Timeline for symplectic form on an algebraic family
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Dec 6, 2017 at 10:38 | comment | added | user21574 | End of comment: A degeneration$X\to \Delta$ give you a family of fibers $X_t$ parametrized by the base $\Delta$. Also in some books, family $(X_t,L_t)$ are defined by restricting on $X\times \{t\}$ where $t\in \Delta$ . In my all comments, I have assumed you mean a family via degeneration parametrized by the base which is curve. | |
| Dec 6, 2017 at 5:51 | comment | added | user21574 | My conjecture is that on degeneration of Calabi-Yau fibers , the relative form (semi-flat form introduced by Vafa-Yau ) is semi-positive in the sense of Daniel Barlet, i.e it is strongly Lelong positive. see arxiv.org/pdf/1705.01743.pdf | |
| Dec 6, 2017 at 5:49 | comment | added | user21574 | .....We shall say that a $π$−relative continuous $(q, q)$−form $ω$ on $X$ is strongly Lelong positive at $x_0$ if the hermitian form on $Λ^q(U)$ defined by$ ω$ is a positive hermitian form at each point of $p^{-1}(x_0)$ . For any $q$−plane $P$ in the Zariski tangent space $T_{X,x_0}$ which is vertical (i.e. contained in the kernel of $T_{π,x_0}$ ) then $ω_{x_0}[v_1 ∧ · · · ∧ v_q] > 0$ when $v_1, . . . , v_q$ is a basis of $P$ | |
| Dec 6, 2017 at 5:43 | comment | added | user21574 | For the relative forms, we have a new notion of positivity introduced by Daniel Barlet. Let me explain it here and state my conjecture.Let $π: X → S$ be a surjective holomorphic map between two irreducible complex spaces. Let $ p : Gr_q(X/S) → X$ be the Grassmannian of $q$−planes in $T_{X/S}$, $U$ the universal $q$−vector bundle on $Gr_q(X/S)$, and $θ : Λ^q(U) → Gr_q(X/S)$ the line bundle which is the determinant of $U$. A $π$−relative continuous$ (q, q)$−form on $X $ defines a continuous hermitian form on $Λ^q(U).$ | |
| Oct 31, 2017 at 7:58 | vote | accept | Nick L | ||
| Oct 31, 2017 at 5:08 | comment | added | S. Carnahan♦ | @HassanJolany In general, if you feel like writing a long sequence of comments, you should consider writing an answer instead. In fact, it might be better to write an answer to this question, and delete these comments. | |
| Oct 31, 2017 at 1:00 | comment | added | user21574 | My letter to Gang Tian can give more effectiveness of it hal.archives-ouvertes.fr/hal-01551080/document | |
| Oct 31, 2017 at 0:07 | comment | added | user21574 | positivity of such fiberwise Kahler-Einstein metric is highly non-trivial. It is still open question. When fibers are of general type there is a paper of Schumacher,in Invent math(with two erratam ) but at the moment I couldn't yet accept his proof. So such forms are not Kahler in general . In my point of view Kontsevich-Soibelman program (which they have copy pasted from Barlet formula(in Invent Math) and Grifiths)is equivalent with positivity of such semi ricci flat forms . See mathoverflow.net/questions/261281/… | |
| Oct 30, 2017 at 23:50 | comment | added | user21574 | references: Schumacher, Georg: Moduli of polarized K¨ahler manifolds, Math. Ann. 269, 137–144 (1984). and Fujiki, Akira; Schumacher, Georg: The moduli space of extremal compact K¨ahler manifolds and generalized Weil-Petersson metrics. Publ. Res. Inst. Math. Sci. 26, 101–183 (1990). | |
| Oct 30, 2017 at 23:36 | comment | added | user21574 | Closedness of $(1,1)$− current when you extend from fibers to whole of total space in singular setting is more complicated see Theorem 1.26, of thichthichiu.files.wordpress.com/2011/07/… . | |
| Oct 30, 2017 at 23:29 | comment | added | user21574 | Note that such fiberwise Kahler metric is not in $c_1(TX)$, it is in $c_1(K_{X/Y})$. i.e. relative first Chern class $c_1(K_{X/Y})=-dd^c\log \Omega_{X/Y}$ where $\Omega_{X/Y}$ is relative volume form | |
| Oct 30, 2017 at 23:24 | comment | added | user21574 | But $\omega_{X/Y}$ on $\pi:X\to Y$ may no longer be closed and we need to finite mass to get closedness (due to being non-smooth) . But the induced of fibers are K\"ahler due to Kodaira-Spencer map | |
| Oct 30, 2017 at 23:23 | comment | added | user21574 | As remark, fiberwise Ricci flat metric and in general fiberwise Kahler-Einstein metrics are strong relative K\"ahler form. In fact any fiberwise K\"ahler-Einstein metric can be introduced locally by this definition. | |
| Oct 30, 2017 at 23:20 | comment | added | user21574 | A weak relative K\"ahler form is given in a similar way, where the $u_j-u_k$ need only be harmonic on the fibers $X_s\cap U_j\cap U_k$ ( $\partial_s$ and $\bar{\partial_s}$ denote derivatives in fiber direction) | |
| Oct 30, 2017 at 23:20 | comment | added | user21574 | Let $\pi:X \to S$ be a family of complex compact manifolds. Then a (strong) relative K\"ahler form $\omega_{X/S}$ is a relative $(1,1)$-form , which is given by $\omega_{X/S}|_{U_j}=\sqrt[]{-1}\partial_s\bar{\partial_s}u_j$ with respect to a suitable open covering $\{U_j\}$ of $X$. The functions $u_j$ have to be strictly plurisubharmonic on $U_j\cap X_s$, $s\in S$ and $u_j-u_k$ are harmonic on $U_j\cap U_k$. | |
| Oct 30, 2017 at 23:19 | comment | added | user21574 | You are facing with the two different notion of relative Kahler metrics introduced by Fujiki and others (Koiso and before him another person(I forgot his name a Japanese mathematician )) | |
| Oct 30, 2017 at 23:08 | answer | added | macbeth | timeline score: 1 | |
| Oct 30, 2017 at 18:37 | history | asked | Nick L | CC BY-SA 3.0 |