Abstract
Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous and such that every point has a symmetry preserving the Hermitian structure. The aim of these notes is to present an introduction to this important class of manifolds, trying to survey the several different perspectives from which Hermitian symmetric manifolds can be studied.
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Notes
- 1.
Cartan’s original notation permutes Type III and Type IV.
- 2.
Usually, one defines \(\mathrm{SO}(2n,\mathbb{C})\) with respect to the standard bilinear symmetric form on \({\mathbb{C}}^{2n} \times {\mathbb{C}}^{2n}\) given by \(x_{1}y_{1} + \cdots + x_{2n}y_{2n}\). However, for our purposes it will be more convenient to use this alternative presentation.
- 3.
Note that \({\mathfrak{s}\mathfrak{o}}^{\mathrm{nc}}(2n)\) is isomorphic to the classical real Lie algebra \({\mathfrak{s}\mathfrak{o}}^{{\ast}}(2n) = \mathrm{Lie}{\mathrm{SO}}^{{\ast}}(2n)\) via the same conjugation map as in (32).
- 4.
As it is seen from Table 5, we could have chosen the (n − 1)-th simple root and we would have gotten an isomorphic (although non conjugate) parabolic subgroup.
- 5.
Note that \({\mathfrak{s}\mathfrak{o}}^{\mathrm{c}}(2n)\) is isomorphic to the classical real Lie algebra \(\mathfrak{s}\mathfrak{o}(2n) = \mathrm{Lie}\mathrm{SO}(2n)\) via the same conjugation map as in (32).
- 6.
Note that \({\mathfrak{s}\mathfrak{p}}^{\mathrm{nc}}(n)\) is isomorphic to the classical real Lie algebra \(\mathfrak{s}\mathfrak{p}(n,\mathbb{R})\) via the same conjugation given in formula (44).
- 7.
The Lie group Sp(n) admits another natural description in terms of matrices with coefficients in \(\mathbb{H}\). Namely, there an isomorphism of Lie group
$$\displaystyle\begin{array}{rcl} & & \qquad \mathrm{Sp}(n)\stackrel{\mathop{\cong}}{\longrightarrow }\mathrm{U}(n,\mathbb{H}):=\{ g \in \mathrm{GL}(n,\mathbb{H})\::\:{ \overline{g}}^{t}g = I_{n}\} {}\\ & & \left (\begin{array}{*{10}c} A &B\\ -\overline{B } & \overline{A} \end{array} \right )\mapsto A - j\overline{B}.{}\\ \end{array}$$
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Acknowledgements
These notes grew up from a Ph.D. course held by the author at the University of Roma Tre in spring 2013 and from a course held by the author at the School “Combinatorial Algebraic Geometry” held in Levico Terme (Trento, Italy) in June 2013. The author would like to thank the students and the colleagues that attended the above mentioned Ph.D. course (Fabio Felici, Roberto Fringuelli, Alessandro Maria Masullo, Margarida Melo, Riane Melo, Paola Supino, Valerio Talamanca) for their patience, encouragement and interest in the material presented. Moreover, the author would like to thank the organizers of the above mentioned School (Giorgio Ottaviani, Sandra Di Rocco, Bernd Sturmfels) for the invitation to give a course as well as all the participants to the School for their interest and feedbacks. Many thanks are due to Jan Draisma for reading a preliminary version of this manuscript and to Radu Laza for suggesting some bibliographical references.
The author is a member of the research center CMUC (University of Coimbra) and he was supported by the FCT project Espaços de Moduli em Geometria Algébrica (PTDC/MAT/111332/2009), by the FCT Project PTDC/MAT-GEO/0605/2012 and by the MIUR project Spazi di moduli e applicazioni (FIRB 2012).
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Viviani, F. (2014). A Tour on Hermitian Symmetric Manifolds. In: Combinatorial Algebraic Geometry. Lecture Notes in Mathematics(), vol 2108. Springer, Cham. https://doi.org/10.1007/978-3-319-04870-3_5
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