Timeline for symplectic form on an algebraic family
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 10, 2017 at 10:36 | comment | added | user21574 | In final let me mention the ${\text{Demailly-Paun conjecture}}$ for the global stability of Kahler structures: Let $X \to S$ be a deformation of compact complex manifolds over an irreducible base $S$. Assume that one of the fibers $X_{t_0}$ is Kahler. Then there exists a countable union $ S'\subset S$ ( $S$ of analytic subsets in the base such that $X_t $ is Kahler for $t ∈ S \setminus S'$ . Moreover, $S$ can be chosen so that the Kahler cone is invariant over $S \setminus S′$ , under parallel transport by the Gauss-Manin connection. annals.math.princeton.edu/2004/159-3/p05 | |
| Nov 19, 2017 at 8:54 | comment | added | user21574 | More comment: If central fibre $X_0$ be Kahler on $π:X→Δ $, then from Kodaira theorem the fibers $X_s$ when $s→0$ are Kahler also, but Kodaira didn't claim that when such $X_s$ could be projective. There are some theorems related it developed by Junyan Cao in his Ph.D. thesis. For instance, if central fiber be compact Kahler Calai-Yau manifold, or tangent bundle $T_{X_0}$ be "nef", or the hermitian metric of $−K_{X_0}$ be semi-positive then $X_s$ are projective when $s→0$(as a sequence) see hal.inria.fr/file/index/docid/749923/filename/Def_pub.pdf | |
| Nov 17, 2017 at 17:01 | comment | added | user21574 | @macbeth : Anyway, I don't think your "Sketch proof" is fine, you may look into the theorem of Kodaira I mentioned in my previous comment and edit your proof. | |
| Nov 17, 2017 at 16:58 | comment | added | user21574 | Moreover, In symplectic geometry we have two notions of Inflation and Deflation which developed by McDuff and V.Shevchishin see Theorem 0.3, and Theorem of 0.4 arxiv.org/pdf/1708.01518.pdf , Let $J$ be an $ω_0-$ tamed almost complex structure on a symplectic 4-manifold $(X,ω_0)$ that admits an embedded $J$-holomorphic curve $C$ with $[C]·[C]>0$. Then there is a family $ω_s$, for $s≥0$, of symplectic forms that all tame $J$ and have cohomology class $$[ω_s] = [ω_0] +sPD([C]),$$ where $PD([C])$ is Poincare dual to $[C]$ | |
| Nov 6, 2017 at 5:09 | comment | added | user21574 | Let $φ : X → B$ be a family of complex manifolds, $0 ∈ B$. If the central fibre $X_0$ is Kahler, then so is $X_b$ for all $b$ sufficiently near $0.$, For proof see Theorem 9.23 of the book of (Claire Voisin Hodge Theory due to Kodaira)wisdom.weizmann.ac.il/~dnovikov/Manifolds5775/… | |
| Oct 31, 2017 at 14:47 | history | edited | macbeth | CC BY-SA 3.0 |
added 649 characters in body
|
| Oct 31, 2017 at 14:24 | history | edited | macbeth | CC BY-SA 3.0 |
deleted 120 characters in body
|
| Oct 31, 2017 at 7:58 | vote | accept | Nick L | ||
| Oct 31, 2017 at 5:26 | comment | added | user21574 | My comments must comeback! (they are very important!)(I stop to work on MO from now until my comments appear) | |
| Oct 31, 2017 at 5:13 | comment | added | S. Carnahan♦ | I have deleted some comments that push the boundaries of professionalism. | |
| Oct 30, 2017 at 23:08 | history | answered | macbeth | CC BY-SA 3.0 |