User balazs - MathOverflow

Do mixed Hodge modules form a stack?

2012-10-17T12:22:14Z

The question should be reasonably self-contained: do MHM's form a stack? This is well-known for perverse sheaves, proved already in [BBD(G)]; does it hold for mixed Hodge modules? In other words, if I have a variety $X$ over the complex numbers, an open cover $(U_i)$ and mixed Hodge modules on all the $U_i$ with gluing isomorphisms on overlaps with the obvious cocycle condition, then do I get a unique MHM on $X$? I would like this in the analytic category but a reference for algebraic varities over the complex numbers is likely to satisfy me.

Smooth variety with positive point-count polynomial and odd cohomology

2012-03-30T08:02:06Z

I am looking for an example of a smooth irreducible quasiprojective variety $X$ over ${\mathbb C}$, such that when reduced over finite fields ${\mathbb F_q}$, the number of its points is a polynomial $P(q)$ of $q$ with nonnegative (integer) coefficients, but $X$ has some odd cohomology.

Background: as discussed in an answer to this question, if a variety $X$ is paved by affine spaces, then it only has $(p,p)$ cohomology, and the number of its ${\mathbb F_q}$-points equals $P(q)$, where $P(t)$ is the Poincare polynomial of (compactly supported cohomology of) $X$. Note that the coefficients of $P$ are necessarily non-negative, given by the number of affine cells in the paving of a fixed dimension. In the appendix to this paper, N.Katz proves a kind of converse to this statement: if the number of points of $X$ over a finite field is given by a polynomial $P(q)$ of $q$, then this polynomial determines the so-called $E$-polynomial $E(x,y)$ of $X$ by the formula $E(x,y)=P(xy)$. The $E$-polynomial is a partial Euler characteristic, where we remember the weights of (compactly supported) cohomology but not the degrees.

Of course varieties of the latter type can have odd cohomology; the typical example is $X={\mathbb C}^*$, with point count polynomial $P(q)=q-1$. $X$ of course has odd cohomology. A slighly more complicated example, due to N.Katz, shows that $X$ can also have non-$(p,p)$ cohomology. But in these examples, the polynomial $P$ has some negative coefficients.

Hence the question: can $X$ have positive polynomial count, but still some odd cohomology (which cancels in the $E$-polynomial)? Note that $X$ can't be smooth projective, since then its cohomology would be pure so any odd cohomology would have to show up in the $E$-polynomial.

There may of course be a trivial example which I am missing.

Answer by Balazs for Why is $\mathbb{Q}$-factoriality not local in the étale topology?

2012-03-06T13:30:05Z

A different set of examples, closer to Kollár--Mori, comes from projective geometry over $\mathbb C$. For a projective variety $X$ over $\mathbb C$, $\mathbb Q$-factoriality is a global topological property: it depends on the prime divisors lying on $X$, rather than just on the local analytic type of its singular points, the issue being that local $\mathbb Q$-Weil but not $\mathbb Q$-Cartier divisors may fail to glue to a global non-$\mathbb Q$-Cartier divisor. For example, for a quartic threefold $X=X_4\subset {\mathbb P}^4$, a small number of ordinary double points do not effect the factoriality, even though the threefold ordinary double point is manifestly non-factorial. See this paper with a nice introduction which has a fuller discussion, explicit examples and further references.

Answer by Balazs for How to find central vertex in a graph?

2012-02-24T09:50:27Z

I am not sure your definition of centrality is the most useful one; in what's called "network science", the study of large-scale graphs, I think some measure of centrality which takes an average rather than a maximum will be more useful. Think for example about a graph which has a vertex $v$, edges $v- v_i$ to lots of vertices $v_1, ...,v_N$, and also contains a chain of length say three, $v - w_1 - w_2 - w_3$. According to your definition, $w_1$ is the center of this graph, but I would say it's $v$. The average distance leads to what's called "closeness centrality". There is also "betweenness centrality", based on the number of shortest paths in the graph that pass through the node divided by the total number of shortest paths; this is trying to measure how useful $v$ is to the rest of the graph. Googling these two phrases will bring up lots of algorithms and further references. Here is a random recent one that seems to have a useful intro: Ranking of Closeness Centrality for Large-Scale Social Networks.

Answer by Balazs for nef Cone of a Toric Variety

2012-01-31T11:22:34Z

What you are interested in is toric Mori theory. This was first written down by Miles Reid back in the 80s (Decomposition of toric morphisms, incidentally to the best of my knowledge the first paper which wrote out the main steps of Mori's programme.) If you google "toric Mori theory", there are plenty of other hits; I checked Wisniewski's nice Toric Mori theory and Fano manifolds which may well do for you.

Answer by Balazs for How "frequent" are smooth projective varieties with trivial canonical bundle?

2011-11-24T16:55:02Z

If $X$ is a complex projective manifold with trivial canonical bundle, then by a theorem of Bogomolov, there is a finite unramified cover $\tilde X$ of $X$ which decomposes into a product $A\times X_1\times\ldots\times X_n\times Y$. Here $A$ is an abelian variety; $X_i$ are irreducible holomorphic symplectic manifolds (simply-connected, with a unique non-vanishing holomorphic 2-form) and $Y$ is a "strict Calabi-Yau" (simply-connected, with no holomorphic 2-form but a non-vanishing holomorphic top-form). Quite how frequent they are depends on your definition of frequent; for example, it's not known whether there are finitely many or infinitely many deformation types of strict Calabi-Yaus in three dimensions.

In memoriam Torsten Ekedahl

2011-11-24T16:34:10Z

This is not a question, just very, very, very sad news. Our community has lost one of its most active members. I am posting a letter I received recently.

From Carel Faber and Sandra Di Rocco:

November 23, 2011

Dear colleagues from the Algebraic Geometry community,

Unfortunately, we have some very sad news:

Torsten Ekedahl has passed away.

Torsten collapsed this morning at the Stockholm University Mathematics Department and died shortly thereafter. Attempts of colleagues and ambulance personnel to resuscitate him were unfortunately unsuccessful.

The SU Mathematics Department and the mathematical communities in Stockholm and Sweden have suffered a great loss. All who knew Torsten personally or through his work know that he was an exceptional mathematician, in many different ways. Torsten's main interest was Algebraic Geometry, but his research and knowledge went far beyond our field.

Torsten Ekedahl will be sorely missed by many.

Carel Faber and Sandra Di Rocco

Answer by Balazs for To what extent does Poincare duality hold on moduli stacks?

2011-11-24T16:03:16Z

I know of one very limited case, which will almost certainly not satisfy you, where the "virtual cohomology" mentioned by Angelo exists, with a version of Poincare duality.

Presumably the generality one would like to address this question is for a moduli space (or stack) $M$ carrying a perfect obstruction theory (let me ignore the stackyness everywhere below). Let me restrict right away to virtual dimension 0, and in fact further to the case when the obstruction theory is symmetric (in the sense of Behrend, Behrend-Fantechi). Note that this already excludes Kontsevich moduli spaces $\bar M_{g,n}(X,\beta)$, even for $X$ a Calabi--Yau threefold, but we are still OK with DT- or other sheaf-theoretic moduli spaces.

In this case, we of course have Behrend's result that locally $M$ is cut out in a smooth ambient space $N$ by the zeros of an almost closed one-form. Now specialize further, and assume that in fact $M$ is cut out by the zeros of a closed one-form. Finally, specialize to the case when in fact globally $M=Z(df)\subset N$; here $N$ is a smooth variety, $f\colon N\to {\mathbb C}$ is a smooth function, and $M$ is cut out by the zeros of the exact one-form $df$. Note that there are still reasonably interesting examples, such as the Hilbert scheme of points on ${\mathbb C}^3$ or related toric or quivery examples.

In this case, it seems that the correct "virtual cohomology" to consider is $H^*(M,\phi_f)$, the cohomology of $M$ with coefficients in the perverse $\mathbb Q$-sheaf $\phi_f\in{\rm Perv}_{\mathbb Q}(M)$ of vanishing cycles of $f$ (appropriately shifted). This is called critical cohomology by Kontsevich-Soibelman. For example, this cohomology has the correct Euler characteristic, namely the integral of the Behrend function on $M$. It also carries a mixed Hodge structure, and the Euler characteristic of the weight filtration (which will differ from the degree filtration because the Hodge structure is not pure) gives what seems to be the right "quantization" (or "refinement") of the numerical invariant.

Now the point is that by standard theory, $D_M\phi_f\cong\phi_f$, where $D_M$ is the Verdier duality functor on $M$. This will give you a kind of Poincare duality $$H^n(M,\phi_f)\cong H^{-n}_c(M,\phi_f)$$ albeit between ordinary and compact-support cohomology, since $M$ is of course almost always noncompact. But at least (using the appropriate shifts) it is elegantly symmetric around zero, which is what we would expect in the virtual dimension 0 case.

I don't really know how well this generalizes; almost nothing is known (to me) about gluing, functoriality, etc.

Answer by Balazs for Which rank 1 bundle over $\mathbb{P}^1$ is this exceptional divisor?

2011-07-20T09:13:33Z

Here is another approach, not even referring to exact sequences. Let's do this torically - using Fulton's fan language. The fan of ${\mathbb P}^3$ is given by the cones spanned by the vectors $e_1,e_2,e_3,-e_1-e_2-e_3$ in the vector space spanned by them. The first blowup corresponds to subdividing the fan by inserting the vector $e_1+e_2$, whereas the second blowup corresponds to subdividing the fan further by inserting the vector $e_1+e_3$ - please draw a picture! Staring at the picture will tell you that this last vector will give an edge of the four cones $(e_1, e_1+e_2, e_1+e_3)$, $(e_1+e_2, e_3, e_1+e_3)$, $(e_3, -e_1-e_2-e_3, e_1+e_3)$, $(-e_1-e_2-e_3, e_1, e_1+e_3)$. The geometry of the divisor corresponding to the vector $v=e_1+e_3$, the exceptional divisor $E_2$ you are looking for, is given by projecting these four cones onto the quotient of the vector space by the span of $v$; you can arrange this simply by setting $e_1+e_3=0$ in the above cones, getting the cones $(e_1, e_1+e_2)$, $(e_1+e_2, -e_1)$, $(-e_1, -e_2)$ and $(-e_2, e_1)$ in two dimensions. This is the standard fan of ${\mathbb F}_1$.

Answer by Balazs for The first eigenvalue of a graph - what does it reflect?

2011-05-02T12:10:59Z

I am not at all an expert on this, but I believe that the largest eigenvalue plays a role in several graph processes that people working in "network science" are interested in. A google search for "largest eigenvalue network" brought up this paper whose abstract begins "The largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes (e.g., synchronization of oscillators, percolation on directed networks, linear stability of equilibria of network coupled systems, etc.)"; see references in the first paragraph of the paper to substantiate this claim.

Answer by Balazs for Mathematical "urban legends"

2011-04-14T21:59:07Z

Oral maths exam for engineers, 1960s, Budapest. To prove: there are infinitely many prime numbers. Candidate shuffles in his chair, has no idea really. Professor tries to help: let's recall the definition of prime numbers. Let's talk about some examples. Etc etc. After 15 excruciating minutes, candidate summarizes progress thus: Professor, I now understand that all odd numbers are prime. But I still don't see why are there infinitely many...

Answer by Balazs for Is there a "motivic Gromov-Witten invariant"?

2011-04-05T17:45:55Z

Let me give an answer from a slightly different point of view.

Let $M_k$ be a moduli space as in your question; say it's a (compact) moduli space of sheaves on some (compact) Calabi-Yau threefold. In general, $M_k$ is going to be very singular. However, it carries a so-called perfect deformation-obstruction theory of dimension zero. This gives a virtual fundamental class on $M_k$, and the technical definition of the numerical invariant $e(M_k)$ is that it is the degree of this virtual fundamental class.

In the case of sheaves on CY3s, the deformation-obstruction theory has a duality property: it is a symmetric obstruction theory. In this case, according to a result of Kai Behrend, $e(M_k)$ can also be expressed as an Euler characteristic, albeit a weighted one:
$e(M_k)=\chi(M_k, \nu_{M_k})$, where $\nu_{M_k}$ is the Behrend function of the singular space $M_k$. In other words, one computes an Euler characteristic, but weighted with a numerical measure of how bad the singularities are.

On can hope that this Euler characteristic definition can now be turned into something motivic. What one needs is a way to attach a motivic weight to points of $M_k$. In some specific moduli problems, such as for Hilbert schemes of points where at least locally the moduli space can be expressed as a critical locus of a function on a smooth variety, this can be done using the tool of the motivic vanishing cycle; indeed, this is what our work does in the paper you cite. The general theory of how one attaches motivic weights is discussed in a (partially conjectural) paper of Kontsevich and Soibelman.

The issue with Gromov-Witten theory on a CY3 is that the deformation-obstruction theory in that case, while it is of dimension zero, is not fully symmetric. It is symmetric on the open part corresponding to stable maps which are immersions from a smooth curve, but (as an expert assures me) not on the whole moduli space.

Answer by Balazs for Question on Kähler/ample cone, cone of curves....

2011-02-21T07:57:59Z

There is an interesting special case when the answer to your Question 4 is positive. Assume that $X$ is a smooth fourfold, and $Y$ an ample anticanonical hypersurface (in particular then $X$ is Fano and $Y$ is Calabi-Yau). In this case, the Kahler cones of $X$ and $Y$ coincide. This is proved by Kollar in an appendix to Borcea, Homogeneous vector bundles and families of Calabi-Yau threefolds II, in: Proc. Symp. Pure Math. 52. Part II.

String theory "computation" for math undergrad audience

2010-11-30T09:06:35Z

I am giving a talk on String theory to a math undergraduate audience. I am looking for a nice and suprising mathematical computation, maybe just a surprising series expansion, which is motivated by string theory and which can be motivated and explained relatively easily. Examples of what I have in mind are the results in Dijkgraaf's "Mirror symmetry and the elliptic curve", or the "genus expansion" of the MacMahon function (aka DT/GW for affine three-space), but I am not sure I can fit either into the time I have. Any thoughts?

Answer by Balazs for How do I compute the compact cohomology of a hypersurface?

2010-11-27T09:08:25Z

The classic reference is Danilov-Khovanskii's "Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers". There is subsequent work by Cox, Batyrev, Malvyutov, etc. but they are mainly concerned with more general toric ambient spaces; if you want a hypersurface in the torus then this original paper should have all you need.

Answer by Balazs for Non-Kahler "Calabi-Yau"?

2010-11-18T22:22:53Z

There is a reasonably extensive literature on non-Kahler Calabi-Yau threefolds. They are of interest in string theory; see for example http://xxx.lanl.gov/abs/hep-th/0301161, as well as http://xxx.lanl.gov/abs/0809.4748 for an analogue of Calabi's conjecture in this context.

Answer by Balazs for The diameter of the Erdös component of the collaboration graph

2010-11-10T23:15:27Z

There is a very large literature on this, written by people doing "network science". One of the names you might want to look up is that of Mark Newman, see for example his papers The structure of scientific collaboration networks or Who is the best connected scientist? A study of scientific coauthorship networks as well as Clustering and preferential attachment in growing networks. Another important player in this field is Albert-Laszlo Barabasi, whose group had a model (for the web graph) which is somewhat similar to what you suggest, Emergence of scaling in random networks. Some of this work is reviewed nicely in Statistical Mechanics of Complex Networks. The model by Barabasi et al is studied in mathematical terms by Bollobas-Riordan-Spencer-Tusnady's The degree sequence of a scale-free random graph process.

To answer your specific question on the diameter, I would expect it to be quite small ("six degress of separation"); whether it decreases in time must depend on the relative rate of birth of new vertices versus new edges (collaborations). If you take a finite graph and add completely random links, then already a few links lead to a diameter which is logarithmic in the size of the network. I think this remains the case also if you weigh your edge creation mechanism by the distance of the vertices; I once played with a model where the only new edges were created between 2-step neighbours, which still lead to small diameter.

Answer by Balazs for Algebraic geometry examples

2010-08-02T20:02:52Z

One of my favourite sets of examples, stolen from Miles Reid, is the determination of rings $R=\oplus_n H^0(X, nD)$ for ample divisors $D$ on projective varieties $X$. A nice sequence, where a lot of the general features of the theory already show up, is to take $X=E$ an elliptic curve, and $D=nP$ for $P$ a point on $E$.

$n=1$: generators in degrees $1,2,3$, with a relation in degree 6 (by Riemann-Roch), leading to $E\subset P^2[1,2,3]$ (weighted projective space) a sextic hypersurface given by Weierstrass equation $z^2=y^3 + ax^3 y + bx^6$.

$n=2$: get $E\subset P^2[1,1,2]$, a double cover of $P^1$; $P\in E$ is one of the ramification points.

$n=3$: get $E\subset P^2$, a general cubic; $P\in E$ is an inflection point on the image.

$n=4$: get $E\subset P^3$, a general complete intersection of bidegree $(2,2)$.

$n=5$: get $E\subset P^4$, a non-complete intersection variety, equations are the $4\times 4$ Pfaffians of a general $5\times 5$ skew-symmetric matrix of linear forms, equivalently a linear section of $Gr(2,5)$ in its Plucker embedding.

...

Answer by Balazs for Interesting applications (in pure mathematics) of first-year calculus

2010-08-10T17:04:51Z

Following on from the Galois theory example of Johannes, one straightforward way to produce an explicit polynomial with non-soluble Galois group over ${\mathbb Q}$ is to use an irreducible quintic with exactly three real roots, which necessarily has Galois group $S_5$. To check that an explicit polynomial (such as $x^5-4x+2$ if I am not mistaken, I am typing from memory) has this latter property reduces to standard calculus arguments such as "differentiate, find turning points, estimate values, use intermediate value theorem". I always find this calculus interlude at the end of half a semester of algebra quite amusing.

Answer by Balazs for How much can small modifications change the nef cone?

2010-07-19T21:08:20Z

In the world of Calabi-Yau (as opposed to Fano) varieties, one does not expect miracles of Mori dream space type. A specific example which gives a positive answer to your first question is described in the paper http://xxx.lanl.gov/abs/math/0102055 of Michael Fryers: a Calabi-Yau threefold (a degenerate quintic) which has some small resolutions having finite polyhedral nef cone, and some having an infinitely generated cone, which is locally rational polyhedral away from a single point on the boundary. The geometry, related to the Horrocks-Mumford bundle and abelian surfaces in projective four-space, is very beautiful.

Answer by Balazs for Do you read the masters?

2010-06-15T17:30:54Z

Bolyai: Appendix, the first account of non-Euclidean geometry by one of its inventors. If nothing else, the beauty, clarity and brevity of exposition alone make this a must-read.

Answer by Balazs for What does the ample cone look like?

2010-06-07T12:38:03Z

For K3 surfaces, there is an interesting dichotomy: the cone of curves, and so dually the (closed) cone of ample divisors, is either (locally) polyhedral, or completely circular (has no polyhedral part at all). This was proved in Sándor Kovács's thesis (see "The cone of curves of a K3 surface", Math. Ann. 300). As far as I know, it still remains an amusing open problem whether there exist projective varieties whose cone of ample divisors is partially circular and partially polyhedral, although one expects that there should be plenty of such examples.

Answer by Balazs for Papers that debunk common myths in the history of mathematics

2010-06-01T21:16:40Z

David Fowler's book "The Mathematics Of Plato's Academy: A New Reconstruction" sets out to deconstruct the myth that "the early Pythagoreans based their mathematics on commensurable magnitudes, but their discovery of the phenomenon of incommensurability (the irrationality of the square root of 2) showed that this was inadequate; this provoked problems in the foundation of mathematics that were not resolved before the discovery of the proportion theory that we find in Book V of Euclid's Elements". The arguments and conclusions are complex and interesting; here is a review with some of the main points: http://www.maa.org/reviews/mpa.html.

Answer by Balazs for Quotients of Abelian Varieties by Finite Groups

2010-05-15T12:16:01Z

There is a classification of finite groups acting freely on abelian threefolds, leading to quotients which are Calabi-Yau threefolds, in Oguiso, Sakurai, Calabi-Yau threefolds of quotient type, arXiv:math/9909175.

Comment by Balazs

2012-09-24T11:47:49Z

Picard lattices of families of K3 surfaces by Belcastro, xxx.lanl.gov/abs/math/9809008, may be of interest.

Comment by Balazs

2012-07-09T21:30:21Z

Do you know Simpson's The construction problem in Kähler geometry? It does not seem to be on the arXiv, here is a link: citeseerx.ist.psu.edu/viewdoc/…. It's basically a great list of open problems, with a vast bibliography - not brand new, but the best I know.

Comment by Balazs

2012-03-30T20:05:21Z

The $E$-poly involves a signed summation in the cohomology degree - so there can be cancellation. There is some non-$(p,p)$ stuff that cancels out.

Comment by Balazs

2012-03-30T09:47:25Z

Thanks, that was fast...

Comment by Balazs

2011-11-24T17:06:30Z

Yes indeed, not everybody reads meta, and that includes the site's search function; I of course did a simple search to see whether someone had already posted this, and it didn't find the meta thread.

Comment by Balazs

2011-05-25T06:27:37Z

I did say 1960s (-:

Comment by Balazs

2011-03-12T19:00:37Z

Milne's course on Abelian varieties gives the classical proof due to Weil. Another proof (using GIT) is in Mukai's book on Invariants and moduli.

Comment by Balazs

2011-02-20T20:42:11Z

Copied from a bio of Johann Bernoulli: "Johann Bernoulli had already solved the problem of the catenary which had been posed by his brother in 1691. [...] At this stage Johann and Jacob were learning much from each other in a reasonably friendly rivalry which, a few years later, would descend into open hostility. For example they worked together on caustic curves during 1692-93 although they did not publish the work jointly. Even at this stage the rivalry was too severe to allow joint publications and they would never publish joint work at any time despite working on similar topics."

Comment by Balazs

2011-02-20T20:37:42Z

Whitehead-Russell

Comment by Balazs

2010-11-29T22:31:25Z

Let me second J.M.'s request, can you please elaborate?

Comment by Balazs

2010-09-20T14:28:26Z

This kind of statement is folklore knowledge in some epsilon neighbourhood of Miles Reid; I don't know a reference for this particular statement. But in one direction, a linear section of $Gr(2,5)$ is clearly an elliptic curve of degree $5$. In the other direction, by Riemann-Roch you get an embedding of $E$ into $P^4$, which is projectively Gorenstein; by Eisenbud-Buchsbaum, its equations are the Pfaffians of a skew matrix. Googling, I also found Tom Fisher: Pfaffian representations of elliptic normal curves, in Trans AMS 263 (2010), which does a lot more of this sort of thing.

Comment by Balazs

2010-09-17T20:59:04Z

Yes, indeed! Thanks.

Comment by Balazs

2010-06-16T12:57:29Z

Fernando Rodriguez Villegas has a neat little book "Experimental number theory", which has several sections on Galois groups. I am away from home so I can't check right now whether there is anything in there that's relevant, but there may well be.

Comment by Balazs

2010-06-11T14:50:51Z

As far as I know, there is still no a priori argument known which would prove that a given projective simply-connected Calabi-Yau threefold X contains a rational curve (let alone a balanced one).

Comment by Balazs

2010-06-07T15:27:10Z

Whenever the ample cone is finite rational polyhedral, adding all the primitive integer generators of the cone gives you an ample class which is invariant under automorphisms of your variety. Thus the whole automorphism group embeds in a projective linear group, so e.g. cannot be infinite discrete.