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The 27th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications
August 13, 2019 16:30–17:00, Section II, Krasnoyarsk, Siberian Federal University
 


The Hartogs phenomenon in toric varietries

A. V. Shchuplev

Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk
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Abstract: We say that a complex space $(X,\,\mathcal O)$ admits the Hartogs phenomenon if for any compact subset $K$ of $X$ such that $X\setminus K$ is connected, a restriction homomorphism
$$ H^0(X,\,\mathcal O) \rightarrow H^0(X\setminus K,\,\mathcal O) $$
is an isomorphism.
In toric varieties this phenomenon has been explored by M. Marciniak [1] who related it to properties of corresponding fans:
Theorem. If $X_\Sigma$ is a smooth toric surface with a strictly convex fan $\Sigma$ then $X_\Sigma$ admits the Hartogs phenomenon.
She has also formulated a conjecture for toric varieties of higher dimensions: A smooth toric variety $X_\Sigma$ admits the Hartogs phenomenon if the complement of its fan $\Sigma$ has at least one concave connected component.
We were able to prove it not only for smooth but also for normal toric varieties. Let $X_\Sigma$ be a normal toric variety corresponding to a fan $\Sigma\subset \mathbb R^d=\mathbb Z^d\otimes_{\mathbb Z}\mathbb R$. We shall say that a connected component of $\mathbb R^d\setminus |\Sigma|$ is concave if its convex hull coincides with $\mathbb R^d$.
Theorem. Let $X_\Sigma$ be a normal toric variety. If the complement of its fan $\Sigma$ has at least one concave connected component then $X_\Sigma$ admits the Hartogs phenomenon.
The proof follows from the study of Dolbeault cohomology with compact support of a smooth toric variety where $X_\Sigma$ can be equivariantly embedded.
This is a joint work with S. Feklistov.

Language: English

References
  1. Malgorzata Aneta Marciniak, Holomorphic extensions in toric varieties, http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3365035, ProQuest LLC, Ann Arbor, MI, 2009 , 147 pp.  mathscinet
 
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