This is the end of the preliminary background lectures. This afternoon, we start Hodge Theory itself, with Hodge Structures, Variations, and Hodge Theory of Maps as the three lecture series. Tonight, when I post the notes for the next three lectures here, I’ll also link to my website, where lecture notes for each series should be available in PDF format, and a bit nicer than the wordpress ones.
1. Tu 4 – Sheaves and the Cech-deRham Isomorphism
Version 6 of lecture notes on Summer School website.
Compute of , , , for problem session today.
Problem Session: Does depend on the order on the index set?
A sheaf on a topological space is a presheaf such that for any open set and any open cover of , we have
- Uniqueness: If such that for all , then .
- Gluing: If such that for all , then there exists such that .
- is a sheaf for any manifold.
- , the closed -forms on a manifold is a sheaf.
- is a sheaf for any complex manifold.
- , the locally constant functions to , is a sheaf.
However, the presheaf of constant functions is not a sheaf.
1.2. Cohomology in Degree 0
Let be a sheaf on a topological space . We want to know . Let be an open cover of . Then as is a sheaf, we have the Cech sequence.
So then as which is just where is the restriction map . But this is precisely .
Now, let be a refinement of . Both Cech groups are isomorphic to , and so they are isomorphic.
If is a sheaf on a topological space , then .
1.3. Further Computations
For all and all , . But how do we prove this? The standard method for proving that cohomology is zero is finding a map such that . Applying to a cocycle just gives . Therefore, this induces the zero map on cohomology, and so , which can only happen when . We call a chain homotopy. There is a theorem guaranteeing that such a exists when the cohomology is zero.
Let be an open cover of , and be a partition of unity subordinate to . For , then by .
We must check that for , and then we have the vanishing of the Cech cohomology, when we take . We in fact have for on a complex manifold.
But is not necessarily zero!
1.4. Sheaf Morphism
A morphism of sheaves is a collection of maps, one for each open set, compatible with the restriction maps
We define the stalk of a presheaf at to be . So a presheaf homomorphism by .
1.5. Exact Sequences of Sheaves
A short exact sequence of sheaves induces a long exact sequence .
The sequence of sheaves is exact.
Proof: Look at , and similarly, we can make for all .
So we have , and we can finish the proof by basic homological manipulations.
2. Trang 4 – Sheaves of Modules
Let be a ringed space and a sheaf on such that for all , is an -module and these structures are compatible with the restriction maps.
2.1. Locally Free Modules
is free if for all , is a free -module.
On a scheme , say , then an -module on is given by an -module in the following way. Define by the sections over are such that for all , there exists such that for all , we have . Then . These are the prototypes of “good” sheaves.
A sheaf of -modules is quasi-coherent if there exists covering by open affines such that it’s restrictions are of the form .
is coherent if the are finite -modules.
2.2. Differential Forms
Let be a ring, an -algebra and a -module. Then an -derivation of into is a map such that is additive, for all we have and for all , we have .
There exists a universal object, a module and derivation such that any other derivation factors through it. We can construct it by looking at given by , setting and then , with given by modulo .
Now, letting be a morphism of schemes, we can take and and let be the ideal sheaf of the image of . Then .
Let . Then we have exact sequences and if is an ideal of and , then we have .
The sheaf is well-behaved with respect to base change: let a morphism, and base change along , then .
More generally, let and morphisms of schemes. Then is exact, and, if is a closed subscheme with ideal , we have .
2.3. Nonsingular Varieties
Let affine, with . Then we say that is the Krull dimension of , and we’ll denote it by . Then a point is a nonsingular point of if and only if at . We say that is nonsingular if it is nonsingular at every point.
It is a theorem that is nonsingular if and only if is a regular local ring, so we take this as the definition, on schemes.
What does it mean to be regular? For a Noetherian local ring of dimension , the following are equivalent and taken to define regularity:
- is generated by elements.
- The associated graded ring with respect to is a polynomial ring in variables.
Now, let be a locally ringed space isomorphic to an analytic space. This is locally a local analytic space, and it is Hausdorff, and there exists an analytification functor which takes varieties to analytic spaces.
3. Cattani 5
We’ll be working on a compact Kähler manifold. We know that .
If is harmonic and , then .
Let be the set of classes in such that has a representation in bidegree .
We in fact have .
Now, if is odd, we have , and there is no middle term, so the dimension of is even when is odd.
The cup product will actually respect the bigrading: . Voisin has found examples where every condition is satisfied except this property.
Why should all of these nice things be true? Let . We have a Lefschetz map by increasing degree by . We have . Now, we can define by for , and we have . Now, define on . This is the adjoint of , and it satisfies and , so this actually gives us a representation of the Lie algebra !
In this representation, and are shifts, and the eigenvalues of are the degrees of forms, increases degree and decreases. Then there’s the Lefschetz Theorem which says, first, that is an isomorphism. Moreover, it tells us that there are two types of cohomology classes . We call this the Lefschetz decomposition for forms.
We in fact have the following relations:
- , , , .
These imply that , by just writing it out and putting between the partials. This implies that the Lefschetz theorem holds if we replace forms by cohomology classes. We call this the Hard Lefschetz Theorem.
This gives another topological restriction: the betti numbers (even and odd separately) must be increasing to the middle degree.
Now, let us assume , so we are on a Riemann surface. So then , and by the Dolbeault theorem, , the holomorphic 1-forms, that can be written locally, for holomorphic. Then there is and , so the Hodge numbers are easy: and .
On a complex surface, things are slightly more interesting, and splits into , where subscript of zero will indicate the primitive cohomology.