User reladenine vakalwe - MathOverflow

Computing rational cohomology of smooth (not necessarily compact) toric varieties

2012-12-04T05:47:53Z

The title pretty much says it all. I am looking for references (lecture notes, books, readable articles, suggestions), preferably example laden, that explain how to compute the rational cohomology of a smooth toric variety. I am particularly interested in methods for doing this when the variety is not compact. Implicit here is the assumption that this can be done in a practical way (please correct me if I am wrong, I do not know much about the subject).

Some background: I have a bunch of varieties whose cohomology I would really like to compute. I have reasonably explicit descriptions of these varieties as subsets of $\mathbb{C}^n$ . These descriptions are along the lines of (but more involved) the variety in this question:

http://mathoverflow.net/questions/115141/a-cohomology-computation-request

I realized earlier today that my varieties are toric, and I am hoping this observation will be the answer to my prayers.

If it helps, I can compute the equivariant cohomology of of my varieties (mainly because the Hodge structure on it is pure), but I am presuming this doesn't really completely determine the ordinary cohomology (apart from Hodge-Euler polynomials etc.).

Around the socle filtration of a Verma module

2013-01-06T03:13:03Z

Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$. Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of highest weight $w^{-1}w_0\cdot 0$, where $w_0$ is the longest element in $W$ and $\cdot$ denotes the so-called dot action. So $\Delta_e$ is simple and there is a unique (up to scaling) inclusion $\Delta_v \hookrightarrow \Delta_w$ whenever $v\leq w$. In particular, there is an inclusion $\Delta_v \hookrightarrow \Delta_{w_0}$ for all $v\in W$.

For a module $M$, let $0\subset soc^1M \subset soc^2 M \subset \cdots$ denote the socle filtration. Set $soc_iM = soc^i M/soc^{i-1}M$. For example, in the case of $\mathfrak{sl}_2$: $W=\{e,s\}$ and $soc_1\Delta_s = \Delta_e = L_e$, $soc_2 \Delta_s = L_s$, where $L_w$ denotes the unique simple quotient of $\Delta_w$.

The basic question: Is it true that

a) $soc_1(\Delta_{w_0}/\Delta_w) \subseteq soc_k(\Delta_{w_0})$,

where $k$ is the smallest integer such that $soc^k(\Delta_{w_0})\not\subseteq \Delta_w$.

Why this has hope of being true: 1) It is true in type $A_1$ and $A_2$, I have not checked type $G_2$ yet out of sheer laziness (and hope that an expert will just point me to some place in the literature or tell me that I am overthinking matters). 2) It is true for $w=e$. 3) Miracles sometimes occur in Schubert varieties.

Why this has no hope of being true: 1) It has a sort of ridiculous feel to it (apologies for the cavalier attitude, but perhaps it will be justified by what follows): roughly the statement is saying that simples in Vermas can't move down in the layers of the socle filtration upon quotienting out by Verma submodules. This feels a bit nutty to me. 2) The examples of type $A_1$ and $A_2$ are both multiplicity one situations (ala occurences of simples in Vermas) and aren't really indicative of the general situation.

Some reformulations/related tidbits (that I am aware of but don't see how to leverage into a counterexample or proof):

1') The radical filtration on a Verma coincides (up to shift) with the socle filtration.

1) The statement is equivalent to the socle filtration on $\Delta_{w_0}/\Delta_w$ coinciding (up to shift) with the weight filtration (ala mixed sheaves or graded category $\mathcal{O}$), since the weight filtration on $\Delta_{w_0}$ coincides with the socle filtration (see "Proof of Jantzen conjectures" by Beilinson-Bernstein or "Koszul duality patterns in representation theory" by Beilinson-Ginzburg-Soergel). Note: the radical filtration on $\Delta_{w_0}/\Delta_w$ does coincide with the weight filtration.

2) The statement implies the assertion obtained by replacing $w_0$ with any $x$ such that $w\leq x$, since in this situation $\Delta_w\hookrightarrow \Delta_{w_0}$ factors as $\Delta_w \hookrightarrow \Delta_x \hookrightarrow \Delta_{w_0}$.

3) The question is motivated by trying to understand an analogous question for the anti-dominant projective in category $\mathcal{O}$ (principal block). Namely, let $P_e$ be the (canonical) indcomposable projective cover of $\Delta_e$ (note: $P_e$ is an amazing object, it is self-dual, injective, tilting). Are the following statements true:

b) $soc_1(P_e/\Delta_{w_0}) \subseteq soc_k(P_e)$,

where $k$ is the smallest integer such that $soc^k(P_e)\not\subseteq \Delta_{w_0}$.

c) Same question as b) but replace $w_0$ by arbitrary $w\in W$. This is of course related to a).

Added later: c) is undoubtedly false, as indicated by Dag's counterexample in type $A_1\times A_1$ below.

Added later: c) is also false for type $A_2$.

Here is why one might care: from the short exact sequence

$0\to \Delta_{w_0} \to P_e \to P_e/\Delta_{w_0} \to 0$

one deduces $Ext^1(\Delta_e, \Delta_{w_0}) = Hom(\Delta_e, P_e/\Delta_{w_0})$. Consequently, the purity (ala mixed sheaves/graded category $\mathcal{O}$) of this $Ext^1$ is (unless I am being screwy) equivalent to b).

Unless I am completely misunderstanding things, V. Mazorchuk proves this latter purity statement (in slightly different language) in Theorem 32 of http://arxiv.org/abs/math/0607589.

In fact, Theorem 32 states that $Ext^1(\Delta_v, \Delta_{w_0})$ is pure for arbitrary $v$. Now for $v=e$ this translates to b) above. This is starting to smell like a proof/answer to my questions. However, the problem is that Mazorchuk's proof (which I don't understand very well) seems to be using statements along these lines.

Related also is the fact that granted the purity of $Ext^1(\Delta_v,\Delta_{w_0})$ a downwards induction gives purity of $Ext^1(\Delta_v, \Delta_w)$. This in turn implies that the dimension of these $Ext^1$'s is given by the coefficient of $q$ (modulo sign) in the corresponding Kazhdan-Lusztig $R$-polynomial (these statements start getting me really worried, since they are certainly not true for all $Ext^i$ thanks to Boe's "Counterexample to the Gabber-Joseph conjecture").

Needless to say I am playing fast and loose with a number of things. So the assertions above should be treated with a healthy dose of suspicion (I would be grateful though to people pointing out the errors of my ways).

This of course ties in with a number of toy questions that have been bugging me:

http://mathoverflow.net/questions/116348/morphisms-between-verma-modules

http://mathoverflow.net/questions/115141/a-cohomology-computation-request

Having typed all that, I really hope I didn't make a silly mistake right in the beginning!

Answer by Reladenine Vakalwe for extensions of IC sheaves

2012-12-31T17:12:21Z

There is general convolution algebra type formalism that you can try and use, see Chriss and Ginzburg's "Complex geometry and representation theory". Chapter 8 in particular is very much in line with the your "source of examples".

In general this can be hard. Heck, consider even the case that your local system is trivial and your IC-sheaf is the (shifted) constant sheaf on $X$. Then you are asking to compute the cohomology of the space. This may be a non-trivial endeavor depending on your space.

An ideal example where things work out very nicely is that of flag varieties and the IC-sheaves are those corresponding to Schubert subvarieties. Then these Ext-computations can be carried out combinatorially in the Hecke algebra. Soergel's papers on this and related topics are particularly enlightening. A related point here is that in this situation considering hypercohomology as a functor to graded modules for the cohomology algebra of the flag variety is full and faithful. This result also generalizes to projective varieties with $\mathbb{C}^*$ -actions. This is a result of Ginzburg "Perverse sheaves and $\mathbb{C}^*$-actions". Similar ideas are also worked out in some of Springer's papers on spherical varieties. Related are also the moment graph techniques that can be found in papers of Braden, MacPherson, etc.

Regarding, "replace cohomology with equivariant cohomology". I am assuming you want to compute $Ext$ in the equivariant derived category. Then similar techniques as above can be tried. In the presence of suitable assumptions, often the equivariant calculation reduces to the non-equivariant one due to formality. Instead of trying to flesh this out let me just refer to Soergel's "Langlands philosophy and Koszul duality". A number of examples are worked out in there.

In general though, there is no magic pill that I know of.

Morphisms between Verma modules

2012-12-14T04:59:57Z

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and `$\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".
In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?
Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!
In the same vein as 3). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

http://mathoverflow.net/questions/111682/intersection-of-plus-minus-cells-in-bialynicki-birula-decomposition

http://mathoverflow.net/questions/115141/a-cohomology-computation-request

The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.

Splitting of the weight filtration

2012-12-28T18:00:21Z

All varieties are over $\mathbb{C}$. Notions related to weights etc. refer to mixed Hodge structures (say rational, but I would be grateful if the experts would point out any differences in the real setting).

I am trying to get some intuition for the geometric meaning of/when to expect the weight filtration on the cohomology groups $H^i(X)$ of a variety to split. By the weight filtration splitting, I mean that the Hodge structure on each $H^i(X)$ is a direct sum of pure Hodge structures.

The simplest situation in which this happens is the classical one of smooth projective varieties. The next simplest situation I think of is the "smooth" being weakened to "mild singularities" (for instance, rationally smooth). So at least in the projective case I think of the "mixed" as encoding singularities. This also gels well with the construction of these Hodge structures using resolution of singularities. Are there other helpful perspectives?

Instead of fiddling with the "smooth" one can make the variety non-compact (but still smooth say). Examples: affine $n$-space, tori. Here I don't know how to think of or when to expect the weight filtration to split. Any intuition would be appreciated. The only rough picture I have is that this encodes information about the complement in a good compactification. But I don't find this particularly illuminating.

Generalizing affine space and tori is the situation of toric varieties for which the weight filtration always splits (thanks to a lift of Frobenius to characteristic $0$). In general when should one expect a splitting of the weight filtration to be given "geometrically" by a morphism (or say correspondence in the context of Borel-Moore homology)?

Related is the following: when should one expect the Hodge structure on each $H^i(X)$ to be pure (not necessarily of weight $i$). Here I am again more interested in weakening the "projective" rather than the "smooth".

At the risk of being even more vague, let me add some motivation from left field. There are several situations in representation theory where one expects/knows that the weight filtration on some cohomology groups splits (and is even of Hodge-Tate type). For instance, the cohomology of intersections of Schubert cells with opposite Schubert cells. However, the reasoning/heuristic has, a priori, nothing to do with geometry but more with the philosophy of "graded representation theory" (ala Soergel, see for instance his ICM94 address) I would love to have a geometric reason/heuristic for this. Pertinent to this is also the question of when should one expect the canonical Hodge structure on extensions between perverse sheaves of geometric origin to be split Tate? The only examples I know come from representation theory (see Section 4 of Beilinson-Ginzburg-Soergel's "Koszul duality patterns in representation theory").

Answer by Reladenine Vakalwe for Geometric interpretation of translation through the wall

2012-12-20T17:16:28Z

I am guessing the following is well known to you/not what your are looking for, but nonetheless:

Let $s$ be a simple reflection, $P_s$ the corresponding minimal parabolic, $\pi_s\colon G/B\to G/P_s$ the projection. Translation across the $s$-wall `corresponds' to $\pi_s^*\pi_{s*}$. I use quotation marks because as stated this is clearly not true (translation across the wall is t-exact, $\pi_s^*\pi_{s*}$ is certainly not). However, $\pi_s^*\pi_{s*}$ does correspond to translation across the wall under Koszul duality. This is also the same as convolving with the $IC$-complex corresponding to $s$.

Morally (as you point out), reflection across the wall should correspond to convolving with the corresponding tilting. But there is an annoying issue here: tiltings are not $B$-equivariant. Similar problem occurs if instead of convolution using equivariant derived categories you try to use the standard Fourier-Mukai formalism and try to use an object on $G/B\times G/B$ as a kernel. However, there is a fix that comes at some technical expense. Namely, Bezrukavnikov and Yun's free monodromic sheaves http://arxiv.org/abs/1101.1253. The idea actually goes back to the paper of Beilinson and Ginzburg that you cite (look at Section 5).

A cohomology computation request.

2012-12-02T02:53:12Z

The short: Let

$X= \{(x,y,z) \in \mathbb{C}^* \times \mathbb{C} \times \mathbb{C} \, |\, yz-x\neq 0\}$

Compute $H^*_c(X)$ (say with $\mathbb{C}$-coefficients).

The long: Unless I messed something up, the answer should be

$H_c^3(X) = \mathbb{C}$, $H_c^4(X) = \mathbb{C}^2$, $H_c^5(X) = \mathbb{C}^2$, $H_c^6(X)=\mathbb{C}$, and $0$ otherwise.

However, as will become clear I did this through an extremely convoluted argument and I am hoping someone can explain to me a simple way of doing this.

Some context: this question is closely related to

http://mathoverflow.net/questions/111682/intersection-of-plus-minus-cells-in-bialynicki-birula-decomposition

Namely, $X$ is the intersection of the big Bruhat cell and the big opposite Bruhat cell for $SL_3$.

It is also closely related to

http://mathoverflow.net/questions/83877/are-kazhdan-lusztig-r-polynomials-the-poincare-polynomials-of-the-corresponding

(the Hodge-Euler characteristic of $X$ is the $R$-polynomial corresponding to the identity and the longest element in type $A_2$; this is a special case of a general fact about $R$-polynomials).

My convoluted argument: Considering the alternatives $y\neq 0$ and $y= 0$, one obtains a decomposition of $X$ into

$X = (\mathbb{C}^*)^3 \sqcup \mathbb{C}\times \mathbb{C}^*$ .

This gives rise to a long exact sequence that puts several restrictions on $H^*(X)$ (but doesn't fully determine it, namely $H_c^3(X), H_c^4(X), H_c^5(X)$ aren't fully determined).

So far this is nice, but now the convoluted bit starts. It is not too hard to see that $H^{*-3}(X)$ equals $Ext^*(\Delta_e, \Delta_{w_0})$ where $\Delta_e$ is the unique simple Verma and $\Delta_{w_0}$ is the unique projective Verma in the principal block of the BGG-category $\mathcal{O}$ of $\mathfrak{sl}_3$.

Aside: this a special case of a statement connecting extensions of Verma modules with cohomology of intersections of Bruhat cells and opposite Bruhat cells (and also why I am interested in the cohomology of these intersections).

Now some standard representation theoretic facts about these $Hom$ spaces combined with the decomposition above yield what I claimed the answer to be.

I would love a simpler/more geometric way of going about this computation!

Answer by Reladenine Vakalwe for Non-rigorous reasoning in rigorous mathematics

2012-12-01T02:04:29Z

Unless I am misunderstanding, the Weil conjectures fit into this framework. I believe it took about two decades for the Grothendieck school to formalize Weil's heuristic that his conjectures follow from a Lefschetz fixed point formula for varieties over finite fields. Rather than try to flesh this post out, let me point to the Brian Osserman's article for the PCM: http://www.math.ucdavis.edu/~osserman/math/pcm.pdf. The Wikipedia account of the history also seems to be not bad (but I haven't really read it in detail): http://en.wikipedia.org/wiki/Weil_conjectures. I also seem to remember learning about the history and some of the mathematics for the first time from an article by Steven Kleiman, but cannot remember the precise reference.

Answer by Reladenine Vakalwe for How to understand the Harish-Chandra isomorphism?

2012-11-21T05:33:52Z

Here is one algebraic/representation theory perspective on why such a morphism might exist, although I don't think historically this is the way it went at all. Let's admit for a moment that one might be interested in Verma modules. It is easy to see that the center acts via scalars on these. Verma modules are parametrized by $\mathfrak{h}^*$ and in this way one gets an algebra morphism $Z(\mathfrak{g}) \to Sym(\mathfrak{h})$. Now it is also relatively straightforward to see (if I remember correctly) that for each simple reflection $s$, the Verma module $M(s\cdot \lambda)$ occurs as a submodule of the Verma module $M(\lambda)$, for $\lambda$ integral. It follows that the algebra morphism constructed earlier lands in $W$-invariants.

Intersection of plus/minus cells in Bialynicki-Birula decomposition

2012-11-06T20:55:09Z

Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$:

$X = \bigsqcup_{w\in W} C_w$ and $X = \bigsqcup_{w\in W} C^w$,

where

$C_w = \{x\in X | \lim_{t\to 0} t\cdot x = w \}$ and $C^w=\{x\in X | \lim_{t\to \infty} t\cdot x = w\}$ ,

the so-called plus and minus Bialynicki-Birula decomposition. Assume that both of these decompositions are in fact stratifications (although this latter bit may be irrelevant to the question).

My (slightly vague) question: Are there any general results about the structure of the intersections $C_w \cap C^v$? When non-empty are they smooth in general? My favorite example of the flag variety, these are smooth (I think). It would make me extremely happy if someone could point me to towards a general description of the cohomology groups of these intersections?

Note: In the special situation of flag varieties I am familiar with Deodhar's and Curtis' results describing these intersections. But, unless I am missing something, their results don't help in describing the cohomology groups.

Answer by Reladenine Vakalwe for Do mixed Hodge modules form a stack?

2012-10-17T17:09:47Z

This is an extremely partial (and sketchy) answer that I haven't fully thought through. But, I am essentially just running the proof for perverse sheaves. Let me first construct the required MHM if the cover just consists of two elements. In this case the required MHM can be define as the cone in the Mayer-Vietoris distinguished triangle. Now use induction to prove the statement for a countable cover. For arbitrary covers I am not quite sure how to proceed. On the other hand, all I am saying is that define the MHM as the cohomology of the Cech complex given by your data. Unless I am missing something, this (modulo some technicalities with finiteness assumptions) gives the construction. Now for uniqueness, probably one can just observe that the usual isomorphism that shows uniqueness for perverse sheaves is a morphism of MHM. Since it is an isomorphism on underlying perverse sheaves, it is an isomorphism on the MHMs. In the case of a 2-element cover this also follows from the fact that there are no negative Exts between MHMs (which leads to the cone in the Mayer-Vietoris triangle being unique).

Polarizable variations of (mixed) Hodge structures

2012-10-14T23:53:26Z

I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be grateful if the experts would point out any subtleties with my questions if rational is replaced by real. Let me at the moment focus on questions related to polarizability:

Let $\mathcal{L}$ be a local system underlying a polarizable variation of Hodge structure on a smooth variety. Does the polarizability imply $\mathcal{L}$ is self-dual? If yes, then does every direct summand of this local system also have to be self-dual (I am guessing no to the latter)?
More generally, let $M$ be a polarizable mixed Hodge module on some variety (not necessarily smooth). Is polarizability equivalent to Verdier self-duality (up to Tate twist) in the derived category of mixed Hodge modules? If not equivalent, does it at least imply Verdier self-duality?

Mixed structures on Hom spaces induced by mixed sheaves

2012-10-06T06:38:35Z

Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let

$rat\colon D^b_m(X)\to D^b(X)$

be the `forgetful' functor. This is t-exact for the perverse t-structure on the right. Write $MHM(X)$ for the abelian category of mixed Hodge modules on $X$. Then $MHM(pt)$ is the category of graded polarizable mixed Hodge structures, and $rat\colon MHM(pt) \to VectorSpaces$ is the evident forgetful functor.

Now let $M,N\in D^b_m(X)$. Set

$\mathcal{H}om(M,N) = \Delta^!(\mathbb{D}M \boxtimes N)$,

where $\Delta\colon X\to X\times X$ is the diagonal map, and $\mathbb{D}$ is Verdier duality.

Let $a\colon X \to pt$ be the evident map. Then

$rat ( H^0(a_*\mathcal{H}om(M,N))) = H^0(rat(a_*\mathcal{H}om(M,N))) = Hom(rat(M), rat(N))$

and in this way we get a Hodge structure on $Hom(rat(M),rat(N))$. All functors are derived.

My question: If $M,N$ are ~~pure~~ pointwise pure (see Geordie Williamson's comment below), then is the induced structure on $Hom(rat(M), rat(N))$ pure?

My gut answer is no (even if $X$ is complete, the $\Delta^!$ should be messing weights up), but it would make me happier if the answer is yes!

If the answer is no, under what additional conditions (other than requiring $X$ to be smooth and complete plus $M,N$ being the `constant' sheaf) can the answer be converted to yes?

I guess one could also ask the same sort of question for mixed $\ell$-adic sheaves. But I am even less familiar with that setting.

Answer by Reladenine Vakalwe for Describe a topic in one sentence.

2012-10-07T22:00:36Z

Geometric representation theory: keep translating the problem until you run into Hard Lefschetz, then you are done.

About an argument in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel.

2012-09-30T20:25:06Z

I am trying to understand Proposition 3.4.2 in `Koszul duality patterns in representation theory' by Beilinson-Ginzburg-Soergel [BGS]. A copy of the paper can be found at http://home.mathematik.uni-freiburg.de/soergel/

I outline the setup below. Text in bold are my own commentary. I have taken the liberty to change some confusing notation in the paper. However, it is entirely possible that the reason I am confused is that one of my "changes of notation" isn't correct. I have tried to be careful in my changes, but apologies in advance for any additional confusion this may contribute to.

My question is that indicated by the last bold face text below.

Let $X$ be a complex variety with an algebraic stratification by affine linear spaces $X = \sqcup_{w\in W} X_w$. Let $IC_w$ denote the intersection cohomology complex on $X$ corresponding to the constant sheaf on $X_w$.

Let $X = Y_0 \supset Y_1 \supset \cdots \supset Y_r = \emptyset$ be the corresponding filtration by closed subvarieties so that $Y_{p}- Y_{p+1} = X_p$ for some strata $X_p$.

Let

$j_w\colon X_w \to X$

be the inclusion of the strata. Assume parity vanishing, i.e., assume

$H^ij_v^*IC_w = 0$ unless $i = dim(X_v) + dim(X_w) \mod 2$, for all $v,w\in W$.

Here $H^*$ denotes perverse cohomology.

Proposition: Under the assumption of parity vanishing, hypercohomology induces an injection

$Hom^{\bullet}_{D^b(X)}(IC_x, IC_y) \to Hom_{\mathbb{C}}(\mathbb{H}^{\bullet}IC_x, \mathbb{H}^{\bullet}IC_y)$.

Proof: By parity vanishing the spectral sequence $\mathbb{H}^{p+q}j_p^!IC_x \implies \mathbb{H}^n IC_x$ is degenerate (the spectral sequence is defined via the filtration by local hypercohomology along the strata, for details see Section 3.4 of [BGS]). So if $f\in Hom^{\bullet}_{D^b(X)}(IC_x, IC_y)$ is given such that $\mathbb{H}^{\bullet}f = 0$, then necessarily $0 = j_p^!f \in Hom^{\bullet}_{D^b(X)}(j_p^!IC_x, j_p^!IC_y)$ for all $p$. Let

$a_p\colon Y_p \to X$

be the closed inclusion. We have a decomposition

$u\colon X_p = Y_p - Y_{p+1} \to Y_p$,

$i\colon Y_{p+1}\to Y_p$

in an open and a closed subset and a distinguished triangle

Edit: the original distinguished triangle (as stated in [BGS]) wasn't correct, I have now made the fix

$i_*i^!a_p^! \to a_p^! \to u_*u^!a_p^!$

(so this distinguished triangle is the same as $i_*a_{p+1}^! \to a_p^! \to u_*j_p^!$ )

which shows that $a_{p+1}^!f = 0 = j_{p}^!f$ implies $a_p^!f = 0$ (it is this implication that I don't understand, related to my confusion is an earlier question of mine http://mathoverflow.net/questions/108481/showing-morphism-of-sheaves-is-zero )

Hence by induction $j_p^!f = 0$ for all $p$ implies $f = a_0^!f = 0$.

Any comments that would clarify the above would be most appreciated!

Showing morphism of sheaves is zero

2012-09-30T16:16:38Z

I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and let $f\in Hom_{D^b(X)}(M,N)$. Suppose $i^*f = 0$ and $j^*f= 0$.

Then is it true that $f=0$?

My gut answer is yes, and I thought I would be able to lever the canonical distinguished triangle $j_! j^* \to id \to i_* i^* \to j_!j^*[1]$ into a proof. But I have failed so far.

Question regarding a statement in `A proof of Jantzen conjectures'

2012-09-04T04:00:38Z

So I am trying to understand a statement in the proof of Corollary 5.2.3 in `A proof of Jantzen conjectures' (a copy of the paper can be found at http://www.math.harvard.edu/~gaitsgde/grad_2009/).

The starting assumptions of the Corollary are:

`Let $M_1, M_2$ be pure perverse sheaves of weights $w_1, w_2$ that are both $*$- and $!$-pointwise pure. Suppose that $Ext^1_{mixed}(M_1, M_2)\neq 0$. Then ...'

The first line of the proof says:

`Clearly either $Y_1 \subset Y_2$ or $Y_2 \subset Y_1$ (otherwise $Ext^1 = 0$)'

Here $Y_i = Supp(M_i)$. This statement confuses me. The ordinary ( = non-mixed) $Ext^1$ group should be the extensions between the restrictions of $M_1$ and $M_2$ to the intersection $Y_1 \cap Y_2$. I don't see why this should vanish if either $Y_1$ isn't contained in $Y_2$ or vice versa. Anyway, even if the unmixed group does vanish, without vanishing of the unmixed $Hom$ group I don't see how I would get $Ext^1_{mixed}$ vanishes.

Presumably I am making a stupid error here and both of the unmixed groups above do vanish? Any comments would be appreciated.

Answer by Reladenine Vakalwe for Question regarding a statement in `A proof of Jantzen conjectures'

2012-09-04T06:18:10Z

Too long to leave as a comment.

Ben: Let $i_k$, $k=1,2$ be the closed inclusions $Y_i \to X$ (where $X$ is my ambient space). Then $M_k = i_{k*}i_k^*M_k$. So

$Ext^1(M_1, M_2) = Ext^1(i_{1*}i_{1}^*M_1, i_{2*}i_2^*M_2) = Ext^1(i_2^*i_{1*}i_1^*M_1, i_2^*M_2)$

Now let $r\colon Y_1\cap Y_2 \to X$ and $s\colon Y_1\cap Y_2 \to Y_2$ be the closed inclusions and we get:

$Ext^1(i_2^*i_{1*}i_1^*M_1, i_2^*M_2) = Ext^1(s_*r^*M_1, i_2^*M_2) = Ext^1(r^*M_1, s^!i_2^*M_2)$

Ah, so my initial error was to magically convert the $s^!$ to $s^*$, but it still reduces the computation of the Ext group to the intersection (or did I do something screwy again?). On the other hand, I don't see how to sanely deal with the $s^!i_2^*$. Regardless, I still don't see why the mixed Ext group in the original question is vanishing.

Answer by Reladenine Vakalwe for On the definition of regularity

2012-06-28T02:55:52Z

There are some comparison results in Chapter 5 of Bjork's `Analytic D-modules and Applications'. Also see Chapter 8. In particular, I think Thm. 8.7.3 combined with Thm. 5.6.5 (almost) gives (1) iff (3). Further, I think Prop. 5.6.22 gives the equivalence with (2). There are also results in there comparing Deligne's description.

I must admit though that I find Bjork quite notationally dense and am not particularly familiar with it, so I may be quite off with the references above. I am interested in this question also, so please comment/post if you find better references.

Characteristic variety of a D-module along smooth pullback

2012-06-27T18:03:24Z

All varieties are over the complex numbers. Given a smooth variety $X$, write $T^* X$ for its cotangent bundle. For a morphism of smooth varieties $f: X \to Y$ write $f_{\pi}: T^*Y \times_Y X \to T^*Y$ for the projection map, and let $f_d: T^*Y \times_Y X \to T^* X$ be the map dual to the derivative.

Now let $f\colon X\to Y$ be a smooth morphism. Let $M$ be a coherent $D_Y$-module. I am trying to understand the proof of:

$Ch(f^*M) \subseteq f_df_{\pi}^{-1}Ch(M)$,

(where $Ch$ means characteristic variety) as presented in J.P. Schneider's notes http://www.analg.ulg.ac.be/jps/rec/idm.pdf (see pages 34-35).

I need a bit more notation before I get to my exact question: we know $f^*M = O_X\otimes_{f^{-1}O_Y}f^{-1}M$ as an $O_X$-module. Endow $f^*M$ with the good filtration $F_i(f^*M) = O_X\otimes_{f^{-1}O_Y}f^{-1}F_iM$, where the $F_i$ on the right hand side is some good filtration of $M$. Then we have a surjection

$O_X\otimes_{f^{-1}O_Y}f^{-1}gr(M) \twoheadrightarrow gr(f^*M)$

of $gr(D_X)$-modules, where $gr$ denotes associated graded. Write $\tilde{gr}(-)$ for the sheaf on the cotangent bundle associated to $gr(-)$. Then the claim (last two lines of the proof on p35 of the aforementioned notes) is that:

since $f_d$ is finite we obtain a surjection

$f_{d*}f_{\pi}^*\tilde{gr}(M) \twoheadrightarrow \tilde{gr}(f^*M)$

I don't understand this claim. Any clarifying thoughts would be most appreciated.

Unless I made a mistake:

$f_{d*}f_{\pi}^*\tilde{gr}(M) = f_{d*}(O_{T^*Y\times_Y X}\otimes_{f_{\pi}^{-1}\pi_Y^{-1}gr D_Y}f_{\pi}^{-1}\pi_Y^{-1}gr(M))$,

where $\pi_Y\colon T^*Y\to Y$. What is required is that this equal

$O_{T^*X}\otimes_{\pi_X^{-1}gr D_X}(O_X\otimes_{f^{-1}O_Y}f^{-1}gr(M))$

I don't see how to proceed. Presumably I am making this way too complicated and there is a very simple explanation?

Also, it is mentioned at the end of the proof that the estimate of the characteristic variety is in fact an equality. Does anyone know of a reference for this?

C^*-equivariant modules on a vector bundle vs graded modules on the pushforward.

2012-06-21T18:00:19Z

All varieties are over $\mathbb{C}$.

Let $X$ be a variety and $\pi \colon E \to X$ a geometric vector bundle. So $ \pi $ is affine. Then certainly the assignment $ M \mapsto \pi_*M $ defines an equivalence between quasi-coherent $\mathcal{O}_E$ -modules and quasi-coherent $\pi_*\mathcal{O}_E$-modules.

Now $\mathbb{C}^{\times}$ acts on $E$ via dilation of the fibres of $\pi$. So $\pi_* \mathcal{O}_E$ acquires a grading. Is it true that $M \mapsto \pi_* M$ gives an equivalence between $\mathbb{C}^{\times}$-equivariant quasi-coherent $\mathcal{O}_E$ -modules and graded quasi-coherent $\pi_*\mathcal{O}_E$-modules?

If this is true, does it generalize to replacing $\pi$ being a vector bundle with $E$ just equipped with a $\mathbb{C}^{\times}$-action, $\pi$ affine and $\mathbb{C}^{\times}$-equivariant, $\mathbb{C}^{\times}$ acting on $X$ trivially?

Added later (in response to a-fortiori's comment): Perhaps I hadn't done my homework as conscientiously as I thought. Regardless, here are some thoughts. As candidate for the quasi-inverse (is there a more sensible choice?) take

$N\mapsto \mathcal{O}_E \otimes_{\pi^{-1}\pi_*\mathcal{O}_E}\pi^{-1}N$

with $\mathbb{C}^{\times}$-equivariant structure given by

$z \cdot (f(x,v) \otimes n_i) = f(x, z^{-1}v) \otimes z^{-i}n_i$,

where $n_i$ is in the $i$-th component of $N$ and the rest of the notation is (I hope) self-explanatory. Hitting the structure sheaf $\mathcal{O}_E$ (with the trivial/evident equivariant structure) with these functors works fine, so this isn't completely ridiculous. But now I am not even sure whether there are other equivariant structures on $\mathcal{O}_E$ that would make this breakdown.

Non-characteristic maps (ala D-modules)

2012-05-23T17:25:29Z

I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is the setup:

All varieties are over the complex numbers. Given a smooth variety $X$, write $T^* X$ for its cotangent bundle, and $T^*_XX \subseteq T^*X$ for the zero section.

Let $f: X \to Y$ be a morphism of smooth varieties. Write $f_{\pi}: T^*Y \times_Y X \to T^*Y$ for the projection map, and let $f_d: T^*Y \times_Y X \to T^* X$ be the map dual to the derivative. Let $\Lambda \subseteq T^* Y$ be a closed $\mathbb{C}^* $ stable subvariety ($\mathbb{C}^*$ acting on $T^*Y$ in the evident way). Then $f$ is called non-characteristic for $\Lambda$ if

$f_{\pi}^{-1}(\Lambda) \cap f_d^{-1}(T^*_XX) \subseteq T^*_YY \times_Y X$

The `well known' fact: if $f$ is non-characteristic for $\Lambda$, then $f_d$ restricted to $f_{\pi}^{-1}(\Lambda)$ is finite.

I would be grateful if someone could explain the truth of this to me.

Some remarks:

a) The statement is actually an if and only if, but the converse is straightforward, since the fibres of $f_d$ are $\mathbb{C}^* $ stable.

b) I believe I understand how to show that $f_d$ restricted to $f_{\pi}^{-1}(\Lambda)$ is quasi-finite (using the the $\mathbb{C}^* $ stability and the upper semi-continuity of fibre dimension). But that's as far as I have got.

Comment by Reladenine Vakalwe

2013-01-07T00:26:39Z

Also, is Jantzen's Habilitationsschrift a reference to "Moduln mit einem hochsten Gewicht"? That text strikes fear in my heart!

Comment by Reladenine Vakalwe

2013-01-07T00:22:44Z

To be honest, apart from being slightly open to "miracles sometimes occur", I am pretty skeptical about a). I do believe in b) though because it matches with some heuristics I have regarding higher extensions between Vermas (heuristics coming from some brute force computations, but I haven't managed to work out all the extensions in Boe's counterexample, so these may still be "too low rank"). I think I understand Mazorchuk's proof (regarding b)) enough to see that he defers the burden to a result of Backelin. But haven't got my hands on the Backelin paper yet.

Comment by Reladenine Vakalwe

2013-01-06T20:14:36Z

I agree with the type $A_2$ check (although I don't think I could typeset the lattice/diagram). Stroppel's diagram is pretty impressive! I am reasonably sure that b) is true in general (since I believe Mazorchuk's result), but of course I don't understand why.

Comment by Reladenine Vakalwe

2013-01-06T19:36:08Z

Inspired by your example, as long as I did it correctly, c) is false in type $A_2$ also, and I am reasonably sure is essentially always going to fail (with the exception of $\mathfrak{sl}_2$).

Comment by Reladenine Vakalwe

2013-01-06T18:32:37Z

Yes! This is a nice counterexample. Not simple, but that's a minor quibble. Thank you! Any thoughts on b)?

Comment by Reladenine Vakalwe

2013-01-06T16:11:49Z

Dag: Could you elaborate on your second comment regarding the counterexample to c) (possibly as an answer)?

Comment by Reladenine Vakalwe

2013-01-06T16:10:07Z

Dag: Your first comment about the statement after "Note:" in a) being wrong is absolutely correct. I have removed it (sorry strikethrough wasn't showing up correctly). As to how I am choosing $k$. Basically I just want the first layer that isn't completely contained in $\Delta_w$. Does that clarify or am I being screwy?

Comment by Reladenine Vakalwe

2013-01-06T16:02:13Z

Jim: Thanks for pointing out the misprint. I have fixed it now. I agree about the risk of relying on rank 2 examples. But well, it's a start. Figuring out socle filtrations for higher rank is quickly going to start being a pain! Apologies for the length. I had hoped that getting a) out there quickly would alleviate some of the pain. My next step is to ask Mazorchuk, but am still holding out hope that I am missing something silly and MO will offer some instant gratification!

Comment by Reladenine Vakalwe

2012-12-31T18:06:43Z

Not quite in line with your question. But if you were dealing with a reasonable topological space $X$, then $Ext$ groups of the constant sheaf with itself (in the category of constructible sheaves) are the cohomology groups of that space. More generally, extensions from the constant sheaf to any complex of sheaves is hypercohomology with coefficients in the complex.

Comment by Reladenine Vakalwe

2012-12-31T02:25:39Z

As the Mathoverflow bot has so graciously pinged this question, I may as well add the following. The "curious Poincare duality" and "palindromic phenomenon" mentioned above has a high powered explanation. Namely: Koszul duality. This can in turn be used to show the dimension bound asked for. But this is using a blowtorch to light a candle.

Comment by Reladenine Vakalwe

2012-12-29T17:38:08Z

Dan: Thanks for the examples! They are helpful. I wasn't aware of Minhyong Kim's paper. It's really nice.

Comment by Reladenine Vakalwe

2012-12-29T17:31:37Z

Donu: Thanks! Your answer is helpful. At least how I am reading it is that the weight filtration has geometric as well as linear algebraic content to it (split over $\mathbb{R}$ vs. $\mathbb{Q}$). As an aside, related to your comment about varieties coming from linear algebra, most of my examples come from flag varieties and "miracles often happen in flag varieties"!

Comment by Reladenine Vakalwe

2012-12-21T01:39:20Z

Jim: Perhaps I should make this a separate question. But don't Soergel's arguments (which I am implicitly using to justify my answer below) show this? Under Soergel's functor to combinatorics $\mathbb{V}$, translation across the wall corresponds to (roughly) restriction/induction for the coinvariant algebra. The latter depending only on the stabilizer (namely $s$) of $\lambda$. Or am I confused? Of course, $\mathbb{V}$ is not an equivalence but it is full and faithful on maps between projectives/tiltings, but this should be enough to show the desired independence? No?

Comment by Reladenine Vakalwe

2012-12-20T00:52:15Z

Slightly confused about a certain point, why is pushforward along a proper map preserving t-structure? Regarding the question, there is a good estimate on singular support of $f_*M$ if $f_{\pi}\colon f_d^{-1} Ch(M)\to T^*Y$ is finite (I hope the notation is self-explanatory. See Kashiwara's D-Modules and microlocal calculus section 4.7. This condition is analogous to the non-characteristic condition.

Comment by Reladenine Vakalwe

2012-12-17T01:56:20Z

Deleted a previous comment of mine where I thought I had an argument, since it was a pipe dream.