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Math History books
Actually Pedoe's book has several editions. The first edition is called "Geometry for the liberal arts" and the second is called "Geometry and the visual arts". Except for the title the two editions seem to be identical. Our library has a copy only of the first edition but the second edition is still in print so it may be easier to find.
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Exact sequences of bundles on Grassmannians
I may be missing something but it seems that you may be able to get these sequences by applying the standard Schur complex functors (as in section 2.4 of J.Weyman's book "Cohomology of vector bundles and syzygies) to the tautological short exact sequence on the Grassmanian.
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Finite fundamental groups of 3-dimensional Calabi-Yau manifolds
There is a very nice and detailed description of this CY in the Gross-Pavanelli paper arxiv.org/abs/math/0512182. If you have not seen those, you may also want to take a look at this paper arxiv.org/abs/math/0609728 by Borisov-Hua, and the paper arxiv.org/abs/math/0609728 of Bouchard-Donagi.
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Finite fundamental groups of 3-dimensional Calabi-Yau manifolds
There are many possible constructions of the first one: via toric geometry, via elliptic fibrations, via abelian surface fibrations, etc. The elliptic fibration construction is written for instance in my old paper <a href="arxiv.org/abs/hep-th/0410055">http://arxiv.org/abs/…>. You can see there that the group acts freely on the total space but acts with fixed points on the base of the elliptic fibration. The second Calabi-Yau was originally constructed by Gross-Popescu as a pencil of abelian surfaces with polarizations $(1,8)$.
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Finite fundamental groups of 3-dimensional Calabi-Yau manifolds
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Finite fundamental groups of 3-dimensional Calabi-Yau manifolds
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Orthogonal Complements of Root Lattices in E_8
The first time I encountered all this was in the excellent paper: Oguiso, Keiji; Shioda, Tetsuji The Mordell-Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40 (1991), no. 1, 83–99. Among other things they show that both embeddings of all five exceptions do occur as narrow Mordell-Weil lattices of rational elliptic surfaces.
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Orthogonal Complements of Root Lattices in E_8
The original paper is: Dynkin, E. B. Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), 349–462. There is also an English translation: E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras. AMS Translations, volume 6, (1957), 111-244. The part you need is in Table 11 in Chapter II.
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Orthogonal Complements of Root Lattices in E_8
$A_{1}^{\oplus 4}$, ($D_{4}$, $A_{1}^{\oplus 4}$); \linebreak $A_{3}\oplus A_{1}^{\oplus 2}$, ($A_{3}$, $A_{1}^{\oplus 2}\oplus \langle 4\rangle$); \linebreak $A_{3}^{\oplus 2}$, ($A_{1}^{\oplus 2}$, $\langle 4 \rangle^{\oplus 2}$), \linebreak $A_{5}\oplus A_{1}$, ($A_{2}$, $A_{1}\oplus \langle 6 \rangle$); \linebreak $A_{7}$, ($A_{1}$, $\langle 8\rangle$). \linebreak So for the primitive embeddings you get a uniquely determined orthogonal complement.
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Orthogonal Complements of Root Lattices in E_8
According to Dynkin's classification of root sublattices in $E_{8}$, there are only five root subblattices in $E_{8}$ (modulo the Weyl group action) that admit more than one embedding in $E_{8}$. Moreover each of the five admits exactly two non-equivalent embeddings. The five exceptions (and their possible complements) are:
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Non-Kahler manifolds and the dd^c-lemma
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