User mohammad f.tehrani - MathOverflow

$P^1$ minus k points

2013-05-22T20:40:37Z

For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write

$$ P^1 \setminus S \cong \mathbb{H}/G $$

where $\mathbb{H}$ is the upper-half plane and $G\subset PSL(2,\mathbb{R})$ is a representation of $\pi_1(\mathbb{P}^1\setminus S)$. G can be generated by $(k-1)$-elements.

Is there an explicit description of how $G$ looks? In that case, the parameters describing such $G$ can give coordinates on $\mathcal{M}_k$.

flatness in complex analytic geometry

2011-09-13T14:42:11Z

It is always a pain to move back and forth between definitions in algebraic geometry and complex analytic geometry. Dictionary is much easier when are working with (family of) smooth varieties but the pain grows exponentially when we include singular varieties.

Here is some of that:

Suppose $f: X\rightarrow Y$ is a map of possibly singular complex analytic varieties, then is there a simpler definition of flatness for $f$ ? this one should be hard to answer but how about following,

---Let $f:X\rightarrow Y$ be a family of curves. there might be multiple fibers, non-reduced fibers, nodal curves, cusp curves,... in the family, but fibers are connected.

---When this fibration is flat? What kind of bad fibers mentioned above are allowed in a flat family?

---Same question for higher dimensional family of analytic varieties?

For simplicity you may assume that the base of fibration is smooth.

Square root for Hamiltonian diffeomorphisms

2012-07-10T15:08:02Z

Let $\psi_t: X\to X$, $t \in [0,1]$, be a path Hamiltonian diffeomorphism on a symplectic manifold $X$, given by functions $H_t$. If $H_t \equiv H$ is independent of $t$ then

$$ \psi_1 = \psi_{\frac{1}{2}}^2 $$

and therefore the Hamiltonian diffeomorphism $\psi_1$ has a Hamiltonian square root.

Is the same thing true for any arbitrary Hamiltonian $\psi_1$, i.e. is there another Hamiltonian $\phi$ such that $\phi^2 = \psi_1$ ?

Ricci flow on Riemann surfaces

2013-03-22T14:48:11Z

Let $g_t$ be the solution of normalized Ricci Flow on a Riemann surface $\Sigma$ of genus g. We know that $g_t$ converges to constant curvature metric. Is it possible for $g_t$ to be of the form $f_t^*g_0$, for a one parameter family of diffeomorphisms $f_t$?

Toroidal embedding

2013-03-13T01:42:01Z

Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit disk whose central fiber is a normal crossing union of (say four for example) some copies of $\mathbb{P}^2$ blown up at 3 points of a triangle; i.e.

$$\pi:X \to \Delta \subset \mathbb{C},$$ such that $ X_t=\pi^{-1}(t), t\neq 0,$ is smooth and and is a complex tori; $X_0= \cup V_i$ and the singular locus of $X_0$ restricted to each $V_i$ is a cycle of 6 minus 1 curves. It is also known that in this case, the dual graph of $X_0$ is $S^1\times S^1$.

Authors mention that this degeneration can be realized via toroidal embeddings (I assume it means using toric varieties) but there is no explicit example.

Does any body know any explicit example, presenting such toroidal embedding?

Just to know: The total space $X$ mentioned above is a Kulikov model whose smooth fibers are complex tori

Kähler structure on cotangent bundle?

2010-06-02T02:15:27Z

The total space of cotangent bundle of any manifold M is a symplectic manifold.

Is it true\false\unknown that for any M, $T^*M$ has Kähler structure?

Please support your claim with reference or counterexample.

Answer by Mohammad F.Tehrani for Kähler structure on cotangent bundle?

2013-03-08T01:32:45Z

In the refernce mentioned by Zemisch, Guillemin and Stenzel prove:

Theorem. For an analytic manifold L and analytic metric g on L, there is a $\sigma$-invariant neighborhood ($\sigma(x,v)=(x,-v)$) of $L\subset T^*L$ with a unique complex structure on that such that

i- $\sigma$ is an anti-holomorphic involution

ii- The one form $Im \bar\partial h$, where $h=|v|^2$ is the square of length of $v$ with respect to $g$, is the standard one-form $\sum y_i dx^i$. (This would imply $\sqrt{-1}\partial \bar\partial h$ is the standard Kahler form).

This is indeed an impressive result.

complexified kahler form

2010-06-17T17:39:36Z

In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli)

Here is the question:

What does $B$ correspond to? what kind of information it has?.... I would also like to know about the moduli of complexified kahler structures on a Calabi-Yau 3-fold. Is it a cone? If yes, what are the walls ? does it have a natural metric?

Are rational varieties simply connected?

2013-01-31T18:46:52Z

Is it true that every smooth rational variety X is simply connected? How is the proof? Would it be still true if X has mild (for example orbifold) singularities?

Homotopy groups of K3

2013-01-28T18:56:46Z

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface. Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ and $H_2(Y)$ are very different.

What are $\pi_2(X)$ and $\pi_2(Y)$?

Answer by Mohammad F.Tehrani for Is there a long exact sequence associated to a ramified covering?

2013-01-28T18:38:40Z

There is a paper "on the homology of double branched covers" by Lee, which is kind of related to your question.

Extending line bundles

2013-01-25T16:27:54Z

Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the rest are smooth.

Is it true that every line bundle over a smooth fiber, after some base-change, extends to the whole family? Do you know any counter example?

This is the true (although not very obvious) for a family of curves, since line bundles are just bunch of points.

Extra assumption: $h^{2,0}=0$ for the smooth fibers, and they are simply connected, so that the ample cone does not move within $H^2$.

Can you get an Enrique surface from quotient of Abelian surface?

2013-01-22T16:27:25Z

Let $A=\mathbb{C}^2/\Lambda^2$, where $\Lambda=\mathbb{Z}+i\mathbb{Z}$, be an abelian surface. Then every body knows that the resolution of the quotient $A/<\pm>$ is a K3 surface.

Question: Is there an easy involution on the resulting K3 surface (may be induced from some action on A) such that the quotient is Enrique surface, i.e. is this K3 surface double cover of some Enrique surface?

I am not sure whether this is some thing easy or well-known.

Known Mirror Calabi-Yau pairs

2011-04-27T16:58:21Z

There is a well known class of Calabi-Yau (3 dimensional) pairs constructed by Batyrev. These are resolutions of Calabi-Yau hypersurfaces in reflexive polytops of dimension 4.

Question: Does any body know any other mirror pair, or a family of them, beside this kind of pairs?

For example, how about Calabi-Yau complete intersections in higher dimensional weighted projective spaces or Fano toric varieties?

Warning: My question is only about closed Calabi-Yau 3-folds.

difference of curve classes

2012-10-25T04:15:58Z

Let $X$ be a smooth protective variety, or just a smooth Kahler manifold. Is it possible to have two curves $C_1$ and $C_2$ in $X$ such that their difference in $H_2(X,\mathbb{Z})$ is a non-trivial torsion class ?

Degeneration of varieties to simple normal crossings

2012-10-25T00:22:17Z

Let $\mathcal{X}\to\Delta$, $\Delta \subset \mathbb{C}$ is the unit disk, be a smooth family of varieties whose fibers over $t\neq 0$ are smooth and the central fiber $\mathcal{X}_0$ is a nice simple normal crossing divisor (in $\mathcal{X}$).

Let $\mathcal{X}_0=\cup X_i$, and define $X_I=\cap_{i\in I} X_i$.

This assumption imposes some relations between the $N_{X_{I}}^{X_{J}}$, $J\subset I$. For example if $\mathcal{X}_0=X_1\cup_D X_2$ ($D=X_{12}$), then $N_D^{X_1}\otimes N_{D}^{X_2}=\mathcal{O}_D$ is trivial, and conversely if these condition holds, then there is a smooth one parameter family realizing that.

Question:

Is it known under what conditions, a simple normal crossing space can be realized as the central fiber of a smooth family? Is it known if those relations are enough for the existence of such family (similar to example above)

In general, is this probelm studied in literature or not?

A singularity waiting for your resolution

2012-10-19T22:14:02Z

Possible Duplicate:
How can I get a small resolution for the binomial fourfold $x_1 x_2 x_3- y_1 y_2=0$ in $\mathbb{C}^5$?

Consider the following non-isolated singularity (I am going to call it $A_{2,3}$, I am not sure if it has a name)

$$uv-xyz = 0 \subset \mathbb{C}^5$$

Question 1) Is this some thing studied before?

Question 2) How would you describe a nice (small) resolution of that? note that this 4-fold is singular along union of x,y,z-axis. Outside the origin its a one parameter family of A_2 singularities, so has a nice small resolution, the main difficulty is at origin. By nice I mean:

symmetric, it would be OK if the resolution is not small at the origin but I want it to be small elsewhere.
description is somehow systematic.

How to construct Enriques surface from Fermat K3

2012-10-10T01:19:56Z

Let $x_1^4+x_2^4+x_3^4+x_4^4=0 \subset \mathbb{P}^4$ be the Fermat K3 surface. Is it possible to start from some involution on $\mathbb{P}^3$, do blow-ups to get rid of fixed points and then quotient with respect to resulting involution to get an Enrique surface inside the quotient of some blow-up of $\mathbb{P}^3$.

can you give an example of an Enrique surface inside a simply-connected 3-fold?

Topology of K3 as a sum of two abelian fibrations.

2012-10-03T20:48:36Z

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).

K3 surfaces is obtained by removing a fiber from two copies of $E$ and gluing along the boundaries.

How do we realize 22 second homology classes of K3, in terms of 10 second homology classes of $E$. I know this is classic but I could not find a reference.

line bundles in families

2012-09-30T01:57:34Z

Let $\pi\colon Z \to \Delta$ be a smooth family of complex (projective) varieties, over a small disk in $\mathbb{C}$ such that $\pi^{-1}(0)$ is the only (normal-crossing) singular fiber, $\pi^{-1}(0)= X\cup_D Y$.

I have some questions (may be equal) about line bundle on the family vs individual fiber.

Under what condition (on $X$, $Y$, $D$), it is true that we extend a line bundle on the singular fiber to a line bundle on the whole family?

Under what condition (on $X$, $Y$, $D$), it is true that there is an isomorphism between the space of line bundles (Picard group) on the singular fiber and the space of line bundles on the smooth fiber?

I general I would like to know about the comparison between the Picard group of $Z$ and its fibers.

For $t\neq 0$, I expect Pic($Z_t$) to be bigger than Pic($Z_0$), can you provide an example of that?

Examples of non-Kahler compact symplectic manifolds.

2012-09-21T18:49:33Z

I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it.

Please avoid giving repetitive examples.

Thanks.

An elementary but confusing question in differential geomerty

2012-07-14T21:58:09Z

Let $f:\mathbb{R}\to \mathbb{R}$ be a function, then looking $f$ as a function between manifolds, $df:T\mathbb{R}=\mathbb{R}^2\to \mathbb{R}^2$ and $d^2f:TT\mathbb{R}=\mathbb{R}^4\to \mathbb{R}^4$ are given by

$$df(x,a)=(f(x),\frac{df}{dx}a)$$ $$d^2f((x,a),(b,c))= ((f(x),\frac{df}{dx}a),(\frac{df}{dx}b,\frac{d^2f}{dx^2} ab+ \frac{df}{dx} c))$$

So the second derivative of $f$, as you know it from calculus, is embedded in the differetial geometric definition of $d^2f$ in a wierd way. There is a similar story in higher orders.

Now suppose $f:M\to N$, $f(x)=y$, is a smooth map between two manifolds. As before, for each k, we have $d^kf: TTT\cdots TN \to TTT\cdots TM$.

Question: For $v_1,\cdots,v_k \in T_xM$, is there an object $\frac{d^kf}{dv_1\cdots dv_k}$ similar to the way we define higher derivatives of a function on $\mathbb{R}$? how is it defined? and where does it belong to? (Is $\frac{df}{dv_1\cdots dv_k}\in T_yN$ ?)

Apologies in advance if the question is obvious or absurd.

Cone of movable curves

2011-10-25T00:08:36Z

Let $X$ be a smooth complex projective variety of dimension $n$.

Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure of ample cone $\overline{Amp}(X)$.

It was proved in 2004 that the closure of cone of effective divisors $\overline{Eff}(X)$ is dual to the closure of cone of movable curves $\overline{Mov}(X)$. A movable curve by definition is a curve class $C \in N_1(X)$ such that $C=\pi_*(H_1 H_2 \cdots H_{n-1})$, where $\pi: X' \rightarrow X$ is a birational morphism and $H_i$'s are ample classes on $X'$.

My question: Let $Q(X)$ be the cone obtained by curve classes $H_1 H_2 \cdots H_{n-1}$ where $H_i$ are ample divisors on $X$ itself. Is it true/false that $\overline{Q}(X)=\overline{Mov}(X)$? i.e. as long as I am interested only in the closure of these cones; do I really miss some curve class if I only restrict my self to intersection of ample classes on $X$ itself.

Can any body give an example where $\overline{Q}(X) \neq \overline{Mov}(X)$?

Meanwhile, I am only interested in $n=3$ case.

geodesics in Lens spaces

2012-07-28T17:08:24Z

For which Lens spaces, all the simple closed geodesics have the same length?

Is it just $S^n$ and $S^n/\mathbb{Z}_2$ or there are more?

Answer by Mohammad F.Tehrani for Square root for Hamiltonian diffeomorphisms

2012-07-10T19:32:35Z

I got this answer from Dusa McDuff (and she got it from some body else):

Suppose given $f:[0,1]\to [0,1]$ such thqt 0 is repelling fixed point and 1 is attracting fixed point and there are no others.

So $f'(0) = \lambda >1$, and $f'(1)=\mu < 1$.

A thm says that in suitable local coords near $0$ $f$ is simply mult by $\lambda$ (this is a linearization them). Therefore f has a unique square root on [0,1). Similarly, it has a unique square root on (0,1].

But in general the coords at the two ends will NOT be compatible so there is no square root on [0,1].

Now consider a smooth $f: S^2\to S^2$ with two non-deg fixed points $p_0,p_1$ with a homoclinic orbit $A$ between them. i.e. there is an arc $A$ which at one end is the unstable manifold of $p_0$ and at the other is the stable manifold of $p_1$. Now restrict f to A.

(There is a stable manifold thm that says that locally these invariant submanifodls exist etc.)

Physicists Euler number conjecture

2012-02-27T00:05:53Z

Physicist's Euler number conjecture says:

If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for $n\leq 3$). Then there exists a basis of $H^*(Y,\mathbb{Q})$ consisting of algebraic cycles in one-to-one correspondence with conjugacy classes of $G$.

I have seen some works (by Reid,...) which date back to 2000. What are the recent results around this conjecture?

See : The McKay correspondence for finite sungroups of SL(3,C), by Miles Reid and Yukari Ito.

When a quotient singularity is toric?

2012-02-25T23:50:38Z

Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order, Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?

Is this manifold orientable?

2012-01-30T21:23:24Z

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy

2) $ a\bar{b}+c\bar{d}=0 $

There is a (component-wise) $S^1$ action on $C$ and let $S$ be the quotient ($S$ is a 3-manifold).

Is $S$ orientable or not ?

Thanks.

A simple question about the degree of some vector bundle over rational curve.

2012-01-26T01:31:46Z

Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, $c(z)=\frac{-1}{\bar{z}}$, and after all suppose we have a commutative diagram (I couldn't draw it here, imagine it yourself)

$\pi: E \rightarrow \bar{E} $

where $\pi$ is an anti-holomorphic (or anti-complex linear) involution covering $c$. In this situation, is it true that $c_1(E)$ should be even!

Answer by Mohammad F.Tehrani for Cone of movable curves

2011-10-25T13:14:53Z

I think following post in the mathoverflow gives an answer:

http://mathoverflow.net/questions/79046/effective-versus-movable-cones-of-curves

There, people mention that there is an example where the Ample cone is rational polyhedral but movable cone is not. But if Ample cone is polyhedral then $\overline{Q}(X)$ would be polyhedral too and so they can not be equal.

Comment by Mohammad F.Tehrani

2013-05-23T15:08:33Z

Sure, but how do the matrices corresponding to these isometries look like? Is it discussed in those references?

Comment by Mohammad F.Tehrani

2013-05-14T15:12:09Z

good to know, thanks

Comment by Mohammad F.Tehrani

2013-03-27T02:57:23Z

This looks fun: grdb.lboro.ac.uk/search/ellk3

Comment by Mohammad F.Tehrani

2013-03-27T02:52:29Z

When talking about j-map above, I am assuming there is a section.

Comment by Mohammad F.Tehrani

2013-03-27T02:07:08Z

Some more questions in this direction: -- @ Jason Starr: What is the weight you are mentioning? --So the number of singular fibers (which I think is the degree of j-map to $\mathbb{P}^1$ can be different in different examples? --Are there examples where the singular fiber is "not" normal-crossing" (or semi-stable)?

Comment by Mohammad F.Tehrani

2013-03-19T19:29:05Z

This and similar papers, even the "Toroidal embeddings I" book, talk about properties of a toroidal embeddings; non of them shows starting from a given polyhedra, how to build an embedding. I know that this is an open problem, but I am just looking for couple of examples for which the procedure is known, e.g. for the one mentioned in my question.

Comment by Mohammad F.Tehrani

2013-03-13T04:12:05Z

OK, I thought X can be embedded in some higher dimensional toric variety (not that I meant X-->\Delta is toric itself) Now can you say how an example like that can be built, that might be enlighting toward the definition of toroidal embeddings.

Comment by Mohammad F.Tehrani

2013-03-07T19:04:46Z

does not your first argument need M to be an analytic manifold? I found the papers mentioned by Zehmisch and the one in the comment below that interesting as well.

Comment by Mohammad F.Tehrani

2013-03-07T19:02:09Z

Also this one: Complex structures on tangent bundles of Riemannian manifolds, Szoke

Comment by Mohammad F.Tehrani

2013-01-31T20:30:17Z

@Tim: My question was about the classical fundamental group. What is the definition algebraic one? @Vesselin: I still believe we can say some thing if the singularities are terminal or something nice.

Comment by Mohammad F.Tehrani

2013-01-25T18:10:03Z

I dont have access to the book, can you recall what the example is. In the setting I have, this is true for the smooth ones. So may be i need to put some restrictions on my family. I will edit my question then.

Comment by Mohammad F.Tehrani

2013-01-23T19:02:31Z

Can you describe the torsion 2nd homology class in some Enrique surface as a difference of two curve classes?

Comment by Mohammad F.Tehrani

2013-01-22T19:43:31Z

good, thanks for the reference.

Comment by Mohammad F.Tehrani

2012-10-29T03:02:49Z

what is torsion here!

Comment by Mohammad F.Tehrani

2012-10-28T16:46:10Z

Is there any new work on this subject (i.e. an attempt to understand normal crossing varieties) in a purely geometric way, i.e. something which avoids monoides and is focused on complex algebraic varieties, rather than formal scheme theoretic point of view. The only work of this type which I know is the thesis of Friedman, which is quit old.