And today, Hodge theory is in full swing. There were some commutative diagrams today, and wordpress is famous for not supporting xy, so I’ll do what I can.
1. ElZein 2
Take a family of elliptic curves: given by for in , for each . We need to compactify:
Look at given by . The fibers are then projective curves of genus 1. Each one is homeomorphic to .
For each , we could look at the fiber as , but we’re going to rely a lot on topology, so we’ll be better served by using the model.
Thus, we have a map and by Ehresmann’s lemma, we have that for a small neighborhood of the point, the preimage is diffeomorphic to .
Calling the projection map , this tells us that . So for every in , we have a homeomorphism, and if we take a different path and get , the two homeomorphisms are the same.
However, if we go around a hole, so that the two paths comprise a nontrivial loop, we can’t homotope one to the other, and so we get a nontrivial map when we go around the loop. This gives us maps , which is not necessarily the identity. We call this monodromy.
The monodromy transformation is not compatible with Hodge structure!
Lemma 1 (Deligne) Let . Then under the projection we have for .
Now, and . However, . However, it is mod . If it’s exactly true, we call the mixed Hodge structure split.
A morphism of mixed Hodge structures is such that , and .
Lemma 2 is strict for and for and the kernel and cokernel are natural mixed Hodge structures.
The category of MHS’s is an abelian category.
Let then the cohomology of this sequence is a MHS.
Let be a compact Kähler manifold of dimension and the holomorphic deRham complex. Then we can set to be , and is a filtered complex. We take a resolution of Filtered complexes the differential forms, a quasi-isomorphism. Then we can define by .
In degree , .
Now, is a quasi-isomorphism, and , and this gives a qis to the Dolbeaut resolution.
and is .
Define ot be . Then we have .
The spectral sequence degenerates at rank 1. That is, .
But even more, .
If is projective, then is a hyperplane section .
The fundamental class of a subvariety of a compact complex manifold of codimension in where is and then this sits inside , which is isomorphic to . This gives us a class and , and so is Poincaré dual to .
Lemma 3 and
Define for a subvariety in , including the possibility that it might be singular.
2. Migliorini 2
Today we’ll be following Chapter 4 of Volume II in C. Voisin’s book.
Let a smooth projective morphism. We want to prove that the Leray Spectral Sequence degenerates at . So then morally, .
The key fact we’ll need is the Hard Lefschetz theorem, and here it tells us that for , we have where is the operator that takes the cup product with the Chern class of a relatively ample line bundle on , is the primitive cohomology, and .
Because is globally define, this decomposition is monodromy invariant.
This decomposition is compatible with the differentials in the LSS. In fact, this decomposition, by Cattani’s first set of notes, holds already at the level of forms.
, cupping with gives a morphism .
EG, prove . Enough to show on the primitive part. There, we have , but the on the right is zero, and on the left is . The same argument gives that .
This splitting, however, is not canonical.
Now, degeneration at implies that
Now, if we have a projective map , the inclusion of an open sense subset such that , the base change, is smooth, then there is a strong generalization of the above.
Let be a nodal cubic, given by . Then , and we can’t expect .
This can be fixed by looking at MHS, as discussed by ElZein. Here we have a finite increasing filtration and a finite decreasing filtration , where induces a pure Hodge structure on each over of weight .
A map of MHS is strict with respect to and .
Let nad then .
The following theorem is actually true:
There is a functorial mixed Hodge structure on the cohomology groups of every open algebraic variety .
In general, if and if .
So on the nodal cubic, the classes are of type .
If is nonsingular, maybe no compact, then if and if is compact, possibly singular, then .
So this implies that the Hodge structure guaranteed by the theorem of Deligne above is pure if is smooth and compact.
Weight Trick: Let compact inside smooth lie inside , smooth and compact. Then .
, then is an isomorphism, but is not, it’s the zero map. The problem here is that the spaces are not algebraic. The Weight trick is a very algebraic phenomenon.
In the algebraic setting, we have for is any smooth compactification of .
Now, let’s prove the Weight trick:
We have . Now, any class of weight in comes from .
From degeneration and MHS theory, we have that is a subHodge structure, and the invariant classes are compatible with the decomposition.
The monodromy representation is completely reducible, when is a smooth projective morphism to a quasiprojective variety.
3. Cattani 2 – Variations of HS, Degenerations of HS
Let be a holomorphic proper submersion of complex manifolds. So by Ehresmann’s Theorem, we can assume that that around each we have a such that .
For each , we have by , and then we can send it along the inverse of the diffeomorphism to , and this map is holomorphic.
Now, assume that is a polycylinder and that .
The naive approach is that sits insider , and this composition is the diffeomorphism .
This gives , with , and so we have a family which give decompositions for all . For , we get the usual decomposition for
So in local coordinates, , we have has a basis of the form .
Thus, if , then we can write it out as .
But of course, this is not all well defined, it is not global. Given , we can consider the expression , and this will kill any holomorphic dependence. So what we get is a map . But this is closed, it’s not just a form, so we actually have . This is called the Kodaira-Spencer Map.
So for eahc , we have , and these fit together into
This gives us a long exact sequence, and in particular, a map . However, the first is simply , and so this gives us again the Kodaira-Spencer map.
Now, assume that the are 1-dimensional. These are all diffeomorphic, so we can say that they are Riemann Surfaces of genus . So then . Now, is constant with respect to , but the decomposition may not be. We still have all the maps , and they give isomorphisms .
So we can view this as having , a fixed vector space with a varying decomposition, satisfying . But we also have the polarization form and and .
So now we have satisfying these conditions. So we can represent these by by matrices, satisfying the conditions. So the conditions guarantee we can make half into and the other , and the condition then becomes and (the imaginary part).
For a Riemann surface, we have that , so we just need to look at the subspace of forms among -forms.
So this whole approach is what we’ll be generalizing to higher dimensional fibers.
In general, given , we get where or . We also have to work with.
So how do these two things relate? Can we go between them without going back to the fibration? We can, via flat vector bundles. (Note: Riemann-Hilbert Correspondence)
Let be the universal cover, and let act on the right, and assume we have a rep . Then we define where this is just the product modulo where for all .
We say that a section of a bundle with connection is flat if .
The follow are equivalent:
- Representations of the fundamental group
- Local systems
- vector bundles with flat connection
4. ElZein 3 – Mixed Hodge Complex (MHC)
Let be a complex irreducible algebraic variety. Then there exists a Zariski open dense subset of smooth point , and its complement is the singular locus.
There exists a diagram
such that is smooth, is a normal crossing divisor on and is isomorphic to .
Consider projective and smooth complete variety, closed in irreducible of codimension , then a desingularization. Then gives . By Poincaré duality, we get .
Lemma 4 , and we will call this space .
Proof: if , and so we take to be the Kähler (1,1)-form on , then we can deduce that , by positivity.
We define to be the set of all formal integer linear combinations of irreducible subvarieties of of codimension . We have a map .
Question: Is this map surjective?
Let be a smooth, open algebraic variety, singular.
Remark: If is smooth and proper complex algebraic variety carries a pure Hodge structure of weight .
There exists projective and which induces an isomorphism on an open dense Zariski subset, and .
Now, the Hodge filtration is algebraically defined. We deduce that the Hodge filtration on is a Hodge filtration of a Hodge structure.
4.2. The Hodge Complex
A Hodge complex of weight is defined as follows: a complex of groups bounded below, a filtered complex of complex vector spaces , finite on each degree, and , a quasi-isomorphism, and the differential on is strict with respect to .
So, for all , , and with induced filtration satisfies , and is a Hodge structure of weight .
Let be a topological space, a CHC of weight on is a complex of sheaves bounded below, a filtered complex of sheaves , a quasi-isomorphism such that is a Hodge complex of weight
A mixed Hodge complex is a complex of groups bounded below, a filtered complex of vector spaces with and bifiltered, an quasi-isomorphism such that for all we have along with is a Hodge complex of weight .
The definition of a cohomological mixed Hodge complex is similar.
Let be a MHC then the filtration on and the filtration on define a MHS.
Proof: Take the spectral sequence . It has . These are HS of weight . Look at . We must prove that this map is compatible with . Then , the cohomology, is a Hodge structure of weight . And then, can be shown to be zero, because it is a morphism of Hodge structures of different weights.
5. Migliorini 3
Summary: a smooth projective morphism. Then is isomorphic in some sense to a complex wsith zero differentials and entries , which are semisimple local systems and is an isomorphism, where .
What about these three facts for a general projective map , nonsingular. Then degeneration fails! But yet, these three facts hold if we replace “local system” with “perverse sheaves”.
Two famous theorems about surfaces:
- Let and a germ of a normal surface singularity, with a resolution of singularities. Then with curves, and the intersection form is negative definite. (Grauert-Mumford Theorem)
- with is a germ of a smooth curve, then is a family of projective curves. Assume is smooth , the inverse image is , and the intersection form is negative semidefinite and the radical is generated by the class of the fiber (Zariski Lemma)
1 gives as a consequence the fact that , which has cohomological version if and
Number 2 gives which is, cohomologically, , and .
The first is the intersection cohomology complex of . The second is the intersection cohomology complex of .
Now, let , which is is small enough retracts onto . Then which both map to the generic fiber. The image lands in l.i.c.t.
Does this hold in general?
, with the first a MHS.
On (smooth compact) it is possible to define a MHS such that the above map is a morphism of MHS.
We call this the limit MHS.
R. Friedman “On the Clemens Schmid exact sequence”
Suppose that , the monodromy operator, is unipotent (that is, is nilpotent) then is defined out of the Jordan form of .
6. Cattani 3
Look in the appendix of the first set of lecture notes, A.5, on the Weight filtration. Also Flat Bundles.
Let projective, . We can look at . Over a point, we have , and is diffeomorphic via to , with inverse , and these give and .
If , we have the set of flat sections.
We look at the Gauss-Manin connection . Then for flat.
Look at then . These terms are all upper semicontinuous, so they cannot decrease in dimension, only increase, and as the total is constant, each must be.
Now, set . For each , we get a map where , taking to . This map is smooth.
So back on the bundle, we get subbundles .
Then the amazing fact is that is holomorphic, so is a holomorphic subbundle.
For notational convenience, we take . Then is zero.
Once we have holomorphicity, there’s Griffiths Transversality, which says that belongs to .
Now, we look at the weight 3 Hodge structure on . We can break it up in two ways. If we take one half ot be , then we get the Weil filtration, and these are polarized, but Griffiths looked at , and lost the polarization in order to get things to vary holomorphically.
Sketch of Griffiths Transversality
Let . Then we have . We can look at . Then we have . Then there exists such that and .
and because the contractions of and with are both zero.
Then, it’s a matter of counting the number of holomorphic and antiholomorphic terms.
We have a map and a map . This gives us by contraction.
For the rest of these talks, we’ll forget geometry and talk about abstract variations of Hodge structures.
Let be a local system of free -modules, and let be the Gauss-Manin connection on . A variation of Hodge structures of weight is an increasing flag of holomorphic subbundles such that and if then .
A polarization of VHS is a flat bilinear form on of parity such that for each , polarizes the Hodge structure on .
For every , is quasi-unipotent, that is, there exists such that is unipotent.
In the geometric case, this theorem goes back to Langlands.
Let’s call the nilpotent part. So the monodromy theorem actually also says that .
In general, the first part goes back to Borel, and the second to Schmid.
The monodromy theorem requires that you have a polarization.
Tomorrow, we’ll take this definition and we’ll look how to map it into some nice space, a period domain, and look at the properties of this map, especially when , and look at limiting behavior as we go to .