The Expanding Zoo of Calabi-Yau Threefolds

Table 2

Manifolds with small Hodge numbers and 𝜋₁≠1.

(𝜒,𝑦)	(ℎ^1,1,ℎ^2,1)	Manifold	𝜋1	Reference
(0,24)	(12,12)	𝑋^20,20/ℤ2	ℤ2	[25]
(−16,18)	(5,13)	(Hypersurface in ℙ^1×ℙ^1×dP4)/ℤ2×ℤ2	ℤ2×ℤ2	[26]
(−20,16)	(3,13)	ℙ^1ℙ^1ℙ^1ℙ^2ℙ^2ℙ^2100100010100001100100011010011001011/ℤ3	ℤ3	[23, 24]
(−12,16)	(5,11)	ℙ^1ℙ^1ℙ^1ℙ^3ℙ^21100011000110000111100111/ℤ3	ℤ3	[23, 24]
(0,16)	(8,8)	𝑋^20,20/ℤ3	ℤ3	[25]
(0,16)	(8,8)	(Toric hypersurface 𝑌^20,20)/ℤ3	ℤ3	[37]
(32,16)	(16,0)	(ℙ^7[^2222]^#/ℤ2)	ℤ2	[29]
(−14,15)	(4,11)	ℙ^1ℙ^1ℙ^1ℙ^2ℙ^2ℙ^2ℙ^210000010010000100010001010010001010010010010010100001100/ℤ3	ℤ3	[23, 24]
(−10,15)	(5,10)	ℙ^1ℙ^1ℙ^1ℙ^1ℙ^1ℙ^1ℙ^2ℙ^2ℙ^2100000100010000100001000100000100010000010010001001010100100001010010001001001001/ℤ3	ℤ3	[23, 24]
(−12,14)	(4,10)	(Toric hypersurface 𝑋^8,26)/ℤ3	ℤ3	[37]
(−8,14)	(5,9)	ℙ^1ℙ^1ℙ^1ℙ^2ℙ^2ℙ^2ℙ^2ℙ^210000000010100000001001000000010010010000100100100001001001000011100000000001110/ℤ3	ℤ3	[23, 24]
(−4,14)	(6,8)	ℙ^1ℙ^1ℙ^1ℙ^1ℙ^1ℙ^1ℙ^2ℙ^2ℙ^2ℙ^21000000010010000001000100000100001000001000010000100100100011001000000010010100000100101000000001100/ℤ3	ℤ3	[23, 24]
(0,12)	(6,6)	𝑋^20,20/ℤ4	ℤ4	[25]
(−10,9)	(2,7)	(Hypersurface in dP5×dP5)/ℤ5	ℤ5	[26]
(2,9)	(5,4)	(Toric hypersurface 𝑋^21,16)/ℤ5	ℤ5	[37]
(−4,8)	(3,5)	(Hypersurface in dP4×dP4)/ℤ4×ℤ2	ℤ4×ℤ2	[26]
(0,8)	(4,4)	𝑋^20,20/ℤ6	ℤ6	[25]
(16,8)	(8,0)	(ℙ^7[^2222]^#/{ℤ2×ℤ2,ℤ4})	ℤ2×ℤ2,ℤ4	[29]
(−6,7)	(2,5)	(𝑋^2,52/ℤ10)^#	ℤ5	Section 3.2.1
(−8,6)	(1,5)	(Hypersurface in (ℙ^1)^4)/ℤ8×ℤ2	ℤ8×ℤ2	[26]
(0,6)	(3,3)	𝑋^20,20/{ℤ8,𝑄8}	ℤ8,𝑄8	[25]
(−6,5)	(1,4)	𝑋^8,44/{Dic3,ℤ12}	Dic3,ℤ12	[22]
(−2,5)	(2,3)	(𝑋^8,44/Dic3)^#	𝑆3	Section 3.2.2
(0,4)	(2,2)	𝑋^19,19/{Dic3,ℤ12}	Dic3,ℤ12	[22, 49]
(0,4)	(2,2)	𝑋^20,20/ℤ12	ℤ12	[25]
(8,4)	(4,0)	(ℙ^7[^2222]^#/𝐺),|𝐺|=8	𝐺	[29]
(0,2)	(1,1)	𝑋^20,20/{𝑆𝐿(2,3),ℤ3⋊ℤ8,ℤ3×𝑄8}	𝑆𝐿(2,3),ℤ3⋊ℤ8,ℤ3×𝑄8	[25]
(4,2)	(2,0)	(ℙ^7[^2222]^#/𝐺),|𝐺|=16	𝐺	[29]

This table complements the one in [2], and briefly describes the manifolds which have 𝑦=ℎ^1,1+ℎ^2,1≤24 and nontrivial fundamental group discovered since that paper appeared in 2008. There should still be a number of other manifolds in this region, including quotients from [23] whose Hodge numbers have not yet been calculated, and manifolds obtained from known quotients by hyperconifold transitions [37], of which only a few have so far been written down explicitly. In the “Manifold” column, 𝑋^20,20 denotes the Calabi-Yau toric hypersurface associated to the 24-cell, discussed in [25] and Section 2.2.3, while 𝑋^19,19 refers to the manifold discussed in Section 2.2.2, and 𝑋^8,44 to that in Section 2.2.1. dP_𝑛 is the del Pezzo surface of degree 𝑛. Multiple quotient groups indicate different quotients with the same Hodge numbers. 𝑋^# denotes a singular member of a generically smooth family, while 𝑋 denotes a resolution of a singular variety 𝑋. The column labelled by 𝜋₁ gives the fundamental group. For each manifold listed here there should also be a mirror, which is not listed.