Relative log Poincaré duality
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Transkript: Englisch(automatisch erzeugt)
00:17
First, I thank organizers for the opportunity.
00:23
My title is Reactive Log Poincare Duality, and the plan is, first, I'll review some
00:55
classical statements, and next, Poincare Duality in log-beta cohomology, and third,
01:29
log-beta cohomology. And the notation throughout the lecture, A denotes a ring, D is non-negative
02:11
integer, and carry 1 is in the derived sense. Okay, first, let F is a continuous map of
03:00
local compact topological space submassive with respect to the dimensional topological manifold,
03:18
which means the upper space is locally the product of the lower space and
03:27
the R-D dimension of the topological space. Then, for any F and G, we have this Poincare
04:05
Duality, and where this is a relative orientation shape and features locally isomorphic,
04:25
the pullback. As a corollary, we have a borrowing statement, and let F be a morphism of complex
05:01
analytic space, submerged of relative dimension D, then since complex manifold always orientable,
05:44
we have this type of duality.
06:28
Yes, parenthesis, D, yes.
06:55
The algebraic corresponding in outer sight, the same statement, the same formula holds.
07:55
Okay, want analog with 0.1 and 0.2 in log geometry. This is the subject of today's lecture.
08:30
Okay, now, is it okay? Oh, sorry, section one. First, we explain the generalization of 0.1.
09:48
Okay, this is the geometry and topology, F-S log analytic space, vertical, exact, and log smooth.
10:20
Recall that exact means this is a Cartesian, and any fiber of F log is F, which is topological manifold,
10:54
as explained soon, and this fiber is of dimension 2D. Then, we have, this is a quick,
12:09
I explained some, briefly, the proof. This theorem is based on the rounding theorem,
12:27
and this means F log is submersive with respect to even dimensional topological manifold.
13:08
To show an essence, we consider an easy, simplest example.
13:21
Consider the diagonal homomorphism from N to N square, then our F is standard log plane to standard log line.
13:56
Then, if we consider the associated real row up space, then the upper space is circle cross circle cross this one.
14:22
And the map is multiplication, then the picture is as follows. This fiber is this, this fiber is this,
14:51
and the special fiber is broken but homeomorphic to general fiber.
15:06
So then, there is a homomorphism like this, so corner is rounded.
15:27
Then, I apply Bell-Jares 0.0 for this map, and the care orientation, then we get 1.1.
16:04
This joint work with Arthur is one of most exciting and fascinating experiences in my life. I explain why. Of course, it is a great honor for me to work with Professor August, but more.
16:26
We know, in general, in a joint work, contribution or role of each author is not necessarily the same. Sometimes, an author does little, another does almost all.
16:44
An author only gives an idea, another only gives proofs, another only writes down, and so on. But in this joint work, I feel our contributions were exactly 50-50,
17:02
because we enjoyed all steps of study together, starting with simple example, this one, and proceeding to more complicated ones, discussing proofs, generalizations, and how to write them up.
17:24
So, I thank my God to arrange such a nice collaboration. Thank you. Now, I proceed to the Alladic analog, which is analog to 0.2.
17:53
If you don't assume vertical, you will have manifold with boundary, I suppose. But if you don't assume, then you will have manifold with boundary, or something like that?
18:04
Yes. And then there is still quadratic duality with using extension by zero from the interior? In that paper, non-vertical cases are also created, and then this is modified, that is another story.
18:36
Today, for simplicity, I restricted all things under the vertical assumption.
18:57
There are two kinds of logator cohomology, and both are due to Professor Cuthoff.
19:10
I briefly review the definition. Let X be an FS log scheme.
19:27
The first one is the ket site. Object is kumma and logator.
19:45
I don't give the definition of kummaness, but it is equivalent to exact and logator. And exact, the definition of the goodness is already given.
20:15
Typical ket covering is such a thing.
20:50
Among two variants of logator cohomology, this ket definition is more similar to the usual eta site.
21:03
For example, ket map is an always open map. Another site is a full logator site.
21:26
Object is all logator. And covering is, sorry, covering, covering subjective family, universal subjective family.
22:14
Exact, exact, projected family, universal subjective family. Exact is not true anymore, right?
22:33
It is all in the FS, the category of FS log structures.
22:55
All the fiber products, ok. Fiber products can be empty.
23:08
The difference lies in log blowup, which is logator and universal subjective, but not exact.
23:33
Not necessarily exact.
23:41
The typical example is like this. If you leave out one of the strict transforms and it's still on the universally surjective, is that right?
24:36
So if we remove some point, then it is not universally surjective, because the base change is the same.
24:47
Base change is the open margin, but not surjective.
25:24
There is a morphism on topo, topoid, from full log to ket.
25:41
This is lemma 2.1, exactification.
26:00
Assume the source is quasi-compact and target has a chart, then in FS log scheme. Then there is a log blowup of the base, such that the base change is exact.
26:41
Sorry, does it not have a frame rather than a chart? Do you happen to know? Oh, frame is enough, I think. And as a corollary of this 2.1, full logator topology generated by ket topology and log blowups.
27:10
The next lemma is pullback and pushforward.
27:39
Then we have the original one.
27:44
And pushforward by kappa and pullback by exact if commute.
28:02
And third, change of the role of kappa and f under the assumption exactness.
28:33
Let p be a log blowup in the ket side, in the ket context.
28:47
This is by Fujiwara Akato.
29:08
And further, if f is locally constant and constructive, f shift away module, then push and pullback is the same as the original one.
29:36
And pushforward is also locally constant and constructive.
30:02
About the full logator, the situation is much better. That is, this gives an equivalence of topology.
30:29
Using this proposition, we can prove various fundamental results, explained soon, by going from ket to full and vice versa.
30:48
By exactifying by log blowup. And do with log blowup with this proposition.
31:04
Under the exactness, we can use this proposition. And so, now I have to explain fundamental theorems.
32:08
I want to tell theorems.
32:50
Ket, proper base change, smooth base change, proper smooth base change,
33:03
cohomological dimension, cohomological dimension. More theorems. Today, these are all necessary.
33:35
It's difficult to read what you write in this part.
33:40
Because? Because there are the shadows. So, proper base change, what's the next one after that? Smooth base change. What did you write about? Okay. Proper smooth base change.
34:01
Looks like cool. Cool. And something new and something old. This is new and this is all new. Published one is this and this, only two.
34:25
And this is, this four theorems due to Professor Kato. And where, I explain. Where did you say not what?
34:40
Not naive. Not naive. Not naive means false and a strange condition as follows.
35:13
Consider the concerned cavitation. Take any point.
35:33
Consider this homomorphism of monoids. And the condition is any element of the associated group of this monoid.
35:55
If f group of a belongs to m bar and g group of a belongs m bar inverse,
36:12
then a is trivial.
36:22
Under this condition, proper base change and smooth base change, ket proper base change and ket SBC hold under this condition. And this is in a sense best possible.
36:42
Not only sufficient but also necessary condition.
37:01
And other statements hold naively with no problem.
37:25
And we remark of the condition. Star, past, either f or g is exact, then this condition holds.
37:56
So, under the exactness condition, we can freely use every base change theorem.
38:10
Next, if condition, ah, sorry, if, sorry?
38:28
If cavitation diagram D in FS logarithmic space satisfies an obvious analog of this condition,
38:51
then D log in topological space is also cavitation.
39:09
This is published. So, it seems a rather strange condition, but something related to a good geometry.
39:35
And CD1, CD1, ah, sorry, CD1 is the following statement.
40:01
Let x, y, let f is compactifiable. Ah, this means underlying is compactifiable. In FS log scheme, a is torsion.
40:24
Let d is a relative dimension. And let r is relative log-rank.
40:53
Then, cohomological dimension is 2d plus r.
41:07
This is the ket in the ket context. And 2d plus 2r in the let context. And this is sharp.
41:22
How do you define proper support, direct image? Because the compactification is only a scheme, it's not a log. Okay, so, this is the definition.
43:01
And CD2.
43:23
Assume further that f is, ah, log flat and, ah, locally finite presentation. Fiber dimension is there.
43:51
Then, we can drop r.
44:04
Of course, there is no time to explain the proof. So, I only state that gives us structure of the proof.
44:29
Ah, structure proof.
44:45
And PSBC is rather complicated. So, log, ah, let PBC plus let SBC implies ket-exact PSBC.
45:03
And this implies let PSBC. And finally, this implies ket-PSBC in general. Like that. People might be surprised to know that log-blob is log-flat. No, no. Log-blob is log-flat. People might be surprised to know that.
45:24
Log-blob is log-flat. Sorry, what is your question? Log-blob is log-flat, right? Yeah, yeah. People might be surprised to hear that. Ah, so, so now, now we come to, ah, sorry.
45:48
Ah, talk reading. State the main theorem. More, more, 14 minutes.
46:27
Particle, proper, no cements. Any fiber equated dimension, no?
47:18
This state, the, the advantages that base is arbitrary.
47:23
But, ah, clearly, there are some problem. Ah, properness shouldn't be, of course. So problem one, ah, non-locally constant F and non-proper F.
47:49
This is okay for the case where y is a standard log point,
48:07
this is an old result 1990, 1997.
48:29
This, ah, this is proved by embedding x locally enter a log log regular, or log log regular, so embedding X as a log locus into a toric
48:55
variety and calculate explicitly. Do you want to take RF floor shrink in the non-verbal case, or RF floor? Is it like a radial duality, so is RF floor shrink?
49:09
On the right hand side, it should be a floor shrink maybe? And problem 2, non-vertical case. This is a different story and I don't give a normal comment.
50:23
And I'm trying the proof.
50:48
This is for the head, or for which type of topology it is? For the usual, or for the head topology, or the full eta topology? Both. Yeah, sorry. Step 1. Assume F is saturated.
51:21
So the main difficulty is to construct trace homomorphism. And there are three steps. First, assume F is saturated. Consider the diagram, the coefficient of getting log.
52:07
And then consider the trace vector sequence, this composition.
52:24
Then, since epsilon is saturated, then R epsilon 0 star lambda is lambda.
53:02
On the cohomological dimension, e2 pq equals 0. If p, not only cohomological dimension, but also some, only this is okay.
53:26
Anyway, this term will vanish. The e2 term only lives here. See this point.
53:55
Then, vector sequence gives e2 2d 0 to the, this to the middle.
54:26
Then this is the subject shown. And by proper base change, this is, by proper base change, this is the pullback of the push forward of this.
54:44
Epsilon rho star, upper star. Then this is classical eta topology, so classical trace in SGA4.
55:07
Then, we can prove that this morphism factors that subject.
55:25
This factorization is non-trivial, but by proper base change, again, reduced to case 4.2.
56:16
So step 2, F is exact. Then, take a ket cover such that F dash, F prime is saturated.
56:54
Then, the underline of this connotation is also connotation.
57:08
Then, trace for F prime group, because the classical, trace for F prime is constructed by classical one.
57:31
So the classical equal to 7, under classical, classical equal to 7, they behave well.
57:41
And finally, the general case reduced to exact case by log blowup. Oh, there is no time. So, thus we have trace morphism.
58:23
And by adjoint, and this should be an isomorphism. Then, problem 1 is to be fixed, but I have no idea to prove this is an isomorphism for the present.
58:49
Anyway, by formal duality in ket, this gives an isomorphism in the statement.
59:06
To prove it, it is an isomorphism. First ket exact case by PSBC, both sides are locally constant-constructive.
59:34
And then by PBC, reduced to log-point base case.
59:45
And applying kappa star, then let exact case. And finally, again by log-blowup.
01:00:00
We have the general case. Thank you very much. Are there questions for Professor Nakayama?
01:00:27
Oh, I have a technical question on step two. If you want to make an exact guide into saturated, why you say that kummer return is enough? Maybe you have to take words not prior to P.
01:00:43
I did think this is needed to give an integral map No, not difficult. Something here, integral, you want to make it integral. I'm not sure about this, but I... Ah, I forgot the detail, but it is possible.
01:01:14
One day I have the same question, and the preprint has a gap, but somehow it is fixed.
01:01:23
So we can discuss it later. Are there other questions? Look, just general questions. When you try to imitate this point in TA4, define F of 3 as a form of a writer joint
01:01:41
and then try to estimate, calculate F of 3 in nice cases. So basically maybe doing the distance is more efficient. So what goes on? Why does the naive imitation of SGA4 not work?
01:02:02
This is a basic point. I think SGA4 is by reduction to curves. You have a huge variety of curves and some passability or property you use something on TA1. So in here maybe you could apply those two curves. I don't know.
01:02:22
So what goes on? Why do you take this sort of complicated times? This is already a problem in the log point case.
01:02:43
Perhaps there is no good vibration in this case. I just wondered if you could write a simple example
01:03:11
where a star isn't satisfied. Star is satisfied? Yes, a simple example where it's not satisfied. Not satisfied. Not satisfied.
01:03:24
We don't do logarithm and logarithm. I hope this is a global reality. Any hope for a local one? Local one?
01:03:44
In fit me? I mean you are in complex. Oh, you are in complex.
01:04:06
It is a problem. Let's thank Professor Takayama again.