We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).Comment: 11 page
We construct an analog of the classical theta function on an abelian variety for the closed 4-dimensional symplectic manifolds that are T 2 -bundles over T 2 with the zero Euler class. We use our theta functions for a canonical symplectic embedding of these manifolds into complex projective spaces (an analog of the Lefschetz theorem).We proposed in [1] a construction of theta functions on the Kodaira-Thurston manifold. Here we present a different approach to their construction on closed 4-dimensional symplectic manifolds, including T 2 -bundles over T 2 with the zero Euler class.From the geometric viewpoint, the classical theta function on an abelian variety is a section of a holomorphic line bundle over the complex torus. The Lefschetz theorem states that a section of a sufficiently high tensor power of the bundle determines a complex-analytic embedding of the abelian variety into some complex projective space.We aim at generalizing this construction to the bundles whose fibers and bases are 1-dimensional complex tori, while the Euler class is zero. We introduce some analogs of the classical theta functions as sections of complex line bundles over these bundles. We have to relinquish the holomorphic embedding since these bundles, generally speaking, lack not only a Kähler structure, but even a complex structure. Nevertheless, they are symplectic manifolds. We will construct theta functions so that a symplectic analog of the Lefschetz theorem holds for them: theta functions with characteristics, which are sections of tensor powers of the defining line bundle, determine a symplectic embedding of the manifold into CP k (for sufficiently high tensor powers).The Kodaira-Thurston manifold can be viewed as a T 2 -bundle over T 2 in two distinct ways (the bundles are not isomorphic), one of which we consider in the article. Therefore, the theta functions on the Kodaira-Thurston manifold introduced in this article differ by construction from the theta functions of [1], and it turns out that they coincide with the theta functions of Kirwin and Uribe [2], whose approach is representation-theoretic. We discuss this coincidence in more detail in Subsection 4.4.In Section 2 we recall the necessary facts of the classical theory of theta functions and in Section 3 we describe the bundles with which we are to work. In Section 4 we define theta functions on bundles and study some of their properties, in Section 5 we construct an embedding of those bundles into complex projective spaces (Theorem 1), and in Section 6 we prove that the embedding is symplectic (Theorem 2).It would be interesting to find out how far the analogy with the classical theta functions goes; for instance, whether the constructed theta functions are related to number theory (see [3] for instance) or nonlinear equations and secant formulas (see the survey [4]).The author is grateful to I. A. Taimanov for stating the problem and A. E. Mironov for useful discussions.
In this note we prove that QR-submanifolds of the hyper-Kähler manifolds under some conditions admit the G 2 holonomy. We give simplest examples of such QR-submanifolds namely tori.We conjecture that all G 2 holonomy manifolds arise in this way.
Summary Post- COVID syndrome refers to the long-term consequences of a new coronavirus infection COVID-19, which includes a set of symptoms that develop or persist after COVID-19. Symptoms of gastrointestinal disorders in post- COVID syndrome, due to chronic infl ammation, the consequences of organ damage, prolonged hospitalization, social isolation, and other causes, can be persistent and require a multidisciplinary approach. The presented clinical practice guidelines consider the main preventive and therapeutic and diagnostic approaches to the management of patients with gastroenterological manifestations of postCOVID syndrome. The Guidelines were approved by the 17th National Congress of Internal Medicine and the 25th Congress of Gastroenterological Scientifi c Society of Russia.
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