-For every , does there exist a proper -dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that is small compared to .

-Do there exist non-algebraizable proper rigid spaces with “arbitrarily bad” singularities?

-Do there exist rigid analytic analogues of Kodaira’s class VII_{0} surfaces?

Pick an open immersion into a compactification ; after a blowup, we can assume that is an effective Cartier divisor. Write for the inclusion of the boundary. By the usual triangle , we reduce to showing that . Filtering by its perverse cohomology sheaves, we reduce further to the case where also is perverse. Cover by open affines such that is the divisor of a function . By an easy Mayer-Vietoras argument, it’s now enough to show that for every open contained in some , .

But now we win: for any choice of such , there is an exact triangle in where is the unipotent nearby cycles functor associated with , and the first arrow is the logarithm of the unipotent part of the monodromy. Since is additive in exact triangles and the first two terms agree, we’re done.

A closer reading of this argument shows that you actually get the following stronger statement: for any , the class is identically zero. From here it’s easy to get a relative version of Laumon’s result.

**Theorem.** *Let be any map of -varieties. Then for any , there is an equality in .*

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It turns out there IS an even better class of rings/schemes, which occurs naturally in some contexts.

**Definition. **A scheme is **marvelous** if it is Noetherian and excellent, and if for every irreducible component and every closed point . A ring is marvelous if is marvelous.

You can easily check that any marvelous scheme is finite-dimensional. Moreover, it turns out that a Noetherian quasi-excellent scheme is marvelous if and only if the function is a true dimension function for (in a certain technical sense). This function is of course the most naive and clean possibility for a dimension function on any given scheme, but it doesn’t always have the right properties.

Unfortunately, marvelous schemes are so marvelous that, unlike excellent schemes, they aren’t stable under many natural operations, not even under passing to an open subscheme! In fact, is marvelous if it is covered by marvelous open affines, but the converse fails. You can check that a scheme as simple as isn’t marvelous, even though is marvelous. So regular excellent schemes aren’t always marvelous, and adjoining a polynomial variable can kill marvelousity. I briefly entertained the hope that any *Jacobson* excellent scheme is marvelous, but this fails too (the scheme considered in EGAIV_{3} (10.7.3) is a counterexample).

It’s not all bad news, though:

- anything of finite type over or a field is marvelous,
- any excellent local ring is marvelous,
- any ring of finite type over an affinoid -algebra in the sense of rigid geometry is marvelous,
- any scheme proper over a marvelous scheme is marvelous; more generally, if is marvelous and is a finite type morphism which sends closed points to closed points, then is marvelous,
- if is a marvelous domain, then the dimension formula holds: for all prime ideals . (Recall that the dimension formula can fail, even for excellent regular domains.)

You might be wondering why I would care about such a stupid and delicate property. The reason is the following. Fix any marvelous scheme and any invertible on . Then there is a canonical potential dualizing complex (in the sense of Gabber) which restricts to on the regular locus of . Here is the (locally constant) dimension of the regular locus, so this numerology is the same as in the case of varieties. Moreover, for any prime invertible on , there is a good theory of -adic perverse sheaves on with the same numerology as in the case of varieties; in particular, the IC complex restricts to on the regular locus. (See sections 2.2 and 2.4 of Morel’s paper for more. Note in particular the hypothesis on in the first sentence of section 2.2: it is exactly the condition that is marvelous.) This discussion all applies, in particular, when for any -affinoid ring . This turns out to be an important ingredient in my forthcoming paper with Bhargav…

(One more comment: Most real-life examples of marvelous schemes, e.g. examples 1. and 3. above, are also Jacobson. It might be more reasonable to consider the class of marvelous Jacobson schemes, because these *are *permanent under finite type maps. But on the other hand we lose excellent local rings when we do this.)

Let be a field with separable closure . Algebraic variety over means separated -scheme of finite type. Let be a prime invertible in . Perverse sheaf means perverse -sheaf.

If is a proper map of algebraic varieties over , the decomposition theorem tells you that *after base extension to * there is a direct sum decomposition

in . Here is some finite set of closed subvarieties, and is a lisse -sheaf on the smooth locus of . (My convention is that where is the natural map, so . Some people have different conventions for shifts here.)

The decomposition is non-canonical. In particular, it is not -equivariant, so it has no reason to descend to an analogous direct sum decomposition of in . Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain *piece* of to when is projective. This turns out to be good enough for some interesting applications.

To present Bhargav’s trick, let me make a definition. (What follows is a slight reinterpretation of what Bhargav told me, all mistakes are entirely due to me.)

**Definition. **Let be a perverse sheaf on a variety . Let be the inclusion of the maximal dense open subvariety such that is a (shifted) lisse sheaf. Then we define the *generic part of * as the perverse sheaf .

Note that is zero if and only if is supported on a nowhere-dense closed subvariety. Also, in general there is no map between and . However, in some cases is a direct summand of :

**Proposition. **Let be a perverse sheaf on a -variety , and suppose that the pullback of to is a direct sum of IC sheaves. Then is a direct summand of .

*Proof.* Let be as in the definition of the generic part of , with closed complement . Our assumptions together with the definition of the generic part guarantee that for some closed subvarieties contained in .

Now look at the natural maps . Set and . Since does not admit any nonzero quotient supported on , the composite map is zero. Thus factors over an inclusion . Moreover, has the same generic part as . This is enough to imply that , so we have a natural inclusion . A dual argument shows that factors over a surjection . It is now easy to see that the composite map is an isomorphism, so is a direct summand of .

**Corollary 0. **Let be a projective map of -varieties. Then is a direct summand of .

*Proof. *The decomposition theorem and the relative hard Lefschetz theorem give a decomposition in . Then is a direct sum of IC sheaves after pullback to , so we can apply the previous proposition.

**Corollary 1. **Let be a projective alteration of -varieties with smooth. Then is a direct summand of .

*Proof. *Check that is a direct summand of by playing with trace maps. Now apply the previous corollary.

**Corollary 2. **Let be a finite extension. Then for any -variety , the -adic intersection cohomology is a de Rham -representation.

*Proof.* Let be a resolution of singularities. The previous corollary shows that is a direct summand of compatibly with the -actions. Since is de Rham and the de Rham condition is stable under passing to summands, we get the result.

Note that we can’t prove this corollary by applying the decomposition theorem directly out of the box: the decomposition theorem does immediately give you a split injection , but this map is not guaranteed a priori to be -equivariant!

**Corollary 3. **Let be a finite extension of or . If satisfies the weight-monodromy conjecture for all smooth projective -varieties , then satisfies the weight-monodromy conjecture for all proper -varieties . In particular, the weight-monodromy conjecture holds for the -adic intersection cohomology of all proper -varieties for finite.

*Proof. *Entirely analogous to the previous proof.

It would be interesting to know whether Corollary 1 has a “motivic” incarnation. Here I will pretend to understand motives for a minute. Suppose we have an assignment on quasi-projective -varieties, where is a suitable triangulated category of constructible motivic sheaves on with -coefficients. This should come with the formalism of (at least) the four operations , and with faithful exact -adic realization functors compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on ? Let denote the symmetric monoidal unit. It then makes sense to ask:

**Question. **In the setting of Corollary 1, is there an idempotent such that for all ?

This would imply that the split injections provided by Corollary 1 can be chosen “independently of ”, i.e. that they are the -adic realizations of some split injection in .

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Two more points:

- We are doing our best to post notes and videos (at the speakers’ discretion) for each talk on the seminar website linked above. Hopefully we will post notes for every talk!
- Bogdan Zavyalov planned to mention the sad case of Azat Miftakhov at the end of his talk. Due to the extended mathematical discussion, this ended up not happening, so I am mentioning it here instead. Please please go here and read more.

The large power output of the Sun is mainly due to the huge size and density of its core (compared to Earth and objects on Earth), with only a fairly small amount of power being generated per cubic metre. Theoretical models of the Sun’s interior indicate a maximum power density, or energy production, of approximately 276.5 watts per cubic metre at the center of the core,

^{[76]}which is about the same rate of power production as takes place in reptile metabolism or a compost pile.

Takeaway: If the sun were a giant ball of lizards, nothing would change.

]]>**Brain teaser a.** Prove that is a locally spatial diamond over with a Weil descent datum to .

Now, let be the division algebra over of invariant , and let be an irreducible representation of whose local (inverse) Jacquet-Langlands correspondent is supercuspidal. Note that acts on by its natural identification with .

**Brain teaser b. **Prove that the geometric etale cohomology of satisfies the following:

if is orthogonal, and if is not orthogonal.

Here denotes the Langlands parameter of .

It is probably not fair to call these brain teasers. Anyway, here is one big hint: the infinite-level Lubin-Tate space for is naturally a -torsor over , by trivializing the bundles .

]]>Let be any map of irreducible complex varieties, and let be a perverse sheaf on . Prove that there is a dense open subset such that for any closed point , the shifted restriction is a perverse sheaf on the fiber .

]]>Please follow the instructions at the seminar website here to get the Zoom link. We will also keep an up-to-date schedule on researchseminars.org here.

All credit to Jared for the name!

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