User daniel pomerleano - MathOverflow

Answer by Daniel Pomerleano for What are the general techniques for proving a variety is not toric?

2013-01-17T17:19:18Z

Nobody has mentioned that you can just classify the complete smooth toric surfaces using some rather simple combinatorics (a fun exercise or chapter 10 of the book by Cox, Little and Schenck). The upshot is that the only complete smooth toric surfaces are obtained from toric blowups of $P^2$, $P^1 \times P^1$, or the Hirzebruch surfaces. Now we are just left to observe that the ones in the original question are not on the list. Of course, I understand people are interested in the more general techniques.

Answer by Daniel Pomerleano for What information is required for SYZ mirror symmetry?

2013-01-17T09:53:20Z

This is more of a comment than an answer, because I think this is a complicated story and it may just be better to give the relevant references. In short, this has been solved partially by (Kontsevich and Soibelman) in

http://arxiv.org/abs/math/0406564

and then completely by (Gross and Siebert) when the singular locus of the fibration has real codimension 2. These works are mostly concerned with the problem of how we can recover the mirror given a singular affine structure with mild singularities. Gross has a big book on his website which surveys this body of work. He also recently released a shorter 65 page survey

http://arxiv.org/pdf/1212.4220.pdf

which also touches on this topic extensively, though I haven't had a chance to look at it closely yet. It seems important to note that many Lagrangian fibrations in nature have codimension 1 singularities. Of course, nobody knows how to write down (special) Lagrangian fibrations with the desired properties on Calabi-Yau threefolds. Gross and Siebert seem to work around these hard analytical issues using toric degeneration and staying in the realm of algebraic geometry.

One could also consult work by Auroux e.g. http://arxiv.org/abs/0706.3207 for concrete examples where one works with actual Lagrangian fibrations and shows how wall-crossing is natural from the point of view of symplectic geometry. Note that Auroux is concerned with a slightly different setup than the OP (that of mirror symmetry in the complement of an anticanonical divisor). I think consulting those sources would be more informative than any short answer.

Answer by Daniel Pomerleano for Questions on how SYZ conjectures is deduced from HMS conjeture.

2013-01-06T04:21:05Z

For your first question, you can see this answer

http://mathoverflow.net/questions/111694/what-is-geometric-intuition-of-special-lagrangian-manifolds

for some idea of how the special condition maybe natural from the point of view of homological mirror symmetry.

For your second question, if we grant this, then the local system is a natural addition, because we can naively expect that $HF((L,c),(L,c))$ is also $H^*(T^3)$. Meanwhile as you have said we expect that there is only a three dimensional family of special Lagrangians, so we need more objects to correspond to the six dimensional family of skyscraper sheaves. It is therefore natural to allow the $U(1)$ flat connections on $L$ and expect that they also correspond to points in our mirror.

For your third question, the answer is that line bundles on the mirror should correspond to sections of the Lagrangian torus fibration. Normally, we fix a section to begin with and just declare that that goes to the structure sheaf $O_Y$. One motivation for this idea is a similar naive reasoning with Exts as the one you mention in the question. Namely $Ext(O_Y, O_y)$ should be isomorphic to $\mathbb{C}$, so we expect that our Lagrangian hits each fiber once. A nice case to consider is that of the elliptic curve. If you examine the functor constructed in Polishchuk and Zaslow's paper on the elliptic curve, you will see that this is in fact how mirror symmetry works in this case, namely points will correspond to local systems over (0,1) curves and line bundles will correspond to (1,n) curves.

family of gerbes over smooth and proper algebraic varieties

2012-11-12T16:13:00Z

Let $X$ be a smooth and proper variety over $\mathbb{C}$. Let $F$ be an $\mathbb{A}^1$ family of $\mathbb{G}_m$ gerbes over $X$. Suppose the fibers over every point away from 0 in $\mathbb{A}^1$ are all isomorphic. Must the fiber over zero be isomorphic to the general fiber $F_t$ ? This is certainly true for line bundles. I could think of stronger results that might be true, but my general ignorance of the subject leads me to pose the question I am interested in. Sorry also if this question belongs on math.stackexchange.

Answer by Daniel Pomerleano for What is geometric intuition of special Lagrangian manifolds?

2012-11-09T13:50:53Z

Since nobody else answered this question, I'll give it a try from the point of view of mirror symmetry. This is probably a very backwards way of telling the story and there are maybe better answers from, say, the point of view of calibrated geometry. A Bridgeland stability condition on a triangulated category $D$ consists of a group homomorphism:

$$ Z : K(D) → \mathbb{C} $$

and a "slicing" $P$ of $D$ such that if $E \neq 0$ in $P(\phi)$ then Z(E) = $m(E)e^{iπφ}$ for some $ m(E) ∈ R>0 $. See Definition 5.1. of the paper which began this subject: http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf

Just as for ordinary stability conditions in geometric invariant theory, there are notions of stability and semi-stability which give us a notion of moduli spaces of objects. There is a conjecture that the derived Fukaya category of the symplectic manifold X carries a Bridgeland stability condition such that the semi-stable objects are Floer theoretically unobstructed special Lagrangians.

Edit: Another important theorem that I totally forgot to mention is that which is due to Thomas and Yau http://arxiv.org/pdf/math/0104197.pdf and which says that there can be at most one special Lagrangian in a Hamiltonian deformation class. We can therefore say that, roughly speaking, the special Lagrangian condition picks out a nice geometric representative in a semi-stable isomorphism class in the Fukaya category. Unfortunately, this is probably only a partial truth since there will probably be semi-stable objects which do not correspond to smooth Lagrangians, but singular ones.

More familiar description of wonderful compactification of SL_n/S(GL_2 \times GL_n-2)

2012-06-23T03:34:40Z

I am trying to learn a bit about spherical geometry and wonderful compactifications. Please correct any misconceptions. If I've understood http://www.springerlink.com/content/x62342v721707828/ properly, wonderful compactifications of rank one homogeneous spaces have more familiar descriptions. For example the wonderful compactification of $SO_n/SO_{n-1}$ is a quadric. I was wondering if the same might be true for some rank 2 homogeneous spaces such as:

$SL_n/S(GL_2 \times GL_{n-2})$, where n is at least 4,

Does the wonderful compactification of this variety show up outside of wonderful geometry at least for low n? If not is there a relatively simple algorithm to at least compute its cohomology?

Center of universal enveloping algebra of nilpotent lie algebra

2012-04-03T20:55:06Z

Let g be a finite dimensional nilpotent lie algebra over a field k of characteristic zero. Let U(g) be the universal enveloping algebra and Z(g) be its center. Denote by Z_1(g) the augmentation ideal of Z(g). If g is not abelian, is the extension of this ideal of infinite index in U(g)? In other words, is the quotient U(g)/(Z_1(g)) infinite dimensional over k?

Answer by Daniel Pomerleano for Noncommutative Fukaya category?

2012-03-10T17:27:26Z

Since no one else has tried to answer, I'll take a shot. It seems to me that there are threads of ideas in this story that in the very distant future might be woven together to give a possible answer.

To begin, we should note that there seems to be a general idea, discussed in this mathoverflow question, http://mathoverflow.net/questions/37420/deformation-quantization-and-quantum-cohomology-or-fukaya-category-are-they. That one could define the Fukaya category as modules over a deformation quantization of $C^{\infty}(X)$ corresponding to the symplectic form $\omega$.

The basic idea is that in two naive respects this category of modules behaves a lot like the Fukaya category. Firstly, the Hochschild cohomology of the deformation quantization is almost by definition the Poisson cohomology of the symplectic form $\omega$, which in turn is known to be isomorphic to $H^*(X)((t))$. As an equation:

$$HH^*(A_\omega,A_\omega) \cong H^*(X)((t)) $$

Second, one can define a reasonable notion of modules with support on a Lagrangian submanifold and for any Lagrangian L, produce canonical holonomic modules supported there. One can compute that $$Ext(M_L,M_L) \cong H^*(L)((t))$$ There is some hope that one can put in the instanton corrections in a formal algebraic way and a fair amount of work has been done in this direction.

This story works best so far for the Fukaya category of $T^*X$ where the deformation quantization is roughly the algebra of differential operators. This is related to more work than I could competently summarize. I'll just mention, work of Nadler and Zaslow, Tsygan and Tamarkin. This approach is used by Kapustin and Witten to incorporate co-isotropic branes into the Fukaya category in their famous study of the Geometric Langlands. There, they are after some enlargement of Nadler's infinitesimal Fukaya category of $T^*(X)$. Note however that this not the same Fukaya category(the wrapped Fukaya category) that one studies in the context of mirror symmetry, but perhaps things will work better in the compact case if that is ever put on firm ground.

This was all a prelude to say that deformation quantization places you firmly in the land of non-commutative geometry anyways. Things like differential operators for non-commutative rings can make sense http://www.springerlink.com/content/r0rqguawu1960qxy/. I've never really looked at Van Den Bergh's work, but perhaps the passage from the sheaf of algebraic functions to the sheaf $C^\infty(X)$ is another stumbling point. One of Maxim's Kontsevich's ideas (see his Lefschetz lecture notes http://www.ihes.fr/~maxim/TEXTS/Kontsevich-Lefschetz-Notes.pdf) is that for any saturated dg-algebra there should maybe exist some nuclear algebra which bears the same formal relationship as the algebra of algebraic functions and smooth functions.

Deducing properness from H^i(X,F) finitely generated over \Gamma(O_X)

2012-03-02T03:46:49Z

Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is there an example where the induced morphism $X \to$ Spec $\Gamma(O_X)$ is not proper?

As the contributer a-fortiori notes in the comments to this question http://mathoverflow.net/questions/89473/is-hix-f-finitely-generated-over-gammao-x-if-f-is-coherent, there is no such example if all the groups $H^i(X,\mathcal{F})$ are known to be finitely generated over $k$. Not being strong in algebraic geometry, I can't off-hand tell whether his argument can be generalized.

Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?

2012-02-25T10:02:37Z

Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\mathcal{X}$. Is $H^i(X,\mathcal{F})$ finitely generated over $\Gamma(O_X)$ if $\mathcal{F}$ is coherent ? This statement is simple enough that I probably would have heard it if it were true.

If it makes a difference, the statement that I really would like to understand is whether $\Gamma(O_X) \to Ext^i(\mathcal{G},\mathcal{G})$ is module finite for any coherent sheaf.

Most of the schemes that I am "friendly with" are either projective or affine. The statements are correct in those particular cases, so probably I just need to learn more examples... Thanks!

Comparison between Hamiltonian Floer cohomology and Lagrangian Floer cohomology of the diagonal

2010-10-23T21:19:51Z

Let X be a compact symplectic manifold with a form $\omega$. And $X \times X$ is equipped with the symplectic form $(\omega,-\omega)$. The diagonal $\Delta:X \mapsto X \times X$ is a Lagrangian submanifold. So, in this question, http://mathoverflow.net/questions/11081/hochschild-cohomology-of-fukaya-categories-and-quantum-cohomology. Tim Perutz says "PSS is a canonical ring isomorphism from QH∗(X) to the Hamiltonian Floer cohomology of X, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal." I have no doubt that this second assertion is straightforward, since I have consulted a couple of references and no one spells this out. But, I don't quite see it. I believe what I am missing is the relationship between holomorphic strips in $X\times X$ and holomorphic cylinders in X. (Edit: I would also like to understand the comparison of the product structures too)

Edit: here is a rough geometric idea which might have something to do with the truth. I want to assume that my Hamiltonian is time independent and that all orbits are actually fixed points. Given a map of a strip into $X\times X$ the two projections give us two strips into X. The idea is to glue the strips together to form a cylinder which is a map into X. Of course, this doesn't take into account issues of compactifications and so on... Anyways, if someone would be happy to spell it out I would appreciate it.

Answer by Daniel Pomerleano for Quantization and noncommutative deformations

2011-08-10T19:00:39Z

Why do deformations always start with a Poisson manifold? This maybe slightly imprecise since I am transporting the story from a knowledge of the deformation theory of algebraic varieties and any corrections are welcome. Here is a small piece of the puzzle that is interesting to me as someone interested in deformation theory. Well there is a general fact in algebra that deformations of a ring or algebra R are governed by the Hochschild cohomology, $HH^*(R)$. That is to say that deformations to first order correspond exactly to classes in $HH^2(R)$ Hochschild cohomology, and deformations to all orders can be understood roughly as placing additional conditions on that class that the deformation extend to all orders, that is to say over $\mathbb{R}[[t]]$. Here the ring in question is $C^\infty(M)$, but it could be the coordinate ring of an algebraic variety as well. Technically speaking we need to think of $C^\infty(M)$ as a topological algebra but let's supress that. Anyways, what's true is that for a $C^\infty(M)$, $HH^2(C^\infty(M)) \cong \Gamma(\wedge^2(T_X))$, that is exactly the bivector fields. The additional condition is exactly that the bivector field be Poisson. Thus Poisson structures are exactly what gives rise to deformations. There is a relation on Poisson structures called gauge equivalence which determines when two deformations are deemed to coincide.

It might be worth noting that this situation is specific to smooth manifolds. In the case of a smooth algebraic variety the situation can be more complicated depending upon what you are trying to deform. There one can as before deform the (sheaf of) functions by algebraic bivector fields, but one can also deform the complex structure of the variety as well. In the algebraic case, we care also about modules over our variety, and there is a third class of deformations of the category of modules, called the gerby deformations. So if you are bored with Poisson structures you can have a look at those!

A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

2011-07-12T17:25:00Z

The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type.

There is an equivalence of Gerstenhaber algebras

$HH^*(C_*(\Omega X,\mathbb{Q}), C_*(\Omega X,\mathbb{Q}) \cong HH^*(C^*(X,\mathbb{Q}),C^*(X,\mathbb{Q})$

On the left hand side we have Pontryagin product on the based loop space and on the right hand side rational cochains. $HH^*$ denotes Hochschild cohomology.

I have never seen anyone speak to the following enhanced statement, which makes me wonder if there is a counterexample or if I am simply missing some literature.

$HCH^*(C_*(\Omega X), C_*(\Omega X) \cong HCH^*(C^*(X),C^*(X))$

The question is: Is this statement true, false or unknown?

Here we are looking at Hochschild cochains in the homotopy category of $B(\infty)$ algebras. For the background police, a $B(\infty)$ algebra is a type of dg-Gerstenhaber structure, that naturally gives rise to a Gerstenhaber structure by passing to homology. For more info, see the paper of Keller mentioned below.

It is possible to prove this theorem when $C^*(X)$ is equivalent to a graded simply connected Koszul algebra( i.e. X is both formal and coformal). I believe this is due to Keller in a paper called the "Derived Invariance of Higher Structures of the Hochschild complex".

Matrix factorization categories beyond the isolated singularity case

2010-08-26T23:59:46Z

In his really nice thesis, Tobias Dyckerhoff proved the following theorems about matrix factorizations(of possibly infinite rank) over a regular local k-algebra R with a function w and residue field k such that the Tyurina algebra, T= $R/(w,dw)$ is finite dimensional. This last condition says that w has an isolated singularity. For further reference, let S denote the ring R/(w).

1) The homotopy category of matrix factorizations has a compact generator as a triangulated category, which he denotes as $k^{stab}$.

2) As a consequence of 1), he derives that there is a natural complex which represents the identity functor thought of as an element of MF($R\otimes R,1\otimes w-w\otimes 1)$ which he denotes as the stabilization of the diagonal $R^{stab}$.

3) MF(R,w) is a Calabi Yau dg-category.

Now my question is how much of the above remains true for when the singularity is non-isolated? In some writings, Kontsevich, while not explicitly saying so, writes as if the homotopy category always has a compact generator and that the category is there by "dg-affine", e.g. equivalent to D(A), the derived category of modules over a dg-algebra. Is this indeed known to be true or false? If not, is there a way to prove 2) without making reference to 1)? I'm asking because I haven't found anything about this stuff in the literature, but a lot of things in this field are not written or written in physics literature that I'm not familiar with.

I've checked a few examples with non-isolated singularities and it appears that for example in the category of factorizations $(k[[x,y]], xy^2)$, that while $k^{stab}$ doesn't generate as Dyckerhoff proves, one has $(k\oplus k[[x]])^{stab}$, which I think does generate. The way I want to argue this is Dyckerhoff's theorem 3.6 that it is enough to show that $Tor_S(k\oplus k[[x]], M)$ implies that $Tor_S(N,M)=0$, where N is a finitely generated T module and M is any S module.Then one does an analysis of finitely generated modules over T(I didn't think about the characteristic 2 case) and does some devissage with short exact sequences. Please let me know if this sounds off. I also think that with a bit more calculation one can prove similarly that in (k[[x,y,z]], xyz) the module $(k[[x]]\oplus k[[y]]\oplus k[[z]]\oplus k)^{stab}$ is a compact generator.

Added: I think the right generalization of the above two examples is the following, in http://websupport1.citytech.cuny.edu/faculty/hschoutens/pdf/finiteprojdim.pdf, the author introduces the notion of a "net". The above method should give a compact generator, whenever the net of finitely generated modules over T is generated as a net by finitely many modules. This happens for example when T has finitely prime ideals. The modules A/p, where p is a prime, generate the net of finitely generated projective modules over T, which is enough to prove the vanishing above. In particular, this should take care of the case when T has dimension 1. A question is what are some conditions on (R,w) which lead to the net of finitely generated modules over T being generated by finitely many objects?

Assuming that this right, I think that to derive 2 and 3 for these examples becomes a formality in view of Dyckerhoff's section 5. One just replaces his compact generator with the new one.

Answer by Daniel Pomerleano for A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

2011-07-18T08:25:02Z

Looking closer at Keller's paper, the result seems to be in there. Namely, in his main theorem in section 3.3, he proves that fully faithful dg-functors $per(A) \to D(B)$ induced by an $A\otimes B^{op}$ module X induce $B(\infty)$ morphisms $\phi_X: HCH^*(B,B) \to HCH^*(A,A)$ . Additionally, in the same theorem, he proves that if the map $per(B^{op}) \to D(A^{op})$ induced by X is also fully faithful, then $\phi_X$ is invertible.

These criterion all apply to M a simply connected space, $X= \mathbb{Q}$ the trivial local system, $A=C_*(\Omega(M))$, and $B= C^*(M)$.

dg-lie structure on $HH^*$ and Koszul duality

2011-07-16T11:25:56Z

This is shamelessly close to my other question: http://mathoverflow.net/questions/70151/a-question-on-koszul-duality-and-b-infty-structures-on-hh. Maybe this one will get a better response. Rather than rewrite that one, I am going to ask about a specific aspect of it in more detail. As in that question, it is known that for a space simply connected space M:

$HH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q})) \cong HH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))$

as Gerstenhaber algebras.

In particular, this implies that $HH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q}))[1] \cong HH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))[1]$ as Lie algebras.

Question:When are the dg-Lie algebra structures on Hochschild cochains: $HCH^*(C_*(\Omega(M),\mathbb{Q}),C_*(\Omega(M),\mathbb{Q}))[1] \cong HCH^*(C^*(M,\mathbb{Q}), C^*(M,\mathbb{Q}))[1]$ quasi-isomorphic.

As I mentioned in that question: this follows from more general results of Keller in the case M is formal and coformal (i.e. the d.g. algebra $C^*(M)$ is equivalent to a graded Koszul algebra).

Now suppose that g is a graded finite dimensional Lie algebra and work over $\mathbb{C}$(or $\mathbb{R}$), which corresponds to M be a $\mathbb{C}$ coformal space, with finite dimensional $\mathbb{C}$ homotopy groups. Let $C^*(g)$ be the Chevalley complex which is a model for $C^*(M)$ . Here is an approach for proving the result:

Step 1. We know from this MO question http://mathoverflow.net/questions/56145/extension-of-the-formality-theorem that $HCH^*(C^*(g), C^*(g)) \cong (T_{poly},[v,])$ as $L(\infty)$ algebras. Here v is a vector field which corresponds to the d on $C^*(g)$ (see that question for a detailed explanation of notation). These notes http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf by Damien Calaque are also extremely useful.

Step 2. now $(T_{poly},[v,])$ is canonically isomorphic as a complex $C^*(g,Sym(g))$ , that is the Chevalley complex in the Lie algebra module $Sym(g)$ .

Step 3. By PBW $Sym(g) \cong Ug$ as g modules.

Step 4. Just as in Step 1, we have an isomorphism between $C^*(g,Ug) \cong HH^*(U(g),U(g))$ . To obtain this we think of Ug as the deformation quantization of $Sym(g)$ given by the Kirillov Poisson structure on $g^*$. Ordinarily, this exists as a formal deformation but just like in Step 1, there is no problem setting the formal parameter t=1. Just as in that question, there is an induced $L(\infty)$ map on tangent cohomology groups that is an iso.

Question: Can these steps be generalized to dg-Lie algebras with finite dimensional homology? Note it follows from the cited question that step one generalizes. A generalization of Step 3 is given here in this paper of Baranovsky http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.1396v1.pdf, but it seems tricky to make this work out with the other steps above.

Answer by Daniel Pomerleano for Is there some textbook for the details of the computation of the homology groups

2011-07-12T20:00:42Z

There is a formula in the commutative case, $HC_1(A) \cong \Omega^1(A)/(dA)$. Namely there is a Connes exact sequence $HH_0(A) \to HH_1(A) \to HC_1(A) \to 0$. In the case of a commutative algebra over a field $HH_0(A) \cong A$ and $HH_1(A) \cong \Omega^1(A)$. The left hand map is d, giving the above formula. For smooth algebras, you have a similar formula for all of the cyclic homology groups. You can surely find all of this in Loday.

How to relate equivariant symplectic cohomology, Contact Homology, Cyclic Homology and String Topology?

2011-06-25T18:35:01Z

I am trying to understand how all the players in the title relate, but with all the grading shifts,and difficult isomorphisms involved in the subject I am having a hard time being sure that I have the picture right. I am going to write what I think is true, and if someone would confirm or deny it, that would be really nice.

The basic jumping off point is that if N is a simply connected manifold, symplectic cohomology of the cotangent disk bundle $D^*(N)$ , symplectic cohomology $SH^*(D^*(N))$ as defined in Seidel's "A Biased View of Symplectic Cohomology" is naturally identified with isomorphic to $H_{n-*}(LN)$ . And symplectic homology is isomorphic to $H^*(LN)$, these are the Hochschild cohomologies and homologies respectively of the algebra $C^*(N)$ .

The reason is that the zero section generates the compact Fukaya category $Fuk^{cpt}(D^*(N))$ and $End(N_0,N_0) \cong C^*(N)$ . There is expected to be a geometric Seidel map from $SH^*(D^*(N)) \to HH^*(Fuk^{cpt}(D^*(N))$ basically one considers cylinders in with a puncture which satisfy a deformed d-bar equation and which are asymptotic to periodic orbits of the Hamiltonian vector field and whose boundary lies on the zero section and is able to deform the compositions in the category to first order.

Question 1) Has anybody checked in this example that Seidel's map is an isomorphism?

Now we take equivariant versions of this, $SH^*_{eq}$ which should be identified with cyclic cohomology $CC^*(C^*(N))$ and we need to consider. Now we move to (linearized) contact homology, which Bourgeois and Oancea claim can be identified with $H_{eq}(LN,N)$ (I give up on the gradings at this point :)) where N is included as the constant loops. Reasonable enough, since those are somehow the generators missing from contact homology. However, they also seem to be making an identification that $SH_* \cong H_*(LN)$ and not the cohomology... I get that it's not really a huge deal as vector spaces to identify a vector space and it's dual, but it adds to the confusion below.

With contact homology one can try a similar map to the Seidel map, namely work in the symplectic completion $T^*(N)$ and consider holomorphic disks with a puncture, boundary on the zero section, and as asymptotic now to a Reeb orbit as $|\rho|\to \infty$ ($|\rho|$ is some norm on$T^*(N)$). This gives a map from$CH_{}\to CC^{} $` which is mysterious because...(or maybe this is a map from "contact cohomology", I'm getting confused). Edit: A reference for this map is in a paper by Xiaojun Chen called "Lie Bialgebras and cyclic homology of A(\infty structures) in topology"

Question 2) We are supposed to somehow map $H_{*,eq}(LN,N) \to H_{*,eq}(LN)$ . I'm not great with topology but this is a strange map, since it seems to go the wrong way. On the other hand one should expect it to be interesting by analogy with the Seidel map. What is this map at least conjecturally supposed to be? The only thing I could think of is the kernel of the map: $C_*(LN) \to C_*(N)$ induced by the map $LN \to N$ but this map doesn't even exist equivariantly.

Answer by Daniel Pomerleano for symplectic form with partition on unity

2011-06-24T11:46:39Z

This was too long for a comment---A conceptual way to understand this is to say for a vector bundle defined using patching, the transition functions live in GL(n,R). GL(n,R) is homotopy equivalent to O(n) by Polar decomposition so one can always define a Riemmanian metric(From this point of you can imagine a vector bundle defined on $X\times I$ and deforming your transition functions by that homotopy equivalence above). The restriction to each end must be isomorphic by a well known theorem.

The symplectic group on the other hand is not homotopy equivalent to GL(n,R) and thus there are obstructions to giving your tangent bundle the structure of a symplectic vector bundle or equivalently giving your manifold a symplectic form. $S^4$ for example is not a symplectic manifold for a simple reason. If $\omega$ were your symplectic form it would necessarily be exact. But for any hypothetical symplectic form on $S^4$, $\omega\wedge\omega$ would be a volume form, so no dice.

Extension of the formality theorem?

2011-02-21T05:44:15Z

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V , d)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an $L_\infty$ quasi-isomorphism between the two.

We can think of the derivation $d$ as corresponding to a vector-field $v$. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism: $$ (T^{poly},[v,-]) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of $L_\infty$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an $L_\infty$ quasi-isomorphism $(T^{poly}[[t]],[tv,-]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same (though I believe it remains a quasi-iso). Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.

Hochschild and cyclic homology of smooth varieties

2010-08-16T16:27:20Z

Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ignore the positive characteristic case. Based upon these sources, I wasn't really sure if this was from a lack of knowledge or because the theorems are just not really that good in characteristic p. Here are a few rather simple(and hopefully correct!) observations about Hochschild homology and cyclic homology of smooth varieties over a field of characteristic p>0 which hopefully get the ball rolling. These are all trivial observations(as long as they are right) but they seem to suggest that there is some interesting math in the characteristic p>0 case, and I was wondering whether I had made a mistake or misunderstood something in the literature/what the opinion of experts was on these questions.

1) If $X$ is a smooth scheme over a field k of characteristic $p>0$, then we can prove the Hochschild Kostant Rosenberg theorem as follows. The basic observation is that to compute $HH_*(X)=Tor_{O_{X\times X}}(\Delta_{*}O_X,\Delta_{*}OX)$ by the adjunction this is the same as $Tor_{O_X}(\Delta^*\Delta_*O_X,O_X)$ but the complex in the first argument is canonically isomorphic to the tangent complex $\bigoplus\Lambda^iT(X)[-i]$ as proven on page 247 in Huybrecht's book on the Fourier Mukai transform.

2)If $A$ is a smooth commutative ring, again over a field k of characteristic $p>d$, the Krull dimension of the ring, then all the arguments that are given in Loday's book regarding the relationship between de Rham cohomology and cyclic homology seem to work exactly the same when the characteristic $p>d$. In particular, the spectral sequence converging to cyclic homology still degenerates on the second page. This should lead to the following scheme theoretic theorem as well. The periodic cyclic homology is isomorphic to $\prod H^*_{dr}$ if the characteristic $p>d$.

3) The above theorems seems to suggest that maybe the above degenerates for smooth algebras over a field, independent of the characteristic. Somewhat independent of that one could wonder if the de Rham cohomology and periodic cyclic homology always agree for smooth varieties over a field. Does anyone know of any counterexamples to this? Again, in the affine case, this might for example follow from a sort of Cartier isomorphism, optimistically a quasi-iso from $(C^*(A,A)((u)),d+uB) \mapsto C^*(A,A)((u)),d)$ . In the case of ordinary de Rham theory, for general schemes there are obstructions to realizing the Cartier isomorphism at the chain level like this--- but I think these obstructions all vanish for affine schemes, hence this guess. Anyways, I have the impression that Kaledin proved something like this, but I haven't had a chance to study it yet, so I thought I'd just ask the MO community.

Answer by Daniel Pomerleano for Quantum cohomology for open varieties

2011-04-06T02:08:14Z

I'm just starting to learn this stuff myself, but here is a possible way to think about Alexander's answer. Given a compact smooth variety X and a divisor D we have the Fukaya category of X and the Fukaya category of X-D. If you look at Seidel's (2002 ?) ICM address, it says that under good circustances we can think of the Fukaya category of X as a deformation of the fukaya category of the open manifold U = X-D (basically the question is under what circumstances there are enough objects of Fuk(X-D) to generate Fuk(X)).

Given an $A(\infty)$ category like the Fukaya category, we can take something called Hochschild cohomology, HH*. If U is say the cotangent bundle of a manifold M, HH*(Fuk(U)) is isomorphic to H_*(LM), but in general it is something known as symplectic homology. HH (Fuk(X)) is (conjecturally in general) quantum cohomology. In principal, if you somehow understand that deformation class really well, there would be some sort of spectral sequence from SH_ (U) to quantum cohomology, but I can't think of a situation where it would be easier to compute this than quantum cohomology itself. I think this paper of Eliashberg and Polterovich might be in the spirit of such a computation from an Symplectic field theory point of view http://arxiv.org/abs/1006.2501... though presumably it does a lot more.

A different idea that is often used is to work the other way around and try to use relative Gromov Witten theory of a bigger manifold e.g. projective space to compute Gromov Witten invariants of a hypersurface. For this you can see work by Andreas Gathmann.

An elementary question in singularities

2011-04-02T17:43:36Z

The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being dumb or I don't have the right tools to understand it. Here it goes, given the ring $\mathbb{C}[x_1,\ldots ,x_n]$ and n functions $f_i(x_1,\ldots, x_n)$ (quasihomogeneous if it matters) such that the ring $\mathbb{C}[x_1,\ldots,x_n]/(f_i)$ is a finite dimensional local artin algebra concentrated at the origin.

Now fix an integer d and consider $\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots y_n]$ with the function $ \widetilde{f}(x_i,y_i)=(\sum f_i(x_1,\ldots x_n)y_i)+p(y_1,\ldots,y_n)$ for polynomials p only depending on y's whose lowest order terms have total degree at least d. Can we find a p such that this function $\widetilde{f}$ has only isolated singularities?

The answer is yes, for d=1 I believe. I am actually interested in knowing if for any collection of f's as above there is some d>1 for which we can solve the problem. If this turns out to be impossible, a nice clean condition where we can find such a p would be great too. This problem seems open to brute force attacks, geometric interpretations, algebraic thinking (like maybe Grobner bases ?) so I am hoping someone will be able to know exactly what is going on. Thanks.

Answer by Daniel Pomerleano for When is a quasi-isomorphism necessarily a homotopy equivalence?

2011-03-24T07:25:16Z

If your complexes are bounded, this is always true for any ring more generally replacing free modules with projectives. The statement is that D^b(A-mod) is equivalent to Ho(Proj-A mod) and you can find it in Weibel Chapter 10.4. If your complexes are unbounded things are more tricky. Then your statement is true in over any ring of finite homological dimension. Basically you have two notions K-projective(which have the property that you want) and complexes of projectives. Bounded complexes of projectives are K-projective, but unbounded ones are not unless you have the finiteness hypothesis(see Matthias' answer). See this post for the injective version of this story http://mathoverflow.net/questions/41642/question-about-unbounded-derived-categories-of-quasicoherent-sheaves. In the cases you are interested in there is no problem.

Answer by Daniel Pomerleano for Is a quasi-iso in Lie algebra cohomology necessarily an iso?

2011-03-21T08:46:35Z

Here are some comments to your answer that I hope will be helpful (it's still sufficiently confusing that I might make some mistakes). Given an $L_{\infty}$ algebra, we can define the "Koszul dual" Chevalley $\it{chains}$ $C_*(L)$(for several reasons it's more natural to think of the Chevalley chains rather than cochains). The ordinary notion of a quasi-isomorphism of $L_{\infty}$ is a morphism of Chevalley complexes whose first Taylor coefficient induces a quasi-isomorphism of complexes. We could ask what happens if we declare two Lie algebras weakly equivalent if there is a quism of Chevalley complexes? We have seen that this notion can differ from the standard one. Another example is when $g_1=sl_2(\mathbb{C})$ and $g_2=\mathbb{C}[2]$ an abelian Lie algebra concentrated in degree 2. From this point of view, your notion is a very natural relaxing of the usual notion of quasi-isomorphism --- it's the notion of weak equivalence transported from Koszul duality. It's worth noting that the two notions will coincide if your dg-Lie algebra is concentrated in positive degree (or more generally with some nilpotency hypotheses on the action of the degree zero piece $g^0$ on your dg lie algebra.) This explains Trial's answer above.

It's useful to have the rational homotopy picture in mind. Given a 1-connected CW complex Xwhich is finite in each degree, you can construct a Lie model L for X whose $H_{*}(L)=\pi_{*}(\Omega(M)) $ with Whitehead bracket. One gets chains on your space by taking $C_{*}(L)$. To go back, you have something known as the cobar construction which spits out $C_{*}(\Omega(M))$, which is U(L), the universal enveloping algebra of L. Again, things start to break down in the non-simply connected case if you don't assume some hypothesis about the action of $\pi_1$ on the higher homotopy groups. From this point of view, the comment at the end of the last paragraph is a reflection of the fact that in these instances, when you have an isomorphism on homology groups, you have an isomorphism on homotopy groups.

Finally when thinking about the idea that the more relaxed notion gives you a derived equivalence, you must define the derived category properly. Let's think about the abelian case. The most classical case of Koszul duality says that given a finite dimensional graded vector space, $D^{+}(SV) \cong D^{+}(\wedge(V^*))$. Here we are considering the derived category of modules which are bounded below. It's pretty easy to see that this cannot be extended an equivalence of unbounded derived categories.

The right notion which makes the whole thing work for unbounded derived categories can be found in Positselski's works and is called the coderived category. Let g be a Lie algebra over $\mathbb{C}$. Then the equivalence between the derived category of modules over U(g) and the coderived category of co-modules over it's Chevalley complex $C_{∗}(g)$ in which $M \to C_{∗}(g,M)$. Under the assumption g is finite, we can consider these co-modules to be modules over C*(g)(now looking at cochains again) and look at the corresponding localization of the category of dg-modules over C*(g). What we see is that we are localizing at a smaller set of weak equivalences than quasi-isomorphisms.

Finite Homological Dimension of R/P for all P for module finite non-commutative rings

2010-11-25T04:00:18Z

I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's center Z(R) which we can assume has finite Krull dimension. One can also assume R integral over Z(R). By a two sided prime ideal, I mean a two sided ideal P where if $xRy$ is contained in P, then either x and/or y are in P. Consider now the abelian category of left modules and suppose we know that $Ext^m(R/P,M)=0$ for any M and $m>n$ for some n (that is independent of P). Now there aren't enough (two-sided) prime ideals in a general non-commutative ring, but there are a fair number in rings such as the one I describe above.

*Question: Does it follow that for any finitely generated left module N, $Ext^m(N,M)=0$ for $m>n'$ which can depend on N *. The issue I am having is that we don't have quite as effective a filtration sequence of any N, we only have a filtration such that $N_i/N_{i-1}=I/P$ for some left ideal and a prime P. That might be the end of the story, but there might be other tricks I am not aware of. Either way, I haven't been able to sort it out or find a good reference. If there is an extra hypothesis that helps the situation, I'd like to know about it.

Quantum polynomial rings and singularities

2010-11-13T08:48:08Z

Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with relations $x_i*x_j=-x_j*x_i$ for $i\neq j$. Set $u_i=x_i^2$ consider central homogeneous functions of the form $f(u_1,...,u_n)$ and the quotient ring R. The question is when, if ever, does Proj(R) have finite homological dimension. By Proj(R) I mean the quotient abelian category Gr(R)/Tor(R), where Gr(R) consists of finitely generated left modules and Tor(R) is the subcategory of torsion modules? Based upon some calculations, I believe the answer should be whenever $C[u_1,...u_n]/(u_idf/du_i)$ is finite dimensional.This is a condition about how the zero locus of f intersecting the coordinate axes $u_i$, which I believe should have a geometric interpretation in terms of the map $Z(Q)=C[u_1,...u_n] \mapsto Q$ when the $u_i$ are not zero, the fibers are matrix algebras which degenerate at the locus when some of the $u_i=0$. By analogy with the ordinary projective space, I'd like to think of this as a smooth hypersurface in the quantum projective space $P_{-1}^{n-1}$ but I can't seem to find this type of stuff analyzed anywhere. Does this ring a bell for anyone? If not, does anyone understand the map on primes $Spec(Q)\mapsto Spec Z(Q)$?

E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product

2010-11-07T07:10:44Z

Let X be a smooth scheme, then an infinity enchancement of QCoh(X) has an E(infinity) structure and in particular an E(n) structure for any n. In this paper, http://arxiv.org/abs/0805.0157 Ben-Zvi, Francis, and Nadler compute E(n) Hochschild cohomology of this category as $T_X[-n]$. I believe there is also separate work of Francis explaining how E(n) Hochschild cohomology (QCoh(X)) gives deformations of these categories. I've been trying to get some kind of idea as to what this result means. The case n=1, I understand pretty well, but I am curious about all of the other odd cases in particular. The thing that I find strange is that the vector space and Lie-structure of the deformation space are very similar for all of the odd n and yet I believe from degree considerations, it must be the case that more often than not, deformations of the E(2k+1) structure are trivial as deformations of the E(2k-1) structure(the notable exception coming from commutative deformations of the underlying scheme).

Just to keep things concrete(I hope), let's just assume X is Spec(A) and n=3. Can these deformations be made reasonably explicit in the same way the n=1 case is say by deforming some sort of associator map? What about in some special case such as that of a function f(these will be some kind of derived deformations in the sense that will break the Z-grading)?

Question about unbounded derived categories of quasicoherent sheaves

2010-10-10T01:04:05Z

This is a bit of a strange question since I more or less want to ask the MO crowd whether I've understood the situation correctly. If you have an unbounded complex of quasicoherent injective sheaves $I$ over a variety over a field k where all local rings are regular (the property that I want to isolate is that every quasicoherent sheaf have a finite injective resolution), is it K-injective in the sense of Spaltenstein? My source for this material is Krause's "The Stable Derived Category of a Noetherian Scheme".

Since the topic of this question maybe of somewhat broad interest(I had mostly ignored unbounded complexes until today and I assume many people have the done same), I'll give a little bit of background. Recall that for a bounded complex of sheaves, one can compute right derived functors by replacing your complex $K$ with a quasi-isomorphic complex of injectives $I$ and apply your functor to this complex. One needs a more nuanced approach. A complex $KI$ is called K-injective if $Hom^{*}(A,KI)=0$ is an acyclic if A is acyclic. Every bounded complex of injectives is K-injective, but this is false for unbounded complexes. A very simple example is given for the dual projective question in the beginning of Spaltenstein's paper. It is a theorem of Spaltenstein that such resolutions always exist.

My question is, in the case of schemes as above, is a complex of injectives K-injective? My reasoning is pretty simple. In the paper above, Krause explains that the inclusion from the homotopy category of injectives $K(Inj)\mapsto K(QCoh)$ has a left adjoint L. Now take an acylic object A viewed as an object in $K(QCoh)$, then we have $Hom_{K(QCoh)}(A, I)= Hom_{K(Inj)}(L(A),I)=0$ because Krause proves that all acyclic injective complexes are zero in $K(Inj)$ when all objects have finite injective resolutions. Edit: We know that L(A) is still acyclic since Krause proves that if $Q_I$ denotes $K(Inj) \mapsto D(QCoh)$(the derived category) and $Q$ denotes the map $K(QCoh) \mapsto D(QCoh)$ we know that $Q$ is isomorphic to $Q_I \circ L$. Now shift A to get the result. Am I missing something? If this breaks down for some reason, does it at least hold when your complex is two periodic? If one only cares about computing derived functors of $\Gamma$, the global sections functor.

A Question about a theorem in Toën's notes "Lectures on dg-categories"

2010-10-01T06:48:33Z

So I am trying to learn a bit about dg categories from Toën's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand proposition 11 on page 52. The reference he cites is around 300 pages and seems to develop a lot more than maybe necessary for this particular statement. I'm happy for the moment to forget about perfect complexes and just try the statement for bounded quasicoherent complexes, although I'd definitely be interested if people wanted to discuss that case too.

Maybe I am talking absolute nonsense, but my vague impression is that the basic statement should be that homotopy pullback is a derived functor of the categorical pullback with respect to localization of the category of dg categories by inverting weak equivalences and that the dg-category $L(X)$ that Toen is discussing is an acyclic replacement for $C(O_X)$ with respect to this functor. Can anyone give clean references for this sort of theorem? Though I appreciate that this theorem probably lies in the realm of $\infty$ categories, I think I'd prefer discussion less fancy and closer in appearance to the above theorem if possible. Ideally, it should be possible to walk me through this theorem by hand? Presumably this all comes down at the end of the day to some relatively simple homological algebra, but my head isn't clear enough with respect to this stuff to work it out.

Comment by Daniel Pomerleano

2013-04-01T08:06:31Z

Not sure I understand the question... what if you have $\mathbb{C}[e]/e^2$ ?

Comment by Daniel Pomerleano

2013-02-23T03:54:16Z

Why not talk to your students about this point before taking them on to get a sense for how they feel ? Also I think a lot of students and (maybe professors) don't have a grasp on what the job market is like, so I think it might be a good idea to make sure they are educated on this point before you take them on ?

Comment by Daniel Pomerleano

2013-02-18T01:24:37Z

Maybe this book will help you make precise your question: mathematik.uni-muenchen.de/~kai/research/…

Comment by Daniel Pomerleano

2013-02-11T01:21:06Z

My "point" is similar to Daniel Litt. When I think about DG, I think about deep differential geometry theorems which involve heavy work in partial differential equations (such as prescribing curvature or Ricci Flow) or relations between curvature and topology such as the sphere theorem. My point is that many people associate DG with some core theorems as well as analytic and geometric techniques. I imagine that for differential geometers, the statement that anything beyond the category of smooth manifolds are an appropriate setting for differential geometry would require a lot of discussion.

Comment by Daniel Pomerleano

2013-02-10T16:27:12Z

It seems slightly biased to speak about any non-trivial differential geometry without differential equations, curvatures and calculus.

Comment by Daniel Pomerleano

2013-01-25T20:24:03Z

Think about your category where you have just one object and that shoud hopefully allow you to straighten yourself out. Make sure you think about everything in detail...

Comment by Daniel Pomerleano

2013-01-18T07:06:35Z

As a small hint: What is the universal property of the one point compactification?

Comment by Daniel Pomerleano

2013-01-16T19:33:44Z

Sure just write down the Serre spectral sequence and it's clear. There is the more classical Leray-Hirsch theorem as well that can work for this situation.

Comment by Daniel Pomerleano

2013-01-09T17:23:28Z

If you can understand math fast enough that it makes a real difference in terms of time whether you read in French or English you must be really smart.

Comment by Daniel Pomerleano

2012-12-26T05:02:28Z

BTW, the Arnold Chord conjecture is different from the Arnold conjecture that is mentioned in the question.

Comment by Daniel Pomerleano

2012-12-26T05:00:07Z

There are many applications due to Vitterbo and others to the Weinstein conjecture in all dimensions, however the results in dimension 3 are really impressive. Taubes proved it completely in dimension 3 (I don't think the proof actually uses Floer homology, though the paper seems to be a first step in the equivalence between Embedded contact homology and Seiberg-Witten Floer homology). There are quantitative improvements due to Hutchings and Cristofaro-Gardiner that use ECH explicitly. The Arnold Chord Conjecture in dimension 3 by Taubes and Hutchings...

Comment by Daniel Pomerleano

2012-12-05T07:50:56Z

Can't one prove this for an arbitrary field of char 0 by extension of scalars? Certainly relaxing condition three will be hopeless.

Comment by Daniel Pomerleano

2012-11-25T14:03:45Z

There is very nice work on etale tubular neighborhoods, beginning with Cox's thesis and which might help you with whatever problem you are considering. You can google "algebraic tubular neighborhood".

Comment by Daniel Pomerleano

2012-11-15T05:21:11Z

why is this question closed? It seems to be related to several interesting things.

Comment by Daniel Pomerleano

2012-11-13T07:21:52Z

Also you use smoothness of X to conclude that the sheaf $R^2 pr_*G_m$ is torsion.