Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1006.33004
Majima, Hideyuki; Matsumoto, Kenji; Takayama, Nobuki
Quadratic relations for confluent hypergeometric functions. (English)
[J] Tohoku Math. J., II. Ser. 52, No.4, 489-513 (2000). ISSN 0040-8735

Let $ \omega$ be a $ 1 $-form on $ \Bbb{P}^1$ with poles $x=\{ x_1, \dots, x_m \}$ of order $ n_1, \dots, n_m$ $(n = n_1 + \cdots + n_m)$, and let $ \nabla_{\pm\omega} = d \pm \omega \wedge $ be the integrable connections on $ X=\Bbb{P}^1 \setminus x$. Consider the twisted cohomology groups $ H^1(\Omega^\bullet(x),\nabla_{\pm\omega}) $ for the complexes $\nabla_{\pm\omega}:\Omega^\bullet(x) \to \Omega^\bullet(x) $, where $ \Omega^k(x) $ is the vector space of rational $ k$-forms with poles at most at $x$ and consider $ u(t) = c \exp (\int^t \omega)$ which satisfies $\nabla_{ - \omega}u(t)=0 $. An integral $ \langle \varphi, \gamma \rangle= \int_\gamma u(t)\varphi $ for some $ \varphi \in H^1(\Omega^\bullet(x),\nabla_\omega) $ is called the hypergeometric integral, where $ \gamma $ is an element of twisted homology group $ H_1(C_\bullet^{\omega}, \partial_\omega)$, which is proved to be dual to $H^1(\Omega^\bullet(x),\nabla_\omega) $ by the pairing $ \langle\ , \ \rangle $. The authors introduce the cohomological intersection pairing for $ (n-2)$-dimensional vector spaces $H^1(\Omega^\bullet(x),\nabla_{\omega}) $ and $ H^1(\Omega^\bullet(x),\nabla_{-\omega}) $, and the homological intersection pairing for $ H_1(C_\bullet^{\omega}, \partial_{\omega}) $ and $ H_1(C_\bullet^{-\omega}, \partial_{-\omega}) .$ For a choice of bases $\{ \varphi^\pm_\mu\}_\mu$ and $\{ \gamma^\pm_\mu\}_\mu$ of the groups $H^1(\Omega^\bullet(x),\nabla_{\pm\omega}) $ and $ H_1(C_\bullet^{\pm\omega},\partial_{\pm \omega}) $, define the four matrices of size $ n-2:$ $$ \Pi^+ = \langle \varphi^+_\mu, \gamma^+_\nu \rangle_{\mu, \nu}, \quad \Pi^- = \langle \varphi^-_\mu, \gamma^-_\nu \rangle_{\mu, \nu}, \quad I_{\text{ch}}= \langle \varphi^+_\mu, \varphi^-_\nu \rangle_{\mu, \nu}, \quad I_{\text{h}}= \langle \gamma^+_\mu, \gamma^-_\nu \rangle_{\mu, \nu}. $$ The main result of this paper is the following. \par Theorem. We have twisted period relations: $$ \Pi^+ {}^t I_{\text{h}}^{-1} {}^t\Pi^- = I_{\text{ch}} $$ which give quadratic relations among confluent hypergeometric integrals.
[H.Kimura (Kamamoto)]

MSC 2000:: *33C15 Confluent hypergeometric functions

Keywords: period relation; confluent hypergeometric function; intersection theory

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]