Timeline for On $\pi_1$ of an algebraic surface
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2012 at 15:51 | comment | added | kla | @Tim Yes, this is exactly my question! | |
| Jul 30, 2012 at 14:48 | comment | added | Tim Perutz | kla: Do I understand right that the question is as follows? Let $X$ be an algebraic surface of general type, and $\omega$ a Kaehler form in class $K_X$. Suppose that $D$ is a connected, oriented surface embedded in $X$ such that $\omega|_D>0$. Assume that the homology class $[D]$ is Poincare dual to $c_1(L)$, where $L$ is an ample holomorphic line bundle (so in particular, ($D \cdot D > 0$). Must $D$ then carry all of $\pi_1(X)$? | |
| Jul 30, 2012 at 12:18 | comment | added | kla | @Misha My question was about the symplectic submanifolds of Kahler surface. I think I made this very clear in my post. | |
| Jul 30, 2012 at 11:09 | comment | added | Misha | ... At least, replace Kahler with "almost Kahler". | |
| Jul 30, 2012 at 11:06 | comment | added | Ben McKay | Is the line bundle a holomorphic line bundle? | |
| Jul 30, 2012 at 11:04 | comment | added | Misha | ... In particular, remove integrability assumption and Kahler assumption. I am also unsure what "general type" would mean in the setting suggested by BS. | |
| Jul 30, 2012 at 9:45 | history | edited | kla | CC BY-SA 3.0 |
deleted 3 characters in body
|
| Jul 30, 2012 at 9:30 | comment | added | BS. | You should edit your question, then. | |
| Jul 30, 2012 at 7:16 | history | edited | kla |
edited tags
|
|
| Jul 29, 2012 at 23:34 | history | edited | kla | CC BY-SA 3.0 |
deleted 16 characters in body
|
| Jul 29, 2012 at 17:43 | comment | added | kla | Yes, BS that is what exactly I am asking. THANK YOU! According to the page 389 of Gompf and Stipsicz an embedded surface $\Sigma$ is a symplectic submanifold if and only if there is a compatiable almost complex structure $J$ such that $\Sigma$ is pseudo-holomorphic. I know there are examples of Kahler surfaces such as $\mathbb{T} \times \mathbb{T}$ which contain a symplectic but non-complex connected curves. For example, there is a connected symplectic surface which represent the homology class $n(\mathbb{T} \times {pt})$ (for any $n \geq 2$), but there exist no such holomorphic curve. | |
| Jul 29, 2012 at 17:21 | comment | added | BS. | I think the question is badly phrased. Maybe the OP asks if a connected symplectic surface $D$ in $X$ whch is Poincaré dual to $c_1(L)$ with ample $L$ and $c_1(L)^2>0$ still verifies Nori's theorem. Am I right ? | |
| Jul 29, 2012 at 16:56 | comment | added | Misha | @kla: Since your almost complex structure is assumed to be integrable, your pseudoholomorphic curve is a complex curve. Hence, you are done by Nori's theorem. | |
| Jul 29, 2012 at 16:28 | comment | added | kla | @Jason I am using the definition on page 389 of R. Gompf and A. Stipsicz "4-Manifolds and Kirby calculus". It defines pseudo-holomorphic submanifold on the almost-complex 4-manifold $X$ as a real 2-dimensional submanifold $\Sigma$ such that if $J$ maps the tangent bundle of $\Sigma$ into itself. If you look the same page, it remarks that if $X(\omega, J, g)$ is an almost-Kahler manifold then a pseudo-holomorphic submanifold $\Sigma$ is alwyas a symplectic submanifold. I didn't know that pseudo-holomorphic should be holomorphic as you remark (in Kahler case). | |
| Jul 29, 2012 at 15:49 | comment | added | Jason Starr | @kla: "Dear Jason: In my question above, I assume that the manifold X is a complex surface of general type." In that case, what is your distinction between "pseudo-holomorphic" and "holomorphic"? A pseudo-holomorphic curve in a <complex> manifold is holomorphic. Once again, I think you should tell us what reference you are using, since you seem to be confused about basic definitions. | |
| Jul 29, 2012 at 14:48 | history | edited | kla |
edited tags
|
|
| Jul 29, 2012 at 13:48 | history | edited | kla | CC BY-SA 3.0 |
added 133 characters in body
|
| Jul 29, 2012 at 13:29 | comment | added | kla | Dear Jason: In my question above, I assume that the manifold $X$ is a complex surface of general type. | |
| Jul 29, 2012 at 13:15 | comment | added | Jason Starr | Dear kla: Could you please give a reference to an article or book that defines "Kaehler form" for a manifold which is not a complex manifold? I am concerned that you are misunderstanding the term "Kaehler form". | |
| Jul 29, 2012 at 12:33 | history | edited | kla | CC BY-SA 3.0 |
deleted 2 characters in body
|
| Jul 29, 2012 at 12:05 | comment | added | kla | Dmitri@ I consider a compatiable almost-complex structure $J$ on $(X, \omega)$, where $\omega$ is a Kahler form. The submanifold $D$ of $X$ is a pseudo-holomorphic if $J$ maps the tangent bundle of $D$ into itself. In my question above, you can assume $D$ is a symplectic submanifold of $X$. I consider the symplectic structure given by the Kahler form. | |
| Jul 29, 2012 at 11:31 | comment | added | Dmitri Panov | Could you please define what you mean by a pesudo-holomorphic curve in your case? In order to do this, you need to define an almost complex structure on your Kahler surface, but have not explained this. Note also, that any Kahler manifold with an ample bundle is projective thanks to Kodaira embedding theorem. | |
| Jul 29, 2012 at 10:23 | history | asked | kla | CC BY-SA 3.0 |