So, I’ve put the actual lecture notes below the fold. Today was a back-to-basics day, to make sure everyone has the same definitions (and I think to give people time to get used to the area before diving into the deep end, and giving me a chance to get used to the 6 hour time difference from home.) Tu talked about manifolds a bit, Cattani spoke a bit about complex manifolds, line bundles and complex structures, and Trang spoke a bit about algebraic varieties. Nothing really exciting. I should note that the lecturer’s notes are, in many cases, on the web already, here, and I’ll be posting brief summaries before the notes each day, and when a lecture series finishes, I’ll be compiling those notes into a single pdf and posting them to my personal website, with links here. So, here are the notes, please, leave comments with any corrections or comments.
1. Topology, cohomology and sheaf theory 1 – Tu
A topological space is locally Euclidean if every has a neighborhood and a homeomorphism , where is an open subset of . We call the pair a chart.
Two charts are compatible if and are functions.
A manifold is a locally Euclidean, Hausdorff, second countable topological space on which there existsa covering by compatible charts.
The circle is given by . Taking the upper and lower semicircles, along with the left and right semicircles, provides the desired cover.
1.2. Tangent Space
Let be coordinates on an open set in and be .
is at if there exists with such that is at .
If and is a chart containing , then we define
, the tangent space at , is the vector space with basis at .
Let and be different sets of coordinates at a point . Then , where .
The proof is a direct computation.
A vector field is a function such that . We can write it at as and we call a vector field if the are all .
1.3. Differential Forms
The cotangent space is . It has basis the duals of ‘s at .
A 1-form is a function such that . We can write them near as and call them if the are.
Let be a real vector space, and let , then .
It then follows that and .
Let be the set of alternating functions on with inputs. If and then and
We define .
A -form on is a map such that . It is called if the coefficient functions are.
We denote to be the -forms on . We have that , and for greater than the dimension of .
1.4. Exterior Derivative
If , define a 1-form on a chart by . More generally, let by , then we define .
The exterior derivative satisfies
1.5. DeRham Cohomology
We define the kernel of to be the closed -forms and the image to be the exact -forms, .
is just , because there is no . So we end up with the set of locally constant functions on , and, because is a manifold, is the set of all functions constant on the connected components.
where is the number of connected component of .
An element of is just where , and element of is . So is every function equal to for some ? Yes, by the fundamental theorem of calculus, so is for and is else.
2. Complex manifolds, Kahler metrics, differential and harmonic forms 1 – Cattani
A complex manifold is a manifold with coordinates holomorphic on rather than on .
What is the difference betwene holomorphic and ?
From the PDE point of view, they must satisfy the Cauchy-Riemann equations: is holomorphic if and only if and .
The fact that this causes the function to be analytic(holomorphic) is what gives the theory a very algebraic flavor.
We get a map and another by stereographic projection. The map from to itself that takes the image of the first to the image of the second is on , which is holomorphic. So what are the holomorphic functions? Any holomorphic function gives a bounded entire function by removing a point, and so must be constant! This is in fact true for holomorphic functions on compact connected complex manifolds.
The only global holomorphic functions on a compact connected complex manifold are the constants.
There are no compact complex submanifolds on of dimension greater than 0.
Set where iff there exists such that . We put charts on it by looking at .
When is a compact complex manifold a submanifold of ?
We will also be interested in
is the space of -dimensional subspaces of .
Two manifolds may be diffeomorphic but not holomorphic.
As a manifold, a torus is just . However, it can have many complex structures. To get one, view it as where is a lattice isomorphic to . There are many different lattices, and so we can get many complex structures this way. It is not hard to see that they are distinct.
The theory of complex manifolds splits into compact and noncompact, we’re only going to look at the compact complex manifolds.
So what do we do? We have no holomorphic functions. A function can be viewed as being a map by . To get something like a function, we can change to something that is only locally this product, that is, a line bundle.
A holomorphic line bundle is where And are complex manifolds and is holomorphic such that the fiber over each point is isomorrphic to and it is locally biholomorphic to .
Taking a trivializing open cover , we get maps (or in the case of more general vector bundles) which we will call , and they will satisfy , these are called the gluing data.
We can get another bundle using , and these will give the dual bundle, whose fibers are naturally the dual of the original, and call it . Similarly, we can take and put the transition functions in blocks, and we get . For line bundles, the transition functions are numbers, not matrices, so their product is another invertible function, so we get a line bundle .
We set and its projection map to makes it a line bundle with transition maps .
So, we replace the notion of a global function with a global section of a line bundle , which is a map such that . This means that on each we have a function and .
Let be a homogeneous polynomial of degree . Then on , set , these form a global section of . But, if we remove the dual, there are no global sections.
3. Algebraic varieties and schemes over any scheme. Non singular varieties 1 – Trang
Let be a field and the polynomial ring with coefficients in . Then we have two objects: polynomials and polynomial functions obtained by turning polynomials into maps . For finite fields, these are very different, though they are the same for infinite fields.
A subset is called an algebraic set if it is equal to the zero locus of a collection of polynomials.
So any collection of polynomials has zero locus determined by finitely many of them.
3.1. Zariski Topology
Closed sets are the algebraic sets, and , and open sets are the complements. This is actually non-Hausdorff.
Let , then if vanishes on an algebraic set , then some power of is in the ideal . In particular, this implies that .
Let where the are irreducible, that is, are not the union of two properly contained algebraic sets. Then, is irreducible if and only if is prime.
If , we call it an algebraic function, and there are also regular functions.
The set of algebraic functions on is isomorphic to .
is regular at if there exists a Zariski neighborhood of such that on it.
is regular on if it is regular at each .
The set of regular functions on is ring isomorphic to .
A morphism of varietes is a continuous function which pulls regular functions back to regular functions .
3.2. Projective varieties
where if there exsits such that . Now, we look at as a graded ring, with gradations the polynomials homogeneous of a given degree .
Let be a set of homogeneous polynomials. Then the algebraic projective set of zeroes. These give a Zariski topology on , and we see that is a homogeneous ideal and we define the coordinate ring to be .
For projective algebraic sets, a regular function is one that is locally with and not vanishing on the neighborhood.
An algebraic variety is then a Zariski open subset of a projective variety, and this gives us a category of algebraic varieties.
Let be a topological space and look at the category of open sets. A presheaf is just a contravariant functor to the category of abelian groups.
A sheaf is a presheaf along with two conditions:
- such that its image in is zero for all is zero.
- If we have which agree when restricted to , then there exists which restricts to each of them.
4. Complex manifolds, Kahler metrics, differential and harmonic forms 2 – Cattani
Example 1.16 and A.1
In mathematics, we often try to reduce problems to linear algebra, because linear algebra is something that we understand very well.
If we set . In fact, we get , and these are the changes of coordinates preserving the matrix .
On a complex manifold , it is possible to define a linear map by and .
Then and so .
Now, take . Call and , the parts with eigenvalue and .
We can actually, for a complex manifold , make into a complex vector space, by defining . And then, this vector space is isomorphic to by taking to .
So what is the tangent bundle of ? Let a curve. Fix . So the curves don’t depend on motion inside the line, thus we end up with , and by changing the lines to subspaces, the same is true of Grassmannians.
Now, we write where the first has basis and the second has basis . Then, .
We now set to be the space of forms whose terms have where and . Then the exterior derivative maps . So , both of which square to zero, and also .
We then have that . The Dolbeault cohomology, then, is is the cohomology of .
Inside of is , which is the collection of the forms with holomorphic, not merely , coefficients.
Now we note that the Dolbealt cohomology is NOT diffeomorphism invariant. It depends on the complex structure of the manifold in question.