Fake quadrics from irreducible lattices acting on the product of upper half planes

Džambić, Amir

doi:10.1007/s00208-014-1031-9

Cited by 8 publications

(17 citation statements)

References 26 publications

(70 reference statements)

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“…Irreducible lattices: here π 1 (X) ⊂ PSL 2 (R) × PSL 2 (R) is an irreducible quaternionic lattice of suitable covolume, see Kuga and Shavel [Sh78], and Džambić [Dz12]. It remains an open question very much in the spirit of the motivation above whether there is a complex fake quadric homeomorphic to CP 1 × CP 1 or at least with finite or trivial π 1 , see Hirzebruch [Hi87] subproblem of Problem 25 on page 779.…”

Section: (B)mentioning

confidence: 99%

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Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

Stix¹,

Vdovina²

2017

Math. Proc. Camb. Phil. Soc.

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We construct an infinite series of simply transitive irreducible lattices in PGL 2 (F q ((t))) × PGL 2 (F q ((t))) by means of a quaternion algebra over F q (t). The lattices depend on an odd prime power q = p r and a parameter τ ∈ F * q , τ = 1, and are the fundamental group of a square complex with just one vertex and universal covering T q+1 × T q+1 , a product of trees with constant valency q + 1.Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over F q ((t)) with ample canonical class, Chern ratio c 2 1 / c 2 = 2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski-Euler characteristic attains its minimal value χ = 1: the surface is a non-classical fake quadric.

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Section: (B)mentioning

confidence: 99%

“…All known examples of fake quadrics X are C-analytically uniformized by H × H, the product of two copies of the upper half plane H, so that π 1 (X ) ⊂ PSL 2 (R) × PSL 2 (R) is a cocompact lattice. [Sh78] and Džambić [Dz14]. The covolume can be computed by Prasad's volume formula, see [Pr89].…”

Section: •1 Fake Quadricsmentioning

confidence: 99%

Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

Stix¹,

Vdovina²

2017

Math. Proc. Camb. Phil. Soc.

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“…There, the author concentrates on arithmetic lattices Γ ⊇ Γ 1 O which are defined by quaternion algebras over real quadratic fields of class number one. More recently, in [7], more examples of quaternionic quadrics associated with quaternion algebras over quadratic fields have been found. In this section we will list all known examples of quaternionic fake quadrics together with their automorphism groups.…”

Section: Quaternionic Fake Quadrics With Non-trivial Automorphism Gromentioning

confidence: 99%

Automorphisms and quotients of quaternionic fake quadrics

Džambić

Roulleau

2014

Pacific J. Math.

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Abstract. A fake quadric is a smooth surface of general type with the same invariants as the quadric in P 3 , i.e. c 2 1 = 8, c2 = 4 and q = pg = 0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using quaternion algebras over real quadratic number fields. We provide examples of quaternionic fake quadrics X with a non-trivial automorphism group and compute the invariants of the minimal desingularisation of the quotient of X by this group. In that way we obtain minimal surfaces Z of general type with q = pg = 0 and K 2 = 4 or 2 which contain the maximal number of disjoint (−2)-curves. We then prove that if a surface of general type has the same invariant as Z and same number of (−2)-curves, then we can construct geometrically a surface of general type with c 2 1 = 8, c2 = 4.

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“…Two-dimensional fake A 1 are also known as fake quadrics; see Hirzebruch [10, page 779f]. There are many known irreducible as well as nonirreducible fake quadrics (see for instance Shavel [19] or the author [8] for the irreducible case and Bauer, Catanese and Grunewald [1] for the nonirreducible case). It is known that no fake products of projective lines of odd dimension are possible.…”

Section: Introductionmentioning

confidence: 99%

Varieties of general type with the same Betti numbers as ℙ¹× ℙ¹×⋯× ℙ¹

Džambić

2015

Geom. Topol.

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We study quotients nH n of the n-fold product of the upper half-plane H by irreducible and torsion-free lattices < PSL 2 .R/ n with the same Betti numbers as the n-fold product .P 1 / n of projective lines. Such varieties are called fake products of projective lines or fake .P 1 / n. These are higher-dimensional analogs of fake quadrics. In this paper we show that the number of fake .P 1 / n is finite (independently of n), we give examples of fake .P 1 / 4 and show that for n > 4 there are no fake .P 1 / n of the form nH n with contained in the norm-one group of a maximal order of a quaternion algebra over a real number field. 11F06, 22E40

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Fake quadrics from irreducible lattices acting on the product of upper half planes

Cited by 8 publications

References 26 publications

Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

Automorphisms and quotients of quaternionic fake quadrics

Varieties of general type with the same Betti numbers as ℙ¹× ℙ¹×⋯× ℙ¹

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Fake quadrics from irreducible lattices acting on the product of upper half planes

Cited by 8 publications

References 26 publications

Simply transitive quaternionic lattices of rank 2 overq(t) and a non-classical fake quadric

Simply transitive quaternionic lattices of rank 2 overq(t) and a non-classical fake quadric

Automorphisms and quotients of quaternionic fake quadrics

Varieties of general type with the same Betti numbers as ℙ1× ℙ1×⋯× ℙ1

Contact Info

Product

Resources

About

Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

Simply transitive quaternionic lattices of rank 2 over_q(t) and a non-classical fake quadric

Varieties of general type with the same Betti numbers as ℙ¹× ℙ¹×⋯× ℙ¹