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Многомерные вычеты и тропическая геометрия
15 июня 2021 г. 14:30–15:30, Секция II, г. Сочи
 


Holomorphic continuation of a formal series along analytic curves

A. S. Sadullaev

National University of Uzbekistan named after M. Ulugbek, Tashkent
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A. S. Sadullaev


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Аннотация: This talk is devoted to a curvilinear analogue of the well-known Forelli theorem [1]: if a function $f$ is infinitelly smooth in a neighborhood of the origin $0 \in \mathbb C^{n} $ $f\in C^{\infty }\{0\},$ and for every complex line $l$ passing through the origin the restriction $f|_{l} $ continues holomorphically into the unit disk $l\bigcap B(0,1),$ then $f$ continues holomorphically into the unit ball $B(0,1) \subset {\mathbb C}^{n}.$
An example of a function
$$ f(z_{1} ,z_{2} )=\frac{z_{1}^{k+1} \bar{z}_{2} }{z_{1} \bar{z}_{1} +z_{2} \bar{z}_{2} } \in C^{k} ({\mathbb C}^{2} ) $$
shows that the condition of infinite smoothness in Forelli's Theorem is essential. The restrictions $f|_{l} $ to complex lines $l \ni 0$ are polynomials, but $f(z_{1} ,z_{2} )$ is not holomorphic.
The following takes place
Theorem 1. Let the unit ball $B(0,1) \subset\mathbb{C}^{n} $ be fibered by a smooth family of analytic curves $A_{\lambda } =ż=p_{\lambda } (\xi )\}, \lambda \in \mathbb{P}^{n-1},$ at the point $0$, where $p_{\lambda } (\xi )=(p_{\lambda }^{1} (\xi ),p_{\lambda }^{2} (\xi ),...,p_{\lambda }^{n} (\xi ))$ is a holomorphic vector function in the unit disk $U= \{|\xi |<1 \}:$ $p_{\lambda } (\xi )=a_{1} (\lambda )\xi +a_{2} (\lambda )\xi ^{2} +..., a_{k} (\lambda )\in C^{1} (\mathbb{C}^{n}),    k=1,2,...,        B(0,1) =\bigcap _{\lambda }A_{\lambda }^ .$ If a function $f\in C^{\infty }\{0\}$ has the property that each restriction $f|_{A_{\lambda } } ,    \lambda \in \mathbb{P}^{n-1} ,$ that is defined in the neighborhood of $0,$ holomorphically continues to the whole $A_{\lambda },$ then $f$ continues holomorphically to $B(0,1).$
Theorem 1 in the following version is also true under a weaker requirements.
Theorem 2. Under the conditions of Theorem 1, if each restriction $f|_{A_{\lambda } } ,    \lambda \in W \subset \mathbb{P}^{n-1} ,$ holomorphically continues to the whole $A_{\lambda },$ then $f$ continues holomorphically to the domain $\hat {O}=ż\in {\mathbb C}^{n} :      | z| \exp V^{*}(\frac{z}{| z | } ,  O)<1\}.$ Here $W \neq \emptyset$ is an open subset of $\mathbb{P}^{n-1},$ ${O} =\bigcap _{\lambda \in W }A_{\lambda }^ , $ $V^{*}(\omega ,  {O} )$ is the Green's function in ${\mathbb C}^{n}.$
In the work [2] Chirka showed the validity of the curvilinear analogue of Forelli’s theorem for $n=2$. Further advances on variations of the Forelli's theorem, were obtained in the works Kim et al. [3-5].

Материалы: Azimbay Sadullaev's slides.pdf (105.0 Kb)

Язык доклада: английский

Website: https://zoom.us/j/9544088727?pwd=RnRYeUcrZlhoeVY3TnRZdlE0RUxBQT09

Список литературы
  1. F. Forelli, “Pluriharmonicity in terms of harmonic slices”, Math. Scand., 41:2 (1977), 358–364  mathscinet
  2. E. M. Chirka, “Variations of Hartogs' Theorem”, Complex analysis and applications, Collected papers, Trudy MIAN, 253, Nauka, MAIK «Nauka/Inteperiodika», M., 2006, 232–240  mathnet  mathscinet  elib; Proc. Steklov Inst. Math., 253 (2006), 212–220  crossref  scopus
  3. K.-T. Kim, E. Poletsky and G. Schmalz, “Functions holomorphic along holomorphic vector fields”, J. Geom. Anal., 19:3 (2009), 655–666  mathscinet
  4. J.-C. Joo, K.-T. Kim and G. Schmalz, “A generalization of Forelli's theorem”, Math. Ann., 355:3 (2013), 1171–1176  mathscinet
  5. Y.-W. Cho, K.-T. Kim, Functions holomorphic along a $C^1$-pencil of holomorphic discs, the presentation, arXiveMath, unpublished, 2020


* ID: 954 408 8727, password: residue

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