User vamsi - MathOverflow

Vector bundles on Stein Manifolds

2013-05-17T01:24:33Z

This might be standard if true (if so, I shall be grateful if provided with a reference). Given a smooth map from a Stein manifold X to Gr(k,n) (the Grassmannian of k planes in C^n), is there a holomorphic map in its homotopy class?

Density of smooth functions in Sobolev spaces on manifolds

2013-04-03T18:01:08Z

Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have been defined as the collection of locally integrable functions whose weak derivatives (w.r.t to the Levi-Civita connection) are in L^p. For compact manifolds, or for manifolds with bounded geometry and positive injectivity radius (like $\mathbb{R}^n$), I can see why these two definitions are equivalent. Are they equivalent for general noncompact manifolds?

Alexandrov-Bakelmann-Pucci maximum principle

2012-09-03T22:33:10Z

The ABP maximum principle states (roughly) that, if $a^{ij} \partial _i \partial _j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient regularity of the coefficients), $\sup _{\Omega} u \leq \sup _{\partial \Omega} u + C (\int _{\Omega} \vert f \vert^n )^{1/n}$. Its proof seems to be quite unconventional (the final inequality is proved by an inequality of the measures of certain sets constructed using the normal mapping). Is there an intuitive explanation of the proof? (Perhaps using some probabilistic ideas or something?) I mean, usually, in order to prove maximum principles, the key idea is to use that at a local max the second derivative is negative-definite. But, this one seems to use some strange ideas..

An example of a complex manifold without a finite open cover

2011-03-20T17:46:11Z

Are there non-compact complex manifolds that a) Don't embed in C^n (holomorphically) and b) Cannot be covered by a finite number of coordinate open sets? If b) can be satisfied, then I think so can a) be by taking a product with a compact complex manifold. If one takes a Riemann surface of infinite genus, one does not have a "good" finite open cover, but I allow non-contractible open covers as well. Apologies in advance for this elementary question.

Hölder estimates for the Complex Monge-Ampere equation

2012-03-26T22:58:00Z

If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{k+2, \alpha}$ ? (I mean, is there an apriori estimate on $u$ with the Hölder exponents of $f$ and $u$ being the same (equal to $\alpha$?)

Atiyah Bott localisation applied to Euler characteristic

2010-02-01T02:26:05Z

Suppose we have a torus action on a compact oriented manifold M. Assume the action has isolated fixed points. Why is it that the equivariant Euler class of the normal bundle at the fixed point (i.e. the tangent space at that point) is (upto a sign) the product of the weights of the action of the lie algebra of the Torus on the tangent space at that point? (Maybe this is obvious but I don't see it).

Monge Ampere equations (concavity)

2012-03-23T07:09:23Z

The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of $\mathrm{Hess}(u)$. This is done for the purpose of $C^{2, \alpha}$ estimates. My question is: Has any work been done in the non-concave setting? (I know of Harvey and Lawson's work that gives us viscosity solutions, but I wish for more regular solutions) Especially with regard to the $C^{2,\alpha}$ estimate?

Bedford-Taylor theory

2012-03-13T21:39:53Z

The Dirichlet problem for the Complex Monge-Ampere equation on a bounded pseudoconvex domain in $\mathbb{C}^n$ was studied in Bedford-Taylor's seminal paper wherein they defined $(dd^{c} u)^n$ for locally bounded plurisubharmonic $u$. But, they don't seem to use it in their Perron-type method, instead using a convex-measure-theoretic construction claiming that the upper envelope is not well-behaved. Now that we know more about psh functions, have people studied the Dirichlet problem without using the measure-theoretic construction of Goffman and Serrin? (and reproved Bedford-Taylor's results)

Answer by Vamsi for Where is number theory used in the rest of mathematics?

2012-03-14T02:37:57Z

If Arakelov geometry counts as number theory, then, http://arxiv.org/pdf/math/0401029v1.pdf demonstrates the computation of the Analytic torsion (a purely analytic object involving the product of determinants of laplacians) using the Arithmetic Riemann-Roch theorem.

Atiyah-Bott Yang-Mills connections

2011-07-10T03:01:50Z

In Atiyah-Bott's paper on Yang-Mills equations on Riemann surfaces, a special case of what they do is to prove that Unitary Yang-Mills connections over a R.S $M$ are in bijective correspondence with unitary representations of a central extension of the fundamental group. To this end, given a representation, they construct a unitary bundle as follows (and therein lies my confusion): Take a degree one $U(1)$ bundle $Q$. Now, pull this bundle back to the universal cover $\tilde{M}$. The new bundle is a $U(1) \times \pi_1(M)$ bundle over $M$. Now apparently, one can "lift the representation of the central extension and thus form a unitary bundle" (with a connection). Why can one do this?

A (non-Kahler) metric on projectivised vector bundles

2011-09-25T00:55:00Z

Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : Firstly, the metric $h$ induces a Fubini-study metric on the fibres of $\mathbb{P}(E)$ (just take local orthonormal frames to get a smooth fibre bundle, use the usual Fubini-study metric and then pullback to the holomorphic bundle). Secondly, the Chern connection of $h$ induces a splitting $T\mathbb{P}(E) = T\mathbb{C}\mathbb{P}^{r} \oplus TX$. Now put the direct sum metric. My question is: This seems like a very natural construction. Has this been studied before (in the sense of curvature properties etc)? If so, I'd be most grateful if a reference is pointed out.

Mathematical software for Chern-Weil theory

2011-12-12T21:11:26Z

I need to compute curvature forms and Chern-Weil forms for a given metric (in local coordinates) on a vector bundle. Is there are software package that does this? If I manage to compute the derivatives involved is there atleast a package that does the algebra part of it?

A specific degeneration of a rank 2 bundle

2011-11-15T16:19:08Z

I wish to know if there is a rank 2 vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{P}^1$ such that $\mathbb{P}(E)$ when restricted to $\mathbb{P}^1 \times [0:1]$ is the $n$th Hirzebruch surface and when restricted to $\mathbb{P}^1 \times [x:y]$ is the $(n-2)$th Hirzebruch surface.

A vector bundle with a given jumping line

2011-11-03T03:15:19Z

I'd like to know if there exists a holomorphic rank 2 sub-bundle of $T\mathbb{P}^3$ which, when restricted to a given line is $\mathcal{O}(-a)\oplus \mathcal{O}(a)$, but is trivial when restricted to all other lines lying in a plane containing this line (i.e. this line is a jumping line of order $a$).

EDIT: From Angelo's answer, we see that there are no subbundles of the tangent bundle satisfying this property. A related question: Is there a vector bundle that has a given jumping line? (which is not a subbundle of the tangent bundle of course).

Learning Arakelov geometry

2011-10-18T14:29:10Z

I have a complex analytic background (Griffiths and Harris, Huybrechts, Demailley etc). Also, I understand some PDE. I want to learn Arakelov geometry (atleast till the point I can "apply" computations of Bott-Chern forms and Analytic torsion to producing theorems of interest in Arakelov geometry). I know almost nothing of schemes or of number theory. I don't how much of these is needed to learn this stuff. I'd be grateful if any good references/suggestions are pointed out.

Regarding Discrete Eigenvalues

2011-10-14T02:29:36Z

For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete. But, suppose we consider the laplacian (of the usual round (Fubini-Study) metric) on $\mathbb{C}\mathbb{P}^1$. If we restrict ourselves to $\mathbb{C}$ and forget about infinity, then we get the eigenvalue problem : $\Delta f = \frac{1}{(1+r^2)^2} \lambda f$ (where $r$ is the distance from the origin in $\mathbb{C}$). Does this have the same eigenvalues as the original laplacian on the sphere? I mean, does forgetting the "boundary" conditions at infinity add new eigenvalues? In general, when is one allowed to "forget" what happens at infinity?

Holomorphic h-principle for compact manifolds

2011-09-10T06:29:43Z

The Oka principle for Stein manifolds says (roughly) that the only obstructions for "things" are topological obstructions (for instance every smooth complex vector bundle admits a holomorphic structure, etc). Is there a similar principle (atleast in some cases) for compact complex manifolds? Or atleast some version of a h-principle for compact manifolds?

When is a given matrix of two forms a curvature form?

2011-08-22T21:14:58Z

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of $1$-forms $A$, given a (smooth) matrix of two forms $F$ which satisfies the condition $dF =B \wedge F - F \wedge B$ for smooth matrix of one-forms $B$ (i.e. the Bianchi identity is satisfied). Notice that this is true for line-bundles (in fact over any convex open set).

Branched covers of compact Riemann surfaces

2011-07-09T19:47:55Z

Let $S$ be a compact R.S of genus $\geq 2$. In the paper "Stable and unitary vector bundles on compact Riemann surfaces" (by Narasimhan and Seshadri), they claim that there is a branched covering map from the upper half plane to $S$ which is ramified at exactly one point (with index $N$) (i.e. $S$ is the quotient of $\mathbb{H}$ by the group $\langle A_i, B_i, C \vert \Pi [A_i, B_i] = C, [A_i,C] = [B_i, C] = I, C^N=I \rangle$). I am not able to access the reference (Grothendieck) pointed out in that paper. Is there any other reference (or any easy proof of this?) ? Also, does a similar fact hold for non-compact Riemann surfaces (compact minus a finite collection of points).

Quillen metric definition

2011-07-01T16:59:00Z

I am confused as to why a regularised determinant is used in the definition of Quillen's metric on the determinant line bundle defined over the space of $\bar{\partial}$ operators on a (hermitian) vector bundle over a compact Riemann surface. I mean, why not use the metric induced by the vector bundle just by itself? Is the determinant introduced so as to define a smooth metric? For the record I am referring to Quillen's paper "On the determinants of Cauchy-Riemann operators" over a Riemann surface.

Answer by Vamsi for Which mathematicians have influenced you the most?

2011-05-23T23:16:32Z

Simon Donaldson. His proofs involve (to quote wikipedia) a creative use of analysis. I loved his proof of the theorem of Narasimhan and Seshadri.

Mehta-Seshadri and Parabolic bundles

2011-05-10T21:42:25Z

In the original paper of Mehta-Seshadri, it seems like they treat the case of zero parabolic degree (i.e. they prove that zero parabolic degree stable parabolic bundles correspond to irreps of the fundamental group of the Riemann surface minus some points). But, as in the Narasimhan-Seshadri theorem, is the nonzero degree case obtained by considering representations of a central extension of the fundamental group? I'd be grateful if someone may point to a reference.

$\partial \bar{\partial}$ lemma for contractible domains

2011-03-24T20:58:08Z

Is every (p,p) ($p\geq1$) closed form in a contractible open set of $\mathbb{C}^n$, $\partial \bar{\partial}$ exact? We know that every d-exact (p,p) form on a compact Kahler manifold is $\partial \bar{\partial}$ exact (by the Hodge theorem), but unfortunately, that can't be applied here...

Griffiths and Harris reference

2010-12-10T23:52:57Z

Trying to read the section on Poincare duality from Griffiths and Harris is a nightmare. I want to know if there is a place where Poincare duality and intersection theory are done cleanly and rigorously in the order that GH do them (usually, one proves Poincare duality for singular cohomology and then defines the intersection pairing by cup product and proves that (using the Thom isomorphism) that indeed the intersection pairing counts the number of points taken with sign (and convert everything to forms using De Rham's theorem). GH on the other hand define intersection pairing and then proceed further. However, one has to wave their hands at relativistic speeds to make some things work here).

Answer by Vamsi for Why were matrix determinants once such a big deal?

2010-12-22T06:57:54Z

1) The Chern-Weil theory of characteristic classes is built upon determinants of functions of curvature forms of vector bundles. 2) Feynman path integrals require determinants (but typically in infinite dimensions).

Answer by Vamsi for Problems where we can't make a canonical choice, solved by looking at all choices at once

2010-12-21T23:07:43Z

Sard's theorem provides such an example. Given a random smooth map between two manifolds (lets say compact and of the same dimension), there is no canonical way of constructing a regular value. But, Sard's theorem looks at this entire set and proves it is of full measure (and hence there exists atleast one regular value). This result enables us to prove many such results like (the transversality theorem for instance).

Answer by Vamsi for Proofs that require fundamentally new ways of thinking

2010-12-11T19:30:03Z

Proving that subgroups of free groups are free requires the knowledge of topology, a completely different field which a priori does not have anything to do with groups.

Chern classes generating cohomology

2010-11-16T00:29:40Z

The fact that Chern classes are Hodge classes (and are rational combinations of algebraic cycles) is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my question is why is the fundamental class of every algebraic variety a rational combination of them?

A non-elliptic PDE

2010-09-30T20:18:31Z

I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X : $\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial \rho \wedge \bar{\partial}\partial \sigma + \bar{\partial}\partial \sigma \wedge \bar{\partial}\partial \sigma = 0$ where $\sigma$ is a given smooth real function on X. (Note that, one has solutions to this in $\mathbb{C}^2$ if $\sigma$ is real analytic). The problem is that the linearisation of this equation is not elliptic. The motivation for solving this is to produce a trivial line bundle with a hermitian metric so that its Chern character forms (not cohomology classes) are zero.

Answer by Vamsi for How do you decide whether a question in abstract algebra is worth studying?

2010-09-29T23:15:00Z

This paper has a very nice introduction (it is on "pointless topology"). So apparently, one may come up with very random definitions for their own sake and hope someone "applies" them to more "concrete" problems. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014

Comment by Vamsi

2013-04-04T06:21:46Z

For bounded domains in Euclidean space, this is true (Evans' book). It gets tricky when one wants to approximate by smooth functions that smooth upto the boundary.

Comment by Vamsi

2012-10-26T15:48:21Z

This isn't a complete answer, but may help. For smooth u_k, u_k >v is an open set. For non-smooth u, this is not necessarily the case. It may have a boundary. Just outside the boundary max(u,v) = v and inside max(u,v) = u. So, whilst testing the current against test functions whose support "ends" at the boundary, it isn't obvious (to me) that dd^c(max(u,v)) = dd^c (u).

Comment by Vamsi

2012-03-23T22:57:23Z

I apologise for (possibly) misunderstanding your reply, but in my case I require the Monge-Ampere equation to be elliptic. The concavity of the equation (as a function of Hermitian matrices) is under question.

Comment by Vamsi

2012-03-21T23:26:31Z

When n=3 and the manifold is orientable, it always does (orientable three manifolds are parallelisable)

Comment by Vamsi

2011-10-18T19:41:27Z

Thanks for the answer. I also want to know if there are any applications of Analytic torsion outside Arakelov geometry. If not, I guess I would have to learn the scheme stuff....

Comment by Vamsi

2011-10-14T15:48:53Z

This is true, but, what if I don't demand that my eigenfunctions lie in L^2 (I just want them to be smooth functions satisfying the differential equation). I mean, in the example I gave, if one can prove that every eigenfunction has a limit as $r\rightarrow \infty$, then it extends to a function on the sphere and hence the eigenvalues will be the same.

Comment by Vamsi

2011-08-22T22:03:18Z

The Bianchi identity guarantees this ($dtr(F^k) = ktr(dF F^{k-1})=ktr([B,F]F^{k-1})=0$)

Comment by Vamsi

2011-07-10T00:13:08Z

Indeed it is a central extension of the fundamental group.

Comment by Vamsi

2011-07-10T00:11:55Z

Thanks! I apologise for my question, but this won't work for a compact R.S minus a finite number of points would it? If it doesn't, then is there an analogue? Also, would the branching index of $S^{'}$ not be $N$ times the number of sheets in $S_1$?

Comment by Vamsi

2011-07-03T11:20:21Z

Yes Paul, I mean the restriction of the $L^2$ metric.

Comment by Vamsi

2011-05-13T15:46:19Z

Yeah, thanks. I realised that yesterday (Biquard gives us a projective representation of the fundamental group which then is a representation of an extension of the same).

Comment by Vamsi

2011-05-10T23:14:25Z

@Mattia: Thanks. Do you happen to have an online copy of the paper? I can't seem to find it at all?

Comment by Vamsi

2011-05-10T23:12:11Z

@Dmitri, I mean : Let $V$ be a stable bundle on a smooth curve $C$ and suppose the parabolic degree (=deg(E) + sum of weights of the flags) is not zero. Then, is it true that one can naturally associate a representation of the central extension of $\pi_1$ of $C$ minus some points to $V$?

Comment by Vamsi

2011-04-08T17:27:05Z

If you have a good finite open cover, the betti numbers would be finite (I think) by a Mayer-Vietoris argument.

Comment by Vamsi

2011-03-21T20:02:04Z

Actually I don't want them to have infinitely many connected components. I want finitely many components. Thanks for the reference anyway.