User shenghao - MathOverflow

Group extensions with a non-commutative kernel

2013-05-05T15:27:56Z

Let

$$ 1\to K\to G\to H\to 1 $$ be an extension of groups. When $K$ is commutative, $H$ acts on $K$ by conjugation; and given groups $K$ and $H,$ with $K$ commutative and $H$ acting on $K,$ such extensions are classified by group cohomology $H^2.$ For instance, if $H$ is profinite and $K$ is a (commutative) discrete $H$-module, then any extension splits potentially, i.e. $G\to H$ has a section over an open subgroup of $H,$ as the continuous group cohomology equals the direct limit of cohomologies of finite groups.

Is there any theory when $K$ is not commutative? In this case, an extension does not induce a "conjugation" action of $H$ on $K.$ For instance, when $K=G(k^s)$ for some algebraic group $G$ over a field $k$ and $H=Gal(k^s/k),$ it seems that one may still ask for "affine extensions with kernel $G$" when $G$ is not a torus, just no $H^2$-interpretation.

Let me ask a real question. Let $K_N$ be the maximal extension of $\mathbb Q$ unramified outside $N$ (for an integer $N>1$), and let $G=Gal(K_N/\mathbb Q).$ Consider its abelianization $H,$ which is $\prod_{p|N}\mathbb Z_p^*$ by CFT. Does the projection $G\to H$ have a section over some open subgroup of $H?$

Thank you.

Where can I find the two books of Weil on abelian varieties?

2013-02-22T06:31:09Z

They are "Sur les courbes algebriques ..." and "Varietes abeliennes et courbes algebriques", published by Hermann in 1948. Is there any link online for these old Hermann books?

Thanks in advance.

Does every regular Noetherian domain have finite Krull dimension?

2013-01-23T07:09:32Z

Background: A Noetherian ring is said to be regular if its localizations at all prime (or maximal) ideals are regular local rings. Without this assumption, there are counter-examples.

Thanks.

roots of analytic functions

2009-10-16T16:59:55Z

Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic continuation to the whole complex plane (so it has at most countably many poles). What do we know about the number-theoretic property of the roots and poles? Are they algebraic numbers? If they are, are they stable under the action of the Galois group of the rationals?

More generally, if the coefficients of $f(z)$ are algebraic numbers, and let $\sigma$ be an automorphism of the algebraic closure of the rationals, then what is the relation between roots of $f(z)$ and roots of $f^{\sigma}(z),$ where $f^{\sigma}(z)$ is the power series obtained by applying sigma to the coefficients of $f(z)?$

Without assuming the finiteness of radius of convergence, $\sin(z)$ gives a counter-example.

Edit: Let me give a second try, by imposing more requirements on $f(z).$ I'm thinking about the case where $f(z)$ is the zeta function of an algebraic stack over a finite field, so let's assume $f(z)$ has an infinite product expansion over $\ell$-adic numbers, like $\prod P_{odd}(z)/\prod P_{even}(z),$ where each $P_i(z)$ is a polynomial over $\mathbb Q_{\ell}$ with constant term 1. Assume they have distinct weights, e.g. reciprocal roots of $P_i(z)$ have weights $i.$ Then can we conclude that all coefficients of $P_i(z)$ are rational numbers? Thanks.

Is there an explicit example of a coefficient sheaf for which hard Lefschetz fails?

2012-08-05T09:05:24Z

Recall that if $X$ is a projective algebraic complex manifold, $L$ is a semisimple $\mathbb C$-local system on $X$ of geometric origin (roughly speaking, this means that $L$ is a cohomology sheaf $R^if_*\mathbb C$ for some algebraic morphism $f:Y\to X;$ see BBD for the precise definition), and $\eta\in H^2(X,\mathbb C)$ is an ample class, then $$ \eta^i\cup-:H^{\dim X-i}(X,L)\to H^{\dim X+i}(X,L) $$ is an isomorphism. This also holds when the projective variety $X$ is allowed to have singularities and $L$ is a perverse sheaf (again semisimple of geometric origin), appropriately shifted.

I'd like to know an example for which this fails; of course, $L$ is no longer of geometric origin. Over finite field, as long as $L$ is assumed semisimple, hard Lefschetz always holds, and conjecturally $L$ is of geometric origin.

References on semismall maps

2011-08-14T15:31:17Z

Where can I find references on semismall maps, in the sense of Goresky and MacPherson? I don't want to restrict to the case where the base is $\mathbb C$ (an arbitrary alg. closed field would be fine), or maps $f:X\to Y$ from a smooth variety $X.$ In particular, I'd like to find the proof (if the statement is correct, which I'm not sure) that $Rf_*$ takes an irreducible (middle) perverse sheaf $F$ supported on $X$ to a perverse sheaf; I can only do this when $X$ is smooth and $F$ is a lisse sheaf, or when all the fibers of $f$ have dimension at most one.

Recall: A proper surjective morphism $f:X\to Y$ is called $semismall$ if $\dim X\times_YX=\dim X.$

Thank you.

The automorphism group of a hyperelliptic curve

2010-11-25T18:05:32Z

Let $C$ be the projective smooth genus 2 curve defined by $y^2=x^5-x$ over $\mathbb F_5.$ What is the order of its automorphism group (automorphisms over $\mathbb F_5$)?

I have seen different answers. In Hartshorne's Algebraic Geometry, p. 306, the answer is $2p(p^2-1)=240.$ In INFORMATION Volume 8, Number 6, pp. 837-844, Isomorphism classes of genus-2 hyperelliptic curves over finite fields $\mathbb F_{5^m},$ by L. Hernández Encinas and J. Muñoz Masqué, theorem 2, the answer is $|A_{4221}|=20$ (using notations there). A professor (let me not to mention the name for now) told me the order is 120.

Maybe I misunderstand some of the references above?

Does S^6 have the structure of an algebraic variety?

2011-11-20T08:11:06Z

More accurately, the question should be: Is it known that $S^6,$ the 6-dimensional sphere, is $not$ a (proper) complex algebraic variety, or algebraic space? And is there a reference? It's easy to see that it cannot be projective (as $H^2=0$), but I don't see how it can violate the usual general properties of algebraic varieties [e.g. by Chow's lemma one can get a projective (and smooth, by resolution of singularities) variety lying over and birational to it, but the cohomology groups can get larger when we go up...]. Maybe one needs some theory of classification of 3-folds (if there is one such theory).

Knowing that it cannot be algebraic will somehow make the analogous question of complex structure much sharper (by the way, it does have an almost complex structure).

Thank you.

A local-global question on group representations

2011-09-11T16:45:11Z

Let $G$ be a group, and let $V$ be a finite dimensional $\mathbb Q$-linear representation of $G.$ By extension of scalars we obtain the $\mathbb Q_l$-linear representation $V_l=V\otimes\mathbb Q_l.$

Question: Is the natural morphism ´$$ H^1(G,V)\to\prod_lH^1(G,V_l) $$ injective? Stated in this generality, the answer is possibly negative. But are there some contexts (e.g. G is a $\mathbb Q$-algebraic group and $V$ is an algebraic representation etc.) in which the answer is yes?

And what about other degrees $H^i$?

Is there a general projection formula for morphisms of ringed topoi?

2010-03-30T23:27:55Z

What's the general projection formula in algebraic geometry, for instance on the level of derived categories of ringed topoi? And what's the reference? I guess it might be in SGA 4, but couldn't find it.

Two examples:

Zariski site, $D^b_{qcoh}$ on schemes. Let $f:X\to Y$ be a proper map of noetherian schemes (maybe there are some other mild conditions), and let $F\in D^b_{qcoh}(X)$ and $G\in D^b_{qcoh}(Y).$ Then $(Rf_*F)\otimes^L G=Rf_*(F\otimes^L Lf^*G)$ .
Etale site, say ringed by a torsion ring like $Z/n.$ Let $f:X\to Y$ be a (seperated; but this condition can be removed. See for instancde Laszlo and Olsson, The six operations on Artin stacks...) map of schemes of finite type over some base $S$ ($S$ may need to satisfy some assumptions, in order for the classical results in SGA 4/4.5 or Gabber's new results on finiteness of $f_*$ and dualizing complexes to work; but let's be sloppy). Let $F\in D^-_c(X,Z/n)$ and $G\in D^-_c(Y,Z/n).$ Then $Rf_!F\otimes^L G=Rf_!(F\otimes^L f^*G).$

We used $f_!$ in example 2 in order to allow $F$ and $G$ to be in $D^-_c$ rather than $D^b_c.$ If one restricts to $D^b_c,$ is it also true for $f_*?$

Lifting varieties from char. $p$ to char. 0 after alterations

2011-05-17T13:11:50Z

The question is related to this MO question:

http://mathoverflow.net/questions/25337/lifting-varieties-to-characteristic-zero

Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an alteration $Y\to X,$ with $Y$ also projective smooth, such that $Y$ lifts to char. 0?

Side remark: over $k=\overline{\mathbb F}_p,$ modulo Tate conjecture, abelian varieties "generate" the motives of all proj. smooth varieties. Since abelian varieties are liftable, one can say that the (irred. components of) motives of any proj. smooth varieties is liftable in some sense. And I wonder if this can be realized geometrically. "Alteration" in my question is just a try; replace it with any reasonable geometric construction if you want. For instance, "a proper surjection" would be fine.

Analogue of Shafarevich-Ogg's theorem over complex numbers

2011-09-08T10:21:18Z

Let $f:E\to D^*$ be a family of complex elliptic curves parametrized by the punctured open disk $D^*.$ Assume that the monodromy on $H^1$ is trivial (i.e. $R^1f_*\mathbb Z$ is a constant sheaf on $D^*$ ). Does this imply that $f$ extends to a family of elliptic curves over the full disk $D?$

Here's an attempt, which I'm not sure if it works: maybe there is an equivalence (Riemann-Deligne?) between families of elliptic curves over a (smooth) base over $\mathbb C$ and variations of $\mathbb Z$-Hodge structures satisfying Griffiths transversality, of rank 2 and weight 1 on the same base. The constant local system certainly extends to $D.$

What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?

2011-08-09T06:03:50Z

I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the morphism to be smooth; e.g. a family of nodal curves $X_t$ with $p_a=2$ and such that the $j$-invariant of the normalizations $\widetilde{X}_t$ is not constant in $t,$ would certainly qualify.

Motivation: If $f:X\to Y$ is a proper morphism of complex algebraic varieties, then by Morse theory, there exists a "stratification of $f";$ in particular, over each stratum of $Y,\ f$ is a $C^{\infty}$-fiber bundle. I wonder if this could be true in char. $p,$ but the first thing is to have an analogous notion.

Does intersection pairing on `$IH^(X)$` agree with cup-product on `$H^(X)$`?

2011-06-16T15:50:06Z

Let $X$ be a proper singular variety over $k=\overline{\mathbb F}_p,$ irreducible of dimension $d.$ Let $H^*(X)$ and $IH^*(X)$ be the $l$-adic cohomology groups and $l$-adic intersection cohomology groups of $X,$ resp. Then, is the natural map $H^*(X)\to IH^*(X)$ compatible with the cup-product on $H^*(X)$ and the intersection product on $IH^*(X)?$

Background and Motivation: Given $X_0/\mathbb F_q$ an $\mathbb F_q$-structure of $X,$ one deduces a Galois action on $H^*(X)$ and $IH^*(X),$ with respect to which they are "mixed" (the 2nd one being pure), and the weight filtrations $W$ on both of them are independent of the choice of $X_0.$ One has a natural morphism $$ H^n(X) \to IH^n(X) $$ which factors as $$ Gr^W_n H^n(X) \hookrightarrow IH^n(X), $$ and this turns out to be injective.

As $X$ is singular, Poincaré duality might fail, i.e. the cup-product $$ H^i(X)\otimes H^{2d-i}(X) \to H^{2d}(X), $$ which is Galois equivariant, may be degenerate. This is the case when $H^i(X)$ (resp. $H^{2d-i}(X)$ ) is not pure of weight $i$ (resp. $2d-i$) for the reason of Galois, and $W_{i-1}H^i(X)$ (resp. $W_{2d-i-1}H^{2d-i}(X)$ ) is contained in the left kernel (resp. right kernel) of the cup-product pairing. I would like to know if this is the only obstruction for Poincaré duality to hold, namely they are exactly the left/right kernel.

Since the intersection pairing $$ IH^i(X)\otimes IH^{2d-i}(X)\to\mathbb Q_l(-d) $$
is perfect (I don't know if the pairing has a geometric definition in char. $p$ --- $D_XIC_X\simeq IC_X$ is the only reason I know), if my question in the beginning has a positive answer, then the cup-product pairing on $Gr^W_*H^*(X)$ will be perfect.

Correction: The argument above for non-degeneracy on $Gr^W_*H^*(X)$ is wrong. Here's a counter-example. Let $X$ be the projective cone of a projective smooth curve of genus $g,$ either over char. 0 or $p.$ Then $H^1(X)=0$ but $H^3(X)$ is of dimension $2g$ and pure of weight 3.

Answer by shenghao for The different types of stacks

2011-06-30T09:46:39Z

Algebraic spaces are non-stacky algebraic (Artin) stacks, and DM-stacks, although stacky, have only finite stabilizer groups (or étale stab. groups; I'm sloppy here) for all points on the "underlying space".

To get to a scheme from an algebraic space, one can either pass to an étale cover or just restrict oneself to an open dense subspace. To get to a scheme from a DM-stack, one can still pass to an étale cover, or, if one prefers finite group actions, pass to a Zariski open, which supports a $G$-bundle with total space an affine scheme, and $G$ is a finite group. The DM-stack is covered by such Zariski opens (though with different groups $G$). But for a general Artin stack there is really a long way to get to a scheme (or alg. space): if schemes are floating on the surface of the ocean, then Artin stacks rest deep in the ocean --- they are "covered" by schemes but the relative dimension are usually positive.

This somehow explains (at least to me) why for algebraic spaces (or more generally, DM-stacks) it suffices to use the étale topology to define and compute cohomology of sheaves, whereas for general Artin stacks, lisse-étale topology is necessary.

Edit: This is actually a comment for 4) in unknowngoogle's answer. It is possible to have a continuously varying family of algebraic groups parameterized by a space (i.e. not iso-trivial, even if one passes to an algebraic stratification of the base space). Therefore, I don't think all algebraic stacks (say over $k$) are locally quotients of $k$-schemes by $k$-algebraic groups: one needs group schemes over a larger base. There exists a stratification of any algebraic stack such that the inertia is flat over each stratum, but one cannot make this flat family a constant family.

Poincaré duality for intersection cohomology

2011-03-07T17:27:11Z

Let $X$ be a projective complex algebraic variety of dimension $d.$ Does it make sense to ask if properties like $$ (x,y)=(-1)^i(y,x) $$ holds, for $x\in IH^i(X,\mathbb Q)$ and $y\in IH^{2d-i}(X)?$ And if it makes sense, is it true?

Edit: For the first question, my concern is that, if the self-duality $D_XIC_X\cong IC_X$ of the intersection complex only ensures that $IH^i(X)$ and $IH^{2d-i}(X)$ are dual, without specifying a particular duality, then it makes no sense to ask if the "product" is graded-commutative. Of course, if it does give a particular isomorphism, then it makes sense.

Edit: As I learned from Gabber, the answer is yes (of course I'd take the responsibility for misunderstanding his comment if it turns out to be ...), and it follows from the symmetric pairing $$ IC_X[-d]\otimes IC_X[-d]\to\mathbb Q_{\ell} $$ normalized so that on the smooth locus, it is the natural identification. I'll be appreciated if anyone can give a reference on this.

self-dual representations

2011-06-12T22:13:03Z

Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of $G$, such that $V\otimes\rho$ is a self-dual irrep. of $G?$ Namely $V\otimes\rho\simeq(V\otimes\rho)^*.$ If not, is there any necessary/sufficient conditions on $V$ for it to be "twisted self-dual"?

If this is always the case, then in particular, if $G$ has no non-trivial linear characters (e.g. $G$ is a simply-connected compact Lie group or a perfect finite group), then every irrep. of $G$ is self-dual.

Thanks.

Irreducibility of a representation of $\Gamma(N)$

2011-06-05T11:14:54Z

Let $Y(N),$ for $N\ge3,$ be the modular curve over $\mathbb C$ with respect to the $\Gamma(N)$-level structure. Let $f:E\to Y(N)$ be the universal elliptic curve. Then $R^1f_*\mathbb Q$ is a semisimple local system on $Y(N).$ Is it irreducible? And how to prove it or where can I find a reference? Thank you.

Answer by shenghao for What is transport of structure in cohomology setting?

2011-05-20T14:57:47Z

I'm not completely sure if this is the answer; it just provides an another way of describing the Galois action.

Let $a:X\to\text{Spec }\mathbb Q$ be the structure map. Then $R^ia_*F,$ being an étale sheaf on $\text{Spec }\mathbb Q,$ corresponds to an $\ell$-adic vector space (its stalk at $\text{Spec }\overline{\mathbb Q}$) with a continuous action of $Gal(\overline{\mathbb Q}/\mathbb Q).$ This vector space is nothing else but $H^i=H^i(X_{\overline{\mathbb Q}},F_{\overline{\mathbb Q}}).$ Think of the sheaf $R^ia_*F$ in terms of its espace étalé, and take fiber products: $$ \begin{bmatrix} H^i & \to & H^i & \to & [R^ia_*F] \end{bmatrix} $$ $$ \begin{bmatrix} \downarrow &&&&& \downarrow &&&&& \downarrow\end{bmatrix} $$ $$ \begin{bmatrix} \text{Spec }\overline{\mathbb Q} & \overset{\to}{\sigma} & \text{Spec }\overline{\mathbb Q} & \to & \text{Spec }\mathbb Q.\end{bmatrix} $$ Since the two maps from $\text{Spec }\overline{\mathbb Q}$ to $\text{Spec }\mathbb Q$ are the same, the two pullbacks are both identified with $H^i,$ and the isomorphism between them in the diagram is how $\sigma$ acts. It seems that the $\overline{\mathbb Q}$-structure of $H^i$ is "transported" by $\sigma.$

Answer by shenghao for Mathematicians who were late learners?-list

2009-11-02T01:06:50Z

Lefschetz didn't move to math until he lost both of his hands in an industrial accident at the age of 23.

What are the automorphism groups of (principally polarized) abelian varieties?

2009-11-01T23:28:52Z

What are the possible automorphism groups of a principally polarized abelian variety $(A,\lambda)$ of dimension $g,$ say an abelian surface ($g=2$) over the complex numbers or algebraic closure of a finite field? The fact that the moduli stack $A_g$ is of finite diagonal (over the integers) implies that the automorphism groups are all finite, but do we know more? Like the size.

When $g=1$ this is given in Silverman I, p.103.

Edit: Let me make the question more specific. Let $(A,\lambda)$ be an $\mathbb F_q$-point of $A_g$ (i.e. an abelian variety $A$ over $\mathbb F_q$ of dimension $g$ and a principal polarization $\lambda$). We want to consider its automorphism group (over $\mathbb F_q$).

Let $\pi:A_{g,N}\to A_g$ be the natural projection, where $A_{g,N}$ is the moduli stack of p.p.a.v. of dimension $g$ with a level $N$ structure (a symplectic isomorphism $H^1(A,Z/N)\to(Z/N)^{2g}$). We always assume $q$ is prime to $N.$ Note that $\pi$ is a $G$-torsor, for $G=GSp(2g,Z/N),$ so it gives a surjective homomorphism $\pi_1(A_g)\to G.$ The sheaf $\pi_*\mathbb Q_l$ on $A_g$ is lisse (even locally constant), corresponding to the representation of $\pi_1(A_g)$ obtained from the regular representation $\mathbb Q_l[G]$ of $G$ and the projection $\pi_1(A_g)\to G.$ For any $\mathbb F_q$-point $x$ of $A_g,$ the local trace $\text{Tr}(Frob_x,(\pi_*\mathbb Q_l)_{\overline{x}})$ is either $|G|$ or 0, depending on $Frob_x\in\pi_1(A_g)$ is mapped to 1 in $G$ or not.

We have isomorphisms $H^i_c(A_{g,N},\mathbb Q_l)=H^i_c(A_g,\pi_*\mathbb Q_l).$ By Lefschetz trace formula, applied to both $\mathbb Q_l$ on $A_{g,N}$ and $\pi_*\mathbb Q_l$ on $A_g,$ we have

$$|A_{g,N}(\mathbb F_q)|=|G|\sum_{x\in S} 1/\#Aut(A_x,\lambda_x),$$

where $S$ is the subset of $[A_g(\mathbb F_q)]$ consisting of points $x$ such that all $N$-torsion points of the abelian variety $A_x$ are rational over $\mathbb F_q$ (i.e. $|A_x[N](\mathbb F_q)|=N^{2g}$), and $(A_x,\lambda_x)$ is the pair corresponding to $x.$ This equation gives some constraints (one for each $N$) that $|Aut(A,\lambda)|$ must satisfy. In particular, when $g=N=2$ and $q=3,$ we have $|A_{2,2}(\mathbb F_3)|=10$ and $|G|=720$ (in this case $G$ is the symmetric group $S_6$), and this becomes a puzzle of solving $$ 1/72 = \sum 1/n_i, $$ and the $n_i$'s satisfy some additional conditions. Any idea on how to solve it? I'm considering the contributions of the two parts in $A_2,$ one for Jacobians of smooth genus 2 curves and one for Jacobians of stable singular ones $E_1\times E_2$. Any suggestion is appreciated.

Edit: Maybe it's easier to solve it over $\mathbb F_5,$ since the (orders of the) automorphism groups of smooth genus 2 curves over finite fields of characteristic 5 is known.

Answer by shenghao for If Spec Z is like a Riemann surface, what's the analogue of integration along a contour?

2009-11-04T19:50:29Z

For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map $$ H^{2d}_c(X,\mathbf{Q}_{\ell})\to\mathbf{Q}_{\ell}(-d), $$ and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.

For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.

So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.

homotopy limits of dg categories

2011-04-27T18:48:49Z

The question is related to the following MO question

http://mathoverflow.net/questions/38993/co-limits-and-fibrations-of-dg-categories

My question is,

how to define the homotopy limit (and colimit) of a system of dg-categories (let's fix a universe and a base ring $k$, and work only with small things...), and
is there an explicit description of the homotopy category of the homotopy limit of dg categories $$ Ho(holim_{i\in I}\mathscr C_i)=? $$ Recall that the homotopy category $Ho(\mathscr C)$ of a dg category $\mathscr C$ is the category with the same objects as $\mathscr C$ and the hom group is the cohomology at degree 0 of the hom complex in $\mathscr C:$ $$ Hom_{Ho(\mathscr C)}(X,Y)=H^0(Hom_{\mathscr C}(X,Y)). $$ One can ask similar questions to "categories" enriched in simplicial sets, which is a slightly more general setting.

I understand (sort of) that there is a model category structure (due to Tabuada) on the category $dg-Cat$ of dg categories such that weak equivalences are what one expects (to be a bit precise, a functor $F:\mathscr C\to\mathscr D$ is a w.e. if $$ Hom_{\mathscr C}(X,Y)\to Hom_{\mathscr D}(FX,FY) $$ is a quasi-isomorphism of complexes, and $Ho(F):Ho(\mathscr C)\to Ho(\mathscr D)$ is essentially surjective). But I don't know how to use this model structure to define homotopy limits.

Maybe one uses cofibrant replacement and the naive $\otimes$-structure on $dg-Cat$ to define a $\otimes^{\mathbb L}$-structure (following Toen) and shows that it is closed, so that one has internal hom $R\mathscr Hom$ on $dg-Cat,$ with which one defines homotopy limits (and colimits) of dg-categories by universal properties. I'm not sure. Both references and direct explanations are appreciated.

Is the weight filtration a topological invariant?

2011-04-20T13:06:28Z

This question is somehow related to (but different from) the following MO question and the one linked from there

http://mathoverflow.net/questions/42744/diffeomorphic-kahler-manifolds-with-different-hodge-numbers

Let $X$ and $X'$ be two complex algebraic varieties that are diffeomorphic to each other. One may or may not assume they are irreducible. Then:

(Weak form) For each pair of integers $(i,n),$ do we always have $$ \dim Gr^W_iH^n(X,\mathbb Q)=\dim Gr^W_iH^n(X',\mathbb Q), $$ where $W$ denotes the weight filtration on the mixed Hodge structures?
(Strong form) Let $f:X\to X'$ be a diffeomorphism. Does $f^*:H^n(X')\to H^n(X)$ respect the $W$-filtrations?

Variants: One can ask similar questions with "diffeomorphic" replaced by "homeomorphic" or "complex analytically isomorphic". For instance, does the weight filtration see the difference between tangential intersections and transverse intersections? One can also add coefficients.

At the first sight, it's not clear to me how to approach the question, since there seems to be no relation between the resolutions of $X$ and $X'.$

Edit: As algori pointed out to me, this is false without the compactness assumption, even for "complex analytic isomorphism". So I will assume $X$ and $X'$ are compact.

cohomology of moduli spaces

2009-10-16T21:59:52Z

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ and $N?$ Like dimensions of the cohomology spaces and the weights. Thanks.

Edit: I'm particularly interested in the weights of these $\ell$-adic cohomology of moduli varieties defined over finite field, or even the precise Frobenius eigenvalues, for the purpose of independence of $\ell$ and automorphy. Therefore I would like to know $H^i$ for all $i,$ in particular the middle cohomology (e.g. $H^3(A_{2,N})$).

"geometric" interpretation of the alternating sum of intersection cohomology groups

2011-03-31T17:35:04Z

Let $X_0$ be a proper variety over a finite field $k.$ For each prime number $\ell\ne p,$ we have the $\ell$-adic intersection cohomology groups $IH^i(X).$ Due to Gabber, the alternating sum of these $$ \sum_{i=0}^{2\dim X_0}(-1)^i[IH^i(X)], $$ regarded as a virtual representation of the Weil group $W(\overline{k}/k)$ of $k$ (i.e. the cyclic group in Gal generated by Frobenius), is independent of $\ell.$ Of course it is also self-dual (up to Tate twist), and (together with Gabber's purity) each summand is independent of $\ell.$

My question is: does this alternating sum have any "geometric" interpretation? Or, to what extent (and in which way) this Euler characteristic depends on the singular locus of $X?$

E.g. when $X$ is in addition smooth, this can be interpreted in terms of the number of $k$-rational points. Being independent of $\ell,$ it's not surprising to expect it to have some "motivic" meaning, e.g. in terms of the geometry. Certainly one could also expect something for the individual $IH^i(X),$ but it seems that the Euler characteristic is always easier to relate to geometry.

Answers/comments in the general setting would be great, and those addressing to Shimura varieties and their Satake compactifications are especially appreciated.

intersection pairing on intersection cohomology

2011-03-16T18:11:48Z

Let $X$ be a projective variety of dimension $d$ over $k=\bar{k},$ with $L$ an ample line bundle on $X$ and $\eta=c_1(L).$ Hard Lefschetz gives an isomorphism (see BBD) $$ \eta^i:IH^{d-i}(X)\to IH^{d+i}(X) $$ with Tate twist ignored, which, together with the intersection pairing between $IH^{d-i}$ and $IH^{d+i},$ gives a non-degenerate bilinear form $$ IH^n(X)\times IH^n(X)\to(\mathbb Q,\mathbb Q_{\ell},\text{ or }\mathbb C...) $$ for each $n.$

Question: Is it $(-1)^n$-symmetric?

This is so when $X$ is non-singular (which follows from the general fact on "cup products"), or when $n=d.$ The question is related to this MO question http://mathoverflow.net/questions/57701/poincare-duality-for-intersection-cohomology. I guess one can probably figured it out by doing some homological algebra on the level of complexes (i.e. before taking hypercohomology groups), and maybe it's written down somewhere.

finite quotients of fundamental groups in positive characteristic

2011-03-16T18:36:06Z

For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental groups.

What about complete smooth curves, or more generally higher dimensional varieties? Are there results or conjectural criteria (or necessary conditions) for finite quotients of their $\pi_1?$ (Definitely, not too much was known around 1990; see Serre's Bourbaki article on this.)

In particular, let $G$ be the automorphism group of the supersingular elliptic curve in char. $p=2$ or $3$ (see http://mathoverflow.net/questions/58455/supersingular-elliptic-curve-in-char-2-or-3 for various descriptions of its structure). Is there (and if yes, how to construct) a projective smooth variety in char. $p$ having $G$ as a quotient of its $\pi_1?$ Certainly there are lots of affine smooth curves with this property (e.g. $\mathbb G_m$), and I wonder if for some of them, the covering is unramified at infinity (so that we win!).

supersingular elliptic curve in char. 2 or 3

2011-03-14T18:29:30Z

Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order 12 (resp. 24) when $p=3$ (resp. 2). Then the $\ell$-adic cohomology $H^1(E,\mathbb Q_{\ell})$ is a 2-dimensional representation of $G.$ Is it irreducible?

Since we know the group structure of $G$ (cf. Silverman's AEC, Appendix A, exercise..), does anyone have a reference for its character table?

Tensor product of simple representations

2011-03-09T21:06:39Z

Let $G$ be a linear algebraic group over some field, and let $V$ and $W$ be two simple rational representations of $G.$ Is $V\otimes W$ semi-simple?

I was trying to convince myself that if $G$ has a faithful semi-simple representation, then $G$ is linearly reductive, and was reduced to the question above. The problem I have in mind is over characteristic 0, but answers addressing char. $p$ is equally appreciated too!

Comment by shenghao

2013-05-06T15:43:55Z

The isomorphism about Tate twists follows from the projection formula $Rf_*(\mathscr F\otimes^Lf^*\mathscr G)\simeq Rf_*(\mathscr F)\otimes^L\mathscr G.$ Then $\mathscr G$ is flat, there's no need to put $"L"$ there.

Comment by shenghao

2013-05-05T17:50:55Z

Isn't $\mathbb Z/n\mathbb Z$ a ring-like object?

Comment by shenghao

2013-02-22T14:13:36Z

Great! Thanks for the practical instruction.

Comment by shenghao

2013-02-22T06:03:22Z

Dear LMN: maybe you want to get a better feeling for general adjoint functors, as in MacLane's book? There're tons of examples there to help. Pullback and pushforward, if defined correctly, should always be "adjoint" to each other in some sense. E.g. given a cont. map f:X -> Y of spaces, one can pullback functions on Y and pushforward measures on X, and the "natural isomorphism" as in adjoint functors becomes an equality of integrals. BTW, the proof seems to be formal: first prove it for presheaves (using univ. property of colimit), then use univ. property of sheafification.

Comment by shenghao

2013-02-18T13:18:42Z

No, I meant that without the regularity assumption, there are examples of noetherian domains of infinite Krull dimension.

Comment by shenghao

2012-09-03T07:51:04Z

B. Conrad's (still unpublished?) book on Ramanujan's conjecture is a (the?) modern exposition on Deligne's 1969 Bourbaki talk. It has all detailed computations. Check Brian's webpage.

Comment by shenghao

2012-09-02T13:53:54Z

The question applies more generally: after knowing the characteristic polynomial of a matrix, why it is important to know its Jordan form? I guess knowing how the space decomposes into generalized eigenspaces will usually be helpful, in various situations. In the case of etale cohomology, as Yuri said, if you know there are many eigenvectors for an eigenvalue, then (modulo Tate conjecture) it will lead to many algebraic cycles, which is usually a difficult thing in alg. geom. This also applies to K3 (a recent result?).

Comment by shenghao

2012-08-06T16:18:01Z

It might be helpful to check LMB and Olsson's article 'Sheaves on Artin stacks', for cotangent complexes on stacks.

Comment by shenghao

2012-08-06T16:07:52Z

I'm not sure if Faltings' work leads to semisimplicity of the $Gal(\bar{\mathbb Q}/\mathbb Q)$-representation, which is a different conjecture.

Comment by shenghao

2012-08-06T16:02:31Z

Also true for K3 surfaces, by Deligne. See Lei Fu's AMS article 'On the semisimplicity conjecture and Galois representations' for more info. For mixed cohomology groups, the weight filtration does not split in general (it would be split if Frobenius were always semisimple), and examples can be found already in dimension 1.

Comment by shenghao

2012-08-05T15:57:28Z

Thanks for the answer, Donu. A somehow related question: what about varieties over $\overline{\mathbb F}_p?$ Namely $X$ is projective (smooth or not) and $L$ is semisimple local system (or perverse sheaf). Without being of geometric origin, they won't have a model over a finite subfield, so that the usual proof breaks.

Comment by shenghao

2012-07-25T18:03:44Z

maybe PD with polarization.

Comment by shenghao

2012-07-21T23:35:57Z

With pleasure: For an abelian variety, $H^1$ determines the others: $H^i$ is the $i$-fold wedge product of $H^1$, so knowing what $H^1$ is gives the full zeta function. For a curve, $H^0$ and $H^2$ are always known, so again $H^1$ determines the zeta function. But for other varieties we are less lucky. I don't know how to tell the cup-product structure (for $X$ a proper variety) from the zeta function either.

Comment by shenghao

2012-07-21T23:20:13Z

The Hodge-de-Rham spectral sequence may not degenerate at E_2, and the Hodge symmetry $h^{pq}=h^{qp}$ may break down, etc.

Comment by shenghao

2012-07-21T10:05:11Z

A related question is to ask for characterization （even conjectural） of Frobenius char. polynomials on cohomology of a fixed degree. For curves and abelian varieties one can use Honda-Tate，and this related question is equivalent to your original one. I believe that very little common feature is known for general schemes （not nece. proper or smooth）. For projective smooth varieties，by weak Lefschetz，the most interesting ones are the middle two， about which again very little is known.

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Does intersection pairing on `$IH^(X)$` agree with cup-product on `$H^(X)$`?