We care about projective algebraic varieties. We know that an analytic subvariety of the complex projective space is algebraic by the theorem of Chow. We want to know if a complex Kahler manifold is going to be algebraic. A an even dimensional riemannian manifold is a Kahler manifold if there is a real endomorphism of the tangent bundle such that , i.e. an almost complex structure satisfying and the form , is closed .
Let be a compact complex manifold and a holomorphic line bundle on it. Any subspace determines a linear system latex M$ by their zero sets. Since is compact, if and only if for some , and is parametrized by . If the linear system has no base point for any point the set of sections vanishing at forms a hyperplane and we have a map . More explicitly let form a basis of and open and a local trivialization of on ,
Clearly is holomorphic and .
The above gives a sketchy description of the way in which one embeds the complex manifold into complex projective space with sections of a line bundle. I will have to come back to work out the details so this is clearer. But I want to get to the deep mathematics issue here, which is Kodaira’s great 1954 embedding theorem.
A line bundle on is defined to be positive if the first Chern class in the sense that its representative in is positive.
Theorem:
Let be a compact complex manifold and a positive line bundle on . There exists such that for the map is an embedding.
In the last episode, we have gone through the argument that if is a character that satisfies (i) it is a finite Fourier series with integer coefficients, (ii) it is skew symmetric under the action of the permutation group , and (iii) for different irreducible representations then . We showed that the tildemap construction given by then in fact any satisfying the conditions (i) and (ii) and (iii) must be of the form for with decreasing values. Now the problem is to show that every that satisfies there conditions occur for satisfying (i), (ii) and (iii) as we vary the representation .
If does not occur, the idea is to produce a contradiction with the completeness of . Consider which is invariant under , belongs to and is orthogonal to for as varies over irreducible representations and this will contradict the completeness of in .
SINGULAR POINTS
The integration formula that Weyl uses for the orthogonality relation is valid for points that are ‘nonsingular’ in a sense. The condition is that the conjugacy class must have maximum possible dimension. At a point we want to understand the condition on the conjugacy class to which belongs, recall that belong to the conjugacy class if and only if there exists such that . We next want to characterize the conjugacy class as exactly the coset space where is the centralizer of , i.e. those that commute with . Let’s be elementary about this, so if the the condition is the same as which occurs if and only if . Therefore we can exactly identify the distinct elements of the conjugacy class of with . The two points of note is first that maps on to the conjugacy class of , and a different argument for distinctness of points of the conjugacy class is implies which implies that $latex y_2^{1} y_1 \in G_x$ and .
Now let us try to understand the dimension of this space . If are distinct eigenvalues of with multiplicities then we want to calculate the dimension of in terms of these multiplicities. Elements of preserve the eigenspaces for implies both $latex as well as . Therefore for a fixed we certainly have that at most components contribute to . Therefore . On the other hand, Varadarajan claims that this is an equality, i.e. that . In other words, that the dimension of matrices that commute with the matrix of the multiplicity eigenspace of eigenvalue has dimension . This is not clear to me yet why. This dimension, seems too large at first glance but I am quite confident that my worries are unnecessary so skipping the issue:
Our original question is when is maximized. We have the constraint that . Therefore .
Now and the two sides are equal when and . Actually this last part tells us that we needed only and not equality.
Since is maximum if and only if is minimum, above can be used to conclude is maximum if and only if has distinct eigenvalues, which is the same as the characteristic polynomial having distinct roots.
THE DISCRIMINANT FUNCTION
The characteristic polynomial with a variable . Now is invariant under permutations and by Newton’s theorem is a function of the symmetric elementary functions where for example and so on. The discriminant is such that latex x$ has distinct roots. We call is regular if has distinct eigenvalues and denote the set of regular values as and call the complement the singular set. The key point is that the singular set is small. More precisely, the Haar measure which is used for all the integration is absolutely continuous with respect to the Lebesgue measure in coordinates and since the singular set is defined as the zero set of an analytic function, it has Haar measure zero.
OUTLINE OF THE PROOF OF WEYL CHARACTER FORMULA
Recall that the Character Formula is that the irreducible characters of are in natural onetoone correspondence with where . The character
where is the character
Define
For an irreducible representation define . Then the following are relatively easily established:
(i) has finite Fourier series with integer coefficients
(ii)
(iii)
Since a finite Fourier series with integer coefficients which is skew symmetric with integer coefficients from the formula while is a symmetric finite Fourier series with integer coefficients (i) and (ii) are clear while (iii) is just the orthogonality relation.
The interesting and quite sophisticated part of the proof is to construct all possible that satisfy the three conditions.
I am going to come back to the pieces maybe tomorrow.
Ok now it is tomorrow, March 18 2018.
For any diagonal character the alternating sum
is a finite Fourier sum and if then . For the first part, we just inserted and then we changed the index of the sum over all the permutations in which shows (ii).
Any character satisfying (i) and (ii) must be of the form
where the coefficients are integers and
If and are in different orbits then . Recall that the characters are elements of and the torus. If and are in different orbits then there are no permutations that can equate . The orthogonality is actually the outcome of a separate result known as ‘the orthogonality relation for characters’: . If the then .
HERE IS ANOTHER ATTEMPT TO FOLLOW THE PROOF:

Recall that is the set of matrices with complex entries with . Here is the conjugate transpose. We denote by the diagonal matrices of Recall that a representation of a Lie group is a group homomorphism . Recall that a character of a representation is a map defined to be the trace of the matrix , so
So here is the theorem I want to understand, the Weyl Character Formula for the group. The work of Weyl is deep and extensive, but I want to go slow so I can understand things clearly.
The irreducible characters of are in natural onetoone correspondence with ntuples with $m_1 > \dots > m_n$. The character is given on the diagonal matrices by
where is the set of permutation of the indices, is the character , and
I am following Varadarajan’s An Introduction to Harmonic Analysis on SemiSimple Lie Groups and follow his presentation.
The permutation group acts naturally on the diagonal matrices as well as the dual group of characters of , homomorphisms of , so .
A conjugacy class is where the equivalence relation on is defined by if and only if there exists such that and since unitary matrices are diagonalizable, every conjugacy class meets the diagonal , and is a single orbit, namely the orbit of permutations of a fixed set of eigenvalues .
For any class function on , i.e. a function that is constant on each conjugacy class, and denote by , obviously and are in onetoone correspondence. Furthermore, must be invariant under the action of permutations .
If is a finite dimensional unitary representation of the restriction of to is a direct sum of characters of .
Let’s unpack the above a little bit more. The PeterWeyl theorem asserts that the irreducible unitary representations of a compact Lie group are all finite dimensional, the irreducible characters form an orthonormal basis of composed of functions on and a subspace is invariant if $\pi(g) L \subset L$ for all . The PeterWeyl theorem also tells us that decomposes as a direct sum of irreducible representations. Although this is using too big a hammer, the PeterWeyl Theorem has the corollary that the restriction of to the diagonal matrices of is a direct sum of characters
with integer coefficients The diagonal elements of a unitary matrix are of form , to put things in context. So the irreducible representations on the diagonal of the group are finite Fourier series with nonnegative integral coefficients, and it is moreover symmetric or invariant.
The idea behind Weyl’s method is to start with the orthogonality relations for . The integral is over the entirety of but Weyl breaks up the integral into diagonal D and then the orbits of the conjugacy as a double integral.
So this is the crux of the matter — if we can decompose the integral in the above manner, the character formula will emerge.
Now I am tired already and will return to the rest of the proof later.
This problem seems equivalent to Weyl’s equidistribution theorem: the points fill up the interval [0,1] if and only if is irrational. This is the idea. Fix and we want to approximate any given by for some integer . Ok actually this is a different proof of the Weyl equidistribution.
The map is obviously surjective. Now implies two things: that and . The first equation implies there exists an integer such that and therefore then which cannot be an integer unless so the second equation cannot hold. Thus is injective. Now we can apply the result that a bijective homomorphism of locally compact Hausdorff topological groups must be a homeomorphism to conclude that is a homeomorphism. Therefore cannot be closed, for otherwise it must be compact, which is impossible since is noncompact. The closure is a compact subgroup of . Assume this closure is onedimensional. Then it must be realized by the exponential map from a onedimensional subalgebra of the Lie algebra of the torus by the exponential map, so there exists such that . This is only possible by the previous argument when . Therefore cannot be onedimensional. Therefore it must be two dimensional. But the only 2dimensional subalgebra of the commutative Lie algebra is the entire space. This implies and is dense in .
Let be a bijective homomorphism between locally compact topological groups. Let which we claim is also a homomorphism: so is a homomorphism.
Let be open which assures us is open. We would like to show is open. Regardless of whether it is open or closed, it must be nonmeager. Then is an open neighborhood of the identity regardless of whether is a priori open. Then is open since is continuous. Now is open. We want to conclude from open to open. Suppose is not open. Then neither is and hence is not open.
If is a leftinvariant metric:
(a)
(b)
(c)
Ok, so the homogeneous spaces are those which occur as for some subgroup and everyone’s favourite homogeneous manifolds are spheres and hyperbolic spaces which have constant curvatures. The map is a riemannian submersion defined by the property that is an isometry.
An interesting result is that if is a riemannian submersion and and is a horizontal lift then is a geodesic if and only if is.
The most interesting homogeneous space is probably the upper half complex plane with consisting of matrices with determinant 1 by Moebius transformations . Grassmannians , projective spaces are homogeneous, etc. The entire Thurston Geometrization program is to decompose compact three dimensional manifolds into homogeneous pieces. Yau likes to emphasize the search for special metrics like Einstein metrics . Most known Einstein metrics with are homogeneous.
If is a holomorphic vector bundle on a complex manifold and is a connection then the curvature is a 2form with coefficients in the . The Chern class is given by the degree closed form where is the polynomial invariant under conjugation that associates the th symmetric function of the eigenvalues. The Chern classes of a complex manifold is defined to be that of the tangent bundle. For the hermitian metric is and . Recall the symmetric polynomials are 1, , and so on. So the first Chern class will be the sum of the eigenvalues of the matrix considered as a matrix . So . Evaluate by contour integration for fixed radius by drawing a circular arc around the origin which will include poles at and the other component the so . The second term is dominated by which will go to zero. So
The poles are of order 4. We have to use the formula .
The residue at is which is a messy computation I don’t feel like doing right now.
For the complex projective space, the total Chern class is where is . The tautological line bundle sheaf, the holomorphic sections of the tautological line bunles is . If we realize a complex line through origin in as a point of then the tautological line bundle is defined to be the subbundle of where at the point the fiber is the vector space . It’s holomorphic sections form the sheaf by definition so then one defines as the dual sheaf and the various are th tensor powers of .
is the standard projection. For any complex line through the origin the tangent space is identified with the linear functions from to its orthogonal complement.
There is an exact sequence . I don’t understand the standard argument yet. where .
Then . By the Whitney formula
where is the generator of given by the negative of the first Chern class of the tautological line bundle using . So .
Now for something slightly more interesting from the point of basic curiosity. What are the Chern classes of a projective variety defined by two of homogeneous equations of degrees . This is a complete intersection. Its degree is . Suppose be the inclusion map. We have the tangent and normal decomposition.
Ok, so I just put in this note a link to an article with discussion of practical computations of Chern classes of algebraic varieties that I am putting down as homework for myself. I don’t understand what Chern classes really mean geometrically and analytically. And I guess my education on actually computing Chern classes is pretty weak, which is the point of this note. In geometry the only examples of things are generally just algebraic objects and homogeneous objects and it’s a little sad that I don’t know how to compute Chern classes for these objects in this late stage of my life.
Step by step, slowly.
If is a complex Kahler manifold and is a line bundle and is a connection on the line bundle, then this defines a holomorphic structure on .
Recall that a Kahler manifold has a hermitian metric , i.e. at each point we have and and the 2form is closed satisfying . Basic properties of these objects and some known results are summarized here. The complex structure satisfied and therefore are the eigenspaces of . Using this decomposition one considers the sections of differential forms which split up into forms. The exterior derivative splits into and the de Rham complex has a corresponding splitting. Hodge theory gives .
A connection is equivalent to a map satisfying .
Ok so in this basic setup, the holomorphic structure defined by the connection is a map defined by .
So this is right, as we can check with some presentation of the basic issue. I am going to be spending some time carefully picking up the pieces of the setting of interest in DonaldsonSun theory because I am extremely rusty on these matters.
So DonaldsonSun theory is taking a class of Kahler manifolds with bounds on volume and so on for which Gromov Compactness applies and then using Sobolev inequality and Hormander’s technique with UNIFORM CONTROL over this parameter space to get control of embedding into a complex projective space. Just as MOSER had looked at NASH embedding and picked out the softer scheme that took over mathematics, I am wondering whether it is a good project for ZULF to look at DONALDSON’s delicate work of genius and pull out a softer UNIVERSAL technique here. You take GromovHausdorff convergence, you ask for what sort of things happen in the limit and control singularities of various types. Let’s remember this idea. This could change mathematics if implemented. WITTEN had looked at Donaldson’s earlier genius work and realized that softer SeibergWitten invariants can reprove and extend the topological results. In the same way, detailed constructions and estimates in his current work is likely to contain some universalizable things. It’s an obvious sort of thought and one that Donaldson himself might be looking for directly himself but sometimes there is value in taking vast distance.
In order to unrustify my mathematics.
Implicit Function Theorem: Let be a map where are Banach spaces. Let and suppose is invertible as an element of . Then there exist so that for all with there is a unique with so that for all with . Moreover and .
Pick any covering of with radius 1 open sets of polydisk type and choose a partition of unity based on this covering. We can use this covering to reduce to proving the coarea formula for supported in a ball of radius 1. So now just assume that support of is contained in such a polydisk, say . Now choose a dense countable set in the ball with the additional property that the implicit function theorem for each of these points to obtain pairs of numbers promised by the theorem. Use the compact closure of the polydisk to choose a subcover. Now we have a finite set of polydisks with centers that cover the support of with the additional property that the implicit function theorem guarantees that . It thus suffices to prove the coarea formula for one of these open polydisks.
Now it’s a matter of interpreting the implicit function theorem. Geometrically it is producing a local parametrization of the fibers of the map $late \phi: \mathbf{R}^n\rightarrow \mathbf{R}$.
The implicit function theorem provides us with such that such that for , the is parametrized by a hypersurface of the polydisk.
We apply this change of coordinates to the integral, and since the fibers are smooth, in this case the Hausdorff measure agrees with the usual hypersurface measure.
For this problem, the application of the implicit function theorem I am happy with but the rest of the argument is too handwavy. So I will come back to it another time.