More of the background stuff. Tomorrow I’ll be making two posts, one around lunchtime here, with the last background material and links to complete notes without having to deal with the vagaries of wordpress formatting (I’ve noticed that theorem, example, proposition etc environments are all gone…just doing a quick latex2wp and then making sure everything compiles…the final notes will be marginally nicer), and one in the evening when we get to actual Hodge theory.

**1. Tu 2 – Computation of deRham Cohomology **

** 1.1. Pullback of Forms **

If is a map, then there is a pullback map .

For , then . In generall,y locally can be written and we define .

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By 3, we know that of a closed form is closed. If , then . Now, , and so exact forms pullback to exact forms. Thus, induces a map (also denoted by on cohomology .

If is a diffeomorphism, then there exists such that and , then and .

If is a diffeomorphism, then is an isomorphism.

** 1.2. Homological Algebra **

A cochain complex is such that . So is defined.

A sequence of vector spaces is exact at if .

A linear map of cochain complexes is a cochain map if (this is the data of a map in each degree).

A cochain map induces .

A short exact sequence of cochain complexes induces a long exact sequence on cohomology .

** 1.3. Mayer-Vietoris Sequence **

Suppose that is covered by open sets so that . Then we have by the restriction maps followed by the difference of restrictions.

The sequence above is exact.

** 1.4. **

We cover by the union of two copies of . Then and is two copies of , and so .

Then the Mayer-Vietoris sequence becomes So by exactness, , because the image of is the diagonal.

** 1.5. Smooth Homotopy **

Two maps are smoothly homotopic if there existsa map such that and (where we say that is if it can be extended to a function in a neighborhood of in ). We write .

has a homotopy inverse if there exists such that and

Then we say that and have the same homotopy type and is a homotopy equivalence.

\underline{Homotopy Axiom}:Homotopic maps induce the same map on cohomology .

has the homotopy type of a point . We have the inclusion of the origin and the unique map to the point. Then is the identity map on the point, and sends every point to . We claim this is homotopic to the identity. Define by to be the homotopy.

If is a homotopy equivalence, then is an isomorphism.

is in degree 0 and else.

**2. Trang 2 – Sheaves **

Let be a topological space, sheaves. A morphism of sheaves is a map for each open set compatible with the restriction maps.

We define the stalk at of to be as abelian groups.

We define the associated sheaf to a given presheaf to be the sheaf such that every map from the presheaf to any sheaf must factor through, and denote it . This is unique up to unique isomorphism.

The associated sheaf has the property that for all .

Now, let be a continuous map and a sheaf on . We define by .

For , and on , we define to be .

** 2.1. Ringed Spaces **

A pair where is a sheaf of rings on is a ringed space. A morphism of ringed spaces is a continuous map and a map .

A locally ringed space is a ringed space such that the stalks of the sheaf are all local rings, and a morphism of locally ringed spaces is required to induce on the stalks maps .

Take where is an algebraic set and is the sheaf of regular functions. This example is the fundamental one in algebraic geometry.

The locally ringed spcae with the sheaf being the sheaf of local holomorphic functions is the fundamental example in analytic geometry.

** 2.2. Local Analytic Spaces **

For , holomorphic for . Then we know is a locally ringed space, look at . Set and then is a sheaf of ideals.

So, we can now distinguish between and , the first is just a point, the second is a double point, and can be viewed as the intersection of a parabola and a line tangent to its vertex.

** 2.3. Affine Schemes **

Let be a ring (commutative with identity). Then is the set of prime ideals of , and the closed sets are given by tkaing an ideal in and setting to be the set of prime ideals containing . The open sets are their complements.

We define on the sheaf , whose stalks at is , and for any open set , we define by is a section if with for all there exists an open neighborhood and such that for all we have and .

The set of morphisms is the same as the set of homomorphisms .

** 2.4. Schemes **

A locally ringed space is a scheme if for every there exists a such that is isomorphic to an affine scheme.

If is a graded ring , then we define a scheme by is the set of homogeneous prime ideals in . We want to set up to be , which is the set of elements of degree zero in , where is the set of homogeneous elements which are not in . We set if such that for all there exists and homogeneous of the same degree such that for all , and .

**3. Cattani 3 **

Let be a real vector space along with an operator . This makes it a complex vector space. We can also say where is the -eigenspace and the -eigenspace. We write along with , which are the and eigenspaces. Then with . So , which are eigenspaces respectively.

So now, we have and for each we get a complex which is exact for a small enough (more precisely, exact as a complex of sheaves.

For , then with holomorphic.

The Following are equivalent

- A symmetric bilinear form such that
- An alternating form such that
- A hermitian form with .

Now, we move to manifolds. Every complex manifold has a positive definite Hermitian structure on the holomorphic tangent bundle, which is equivalent to every complex manifold has a Riemannian metric compatible with .

By this, we mean that on , we have that and .

We define a hermitian structure on to be Kahler if . This implies that .

** 3.1. Symplectic and Kähler Manifolds **

A symplectic manifold is a pair where and , with a 2-form. We’ll assume that is compact. Then being Kähler implies that is symplectic, because , and so , so each is nonzero.

Calabi and Eckmann proved that for , there was a complex structure on , and these can never be Kähler.

In fact, any compact symplectic manifold has an almost complex structure.

is Kähler with metric .

has a Kähler structure, by taking on each the sunftion . On , we have , we then take logs and apply , and we find that . So we set , and the metric we construct is the Fubini-Study metric.

Let . Then for all , there exist coordinates on around Such that is described by . If is Kähler and is a submanifold, then is Kähler.

Thus, if is a submanifold of , then is Kähler. Thus, is necessary.

**4. Tu 3 – Presheaves and Cech Cohomology **

A presheaf on a topological space is a function that assigns to each open an abelian group and to every inclusion a group homomorphism such that , and .

is the -forms on . This is a presheaf on a manifold .

If is an abelian group, for every open , define to be the locally constant functions . Then is a presheaf.

** 4.1. Cech Cohomology of an Open Cover **

Let be an open cover of a topological space indexed by a totally ordered set. We’ll denote intersections by putting the subscripts together.

When gives the cover, we have Mayer-Vietoris, which says .

Now, let be a presheaf on a topological space . We then have a sequence

Define to be the term involving open sets. Then we define by .

It turns out that , and so we define the cohomology of this complex to be the Cech cohomology .

** 4.2. Direct Limits **

A directed set is a set with a binary relation that is reflexive, transitive and such that any two elements have a common upper bound.

Fix . Let be the set of neighborhoods of in and say that iff .

Fix a topological space . Then be the set of all open covers of . An open cover refines if every is contained in some . Refinement gives a directed set structure to the set of covers. A refinement of can be given by a refinement map on the index sets stating which each is contained in.

A directed system of groups is a collection of groups indexed by a directed set such that for all we have a homomorphism satisfying that and .

Let be the neighborhoods of . Then set and say that and are equivalent iff there exists such that . We call these the germs of functions at .

In , let and . Then we say if there exists such that and define .

** 4.3. Cech Cohomology of a Topological Space **

For each open cover, we have , and we have restrictions making it into a directed system of abelian groups. So we define to be the limit of this system.

** 4.4. partitions of unity **

A partition of unity on a manifold is a collection of functions with and .

We define the support of a function to be the set where it is nonzero.

A collection of subsets in is locally finite if every has a neighborhood that meets only finitely many of the .

Given any open cover of a manifold, there exists a partition of unity with each element’s support contained in one of the open sets of the cover.

**5. Trang 3 – Projective Schemes **

Let be a graded ring, look at . Let , then we define to be the homogeneous primes not containing . is isomorphic to .

For any ring , we have and we’ll call it .

** 5.1. Gluing Schemes **

Let and be two schemes wuch that is an isomorphism. Then we can construct a new scheme by identifying them along this map.

** 5.2. Schemes over a scheme **

Let . We call this an -scheme, and often abuse notation by calling an -scheme.

In particular, we will look at schemes over .

** 5.3. Varieties and Schemes **

Any variety is covered by a finite number of affine algebraic varieties.

This means that we can take any variety over , and make a scheme over out of it, by just taking affine varieties to .

Now, we say that a scene is connected if is, irreducible if is, it is reduced if the rings are all reduced (have no nilpotents) and integral similarly.

A scheme is integral if and only if it is reduced and irreducible.

A scheme is locally noetherian if it has a covering by spectra of noetherian rings.

Now, let be a morphisms of schemes.

We say is locally of finite type if is covered by such that with the being -algebras of finite type, that is, are finitely generated as algebras. We say that it is of finite type if the are finite as modules.

An example is the map from a parabola to a line, which induces .

We can construct fiber products, take and , we can get , it’s the unique scheme such that for all maps and that are equal after composing with the maps to , we get a unique map .

This allows us to define base change: If we have a map and another , we can define and we have , the base change of the morphism.

For any , we have a map by using the same map to create the fiber product. If this morphism is closed, then we say that is separated.

We call a morphism proper if it is separated, of finite type and universally closed, and we say that a scheme which is proper over is complete.

** 5.4. Projective Morphisms **

Let be a scheme. We define to be projective space over , where .

We say that a morphism is projective if it factors through and the map is a closed immersion.

Projective morphisms of Noetherian schemes are proper, and quasi-projective morphisms are separated and of finite type.

So a variety over turns out to just be a scheme over which is integral and of finite type.

**6. Cattani 4 **

Let’s look at the real, smooth case. Let be a compact oriented Riemannian manifold.

If is a real vector space which is oriented with an inner product, then has an inner product as well.

Show that .

Volume element given by , where the form an orthonormal basis.

We have a map , and . So then .

Now, is an isomorphism, and it satisfies .

Back to the manifold . We can define an inner product on forms by . This is a positive definite bilinear form.

Now, we define , and it takes -forms to -forms, using this operator. We claim that and are adjoints, that is, .

, but this will just be , up to sign, we can just insert a in front of , and the signs work out.

Now, if , then may not be closed, but it is if and only if .

if and only if .

One direction is simple, for the other, we have , which using the adjoint property proves the result.

We call this operator , the Laplace-Beltrami operator, or the Laplacian, and we call any form with a harmonic form.

.

Why should we hope that every cohomology class has a harmonic form in it?

Heuristically, start with . Then is the set of forms of the form . Then . Now, suppose is a maximum. Then for all we’ll have that and , so we can assume that .

- , the harmonic forms, is finite dimensional
- .

In particular, every form is a harmonic form, plus of something plus of something. So then any form is . But if it’s closed, then , which implies that , so for any closed form, it is of the form .

Thus, .

Take a submanifold , then for any form in , we can restrict it to and integrate to get a map to . By Poincaré duality, this gives us a class .

Now, we define everything in a “hermitian way.” So we take , and note that takes to .

We define and , and these are of type and and match with and . So then we have , and we can write any form as a sum .

And, we leave off with the fact that, on a Kähler manifold .

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