the direct sum of R (p times; p=0 corresponds to the 0-module).
Resolutions of Sheaves
Example 2.10: Let X be a differentiable manifold of real dimension m and let EpX be the sheaf of real-valued differential forms of degree p. Proof: The proof consists of an application of the Poincaré lemmas for the operators d,∂, and ∂¯ (see Examples 2.10 and 2.12).
Cohomology Theory
Before we move on to the consequences of Theorem 3.2, we would like to introduce another class of disks, which will give us many examples of soft disks. Then Theorem 3.2 applies, and the given section of C over S to be extended to all of X comes from a section of B over S, which then extends by softness to all of define the cohomology groups of a space of coefficients in a given bundle.
Remark: Note that in the proof of the previous theorem we did not use the definition of cohomology, but only the formal properties of cohomology as given in Theorem 3.11 (i.e., the same result holds for any other definition of cohomology that satisfies the properties in theorem 3.11). We want to consider a generalization of Theorem 3.17, and for this we need to introduce the tensor product of modules.
Cech Cohomology with Coefficients in a Sheaf ˇ
We will mostly use resolutions of particular curves to represent cohomology, mainly because the techniques we develop are derived from the theory of partial differential equations and applied to differential forms and their generalizations. This chapter is an exposition of some of the basic ideas of Hermitian differential geometry, with applications to Chern classes and holomorphic line bundles. 1 gives us the basic definitions of the Hermitian analogues of the classical concepts of (Riemannian) metric, connection and curvature.
More specific formulas are obtained in the case of holomorphic vector bundles (in Section 2) and holomorphic line bundles (in Chap. In the case of line bundles, we give a useful characterization of which cohomology classes in H2(X,Z) are the first Chern class of a line bundle.
Hermitian Differential Geometry
We prove that Chern classes are the primary barrier to finding trivial subbundles of a given vector bundle and, in particular, to the given vector bundle itself being trivial. Namely, the principal bundle P (E) has fibers isomorphic to GL(r, C) with the same transition functions as the vector bundle E→X. Therefore, (1.1) gives a vector representation for sections ξ ∈E(U, E), and (1.3) shows how the vector transforms when changing the frame for the vector bundle E.
Suppose E→X is a vector bundle endowed with a connection D (as we will see below, every vector bundle admits a connection). Returning to the problem of defining the curvature, let E −→X be a vector bundle with a connection and let θ (f )=θ (D, f ) be the associated connection matrix.
The Canonical Connection and Curvature of a Hermitian Holomorphic Vector Bundle
Recalling Lemma 1.6(a), we see that this is the necessary transformation formula for θ to define a global connection. This theorem gives a simple formula for the canonical compound in terms of the metric h; namely,. Then the choice of A(z) given by (2.6), depending on the derivatives of the metric h, ensures that (2.5) holds.
This lemma allows us to compute the curvature at a specific point without computing the inverse of the local representation for the metric, provided we have the right frame. Alternatively, we can think of (ξ1, . . . , ξn) as homogeneous coordinates for Pn−1, and by the homogeneity of (2.9), we see that the expression in (2.9) induces a well-defined 2-form on all of Pn−1, which corresponds to the 2-form onU mentioned above.
Chern Classes of Differentiable Vector Bundles
It is easy to check that these definitions are independent of the local coordinates used and that α(t )˙ en+b. It follows from Theorem 3.2 that the Chern classes are well defined and are independent of the relation used to define them. The Chern classes are therefore topological cohomology classes in the basis space of the vector bundle.
Note that the Chern class integral over P1 was actually 2, which is the Euler characteristic of P1. In example 2.4, we calculated the curvature in the appropriate coordinate system and obtained
Complex Line Bundles
Let Ai be the set of unordered (i+1)-tuples of different elements of the index set {ϕα}. Note: This is half of the classification theorem for vector bundles, Theorem I.2.17, discussed in Sect. Let H˜q(X,Z) denote the image of Hq(X,Z) in Hq(X,R) under the natural homomorphism induced by the inclusion of constant sheaves Z⊂R [that is, H˜q(X,Z ) is integral cohomology modulo torision].
Moreover, one can show that the Chern classes of the universal bundle Ur,N, suitably normalized, are integral cohomology classes, and therefore a version of Proposition 4.3 is valid for vector bundles (see Chern [2]). Using the functions {gαβ}, one can construct a holomorphic line bundle with these transition functions.
Sobolev Spaces
2 we will discuss the basic structure of differential operators and their symbols, in Chap. Using parametrics, we will show that the kernel (null space) of L is finite dimensional and contains only C∞ sections (regularity). In the case of self-adjoint operators, we will obtain Hodge's decomposition theorem, which states that the vector space of the sheaf sections is the (orthogonal) direct sum of the (finite-dimensional) kernel and range of the operator.
5 we will introduce elliptic complexes (a generalization of the basic model, the de Rham complex) and show that the Hodge decomposition in Sect. As usual, we will denote the compactly supported sections‡ by D(X, E)⊂E(X, E) and the compactly supported functions by D(X)⊂E(X).
Differential Operators
A differential operator is said to be of order k if no derivatives of order≥k+1 occur in a local representation. It is also easy to see that the σk(L), thus defined, is independent of the choices made. Proof: One must show that the k-symbol of a differential operator of order k has a certain form in local coordinates.
We include the ik factor so that the symbol of a differential operator is compatible with the symbol of a pseudodifferential operator defined in Sec. We have given a brief discussion of the basic elements of partial differential operators in a setting suitable for our purposes.
Pseudodifferential Operators
Since the variable thex has compact support and due to the estimate (3.2), we have (as before) the estimate, for any large N, . Proceeding with the proof of (3.5) we immediately obtain from (3.6) and estimate (3.7), allowing C to denote a sufficiently large constant at any estimate. We will see that the smallest possible integer m will in some sense be the order of the pseudodifferential operator on X.
Using the estimates (3.2), we see that the integral on the right converges for N sufficiently large. One of the fundamental results in the theory of pseudodifferential operators on manifolds is contained in the following theorem.
A Parametrix for Elliptic Differential Operators
See Riesz and Nagy [1], Rudin [1], or any other standard reference on functional analysis for a discussion of this, as well as a proof of the above proposition. To show that HLs ⊂E(X, E) is known as the regularity of homogeneous solutions of an elliptic differential equation. We note that S is called a softening operator precisely because of the role it plays in proving the above lemma.
Proof: First we have to solve the equation Lξ = τ, where ξ ∈ W0(E), and then it will follow from the regularity (Theorem 4.9) of the solution ξ that ξ is C∞ since τ is C∞, and we get our desired solution . The Atiyah–Singer index theorem asserts that i(L) is a topological invariant depending only on (a) the Chern classes of E and (b) a cohomology class in H∗(X,C) defined by the top-order symbol of the differential operator L .
Elliptic Complexes
The proof of this theorem is a simple consequence of the fact that a spectral series exists (Fröhlicher [1]). As before, it follows from Theorem 5.2 thathp,q(E) <∞, and we can define the Euler characteristic of the holomorphic vector bundle E to be. This follows from the fact that thecj(E) are the elementary symmetric functions of the(x1, . . , xr) (analogous to the case of the coefficients of a polynomial).
1 we will present a discussion of the exterior algebra on a Hermitian vector space, introducing the fundamental 2-form and Hodge. 5 to prove Hodge's Decomposition Theorem, which expresses a de Rham group as a direct sum of Dolbeault groups (of equal common degree).
Hermitian Exterior Algebra on a Hermitian Vector Space
Note that the definition (1.12) does not depend on the choice of the basis, and is an intrinsic definition of a volume element on E′. Remember that the real operator J representing the complex structure of the vector space F has the property that if v∈ ∧1,0F, then J v=iv, and if v ∈ ∧0,1F, then J v= −iv. We therefore immediately see that J defined above is the natural multilinear extension of the complex structure operator J to the exterior algebra of F.
We now define a linear mapping of the terms of , the fundamental form associated with the Hermitian structure of E, namely, let. It is clear that the result of the ∗action in the expression in parentheses is of the form.
Harmonic Theory on Compact Manifolds
However, the representation we have given here for the inner product in E∗(X) will be very useful, as we shall see. Then we can define the Hodge inner product on E∗(X) with respect to the underlying Riemannian metric and a fixed orientation given by the complex structure (all complex manifolds are orientable). We are now in a position to compute the connections of the various operators with respect to the Hodge inner product.
Additionally, all links in this and subsequent sections of the book will be to Hodge's internal product. Proposition 2.3: Let X be an oriented compact Riemannian manifold of real dimension m and let =dd∗+d∗d, where the attached d∗ is defined with respect to the Hodge inner product on E∗(X). according to the rule for differentiating a product of forms.
A representation of a Lie group (eg, a matrix group)G into a complex finite-dimensional vector space V is a real-analytic homomorphism ρ: G→GL(V), where GL(V) denotes the Lie group of nonsingular endomorphisms of the vector space V. We now have the basic description of an irreducible representation of sl(2,C) in a complex vector space of finite dimensions. Then applying the second "scale operator", we see that. m−r)!vr, and thus we obtain the useful identity.
Now we note that the identity (3.11), which involves both the representation of SL(2,C) and sl(2,C), was derived from this particular explicit representation, but we see from its form that it will be valid on any irreducible representation of SL(2,C)andsl(2,C) on a vector space of dimension m+1. Now consider a specific representation of sl(2,C) on the outer algebra of forms on a Hermitian vector space E.
