K3 surfaces via almost-primes

By Keiji Oguiso. 2001

95 citations recorded by the source corpus.

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard lattices are not isomorphic but their transcendental lattices are Hodge isometric, or equivalently, their derived categories are mutually equivalent. After reviewing some finiteness result, we also give an explicit formula for the cardinality of the isomorphism classes of projective K3 surfaces having derived categories equivalent to the one of X with Picard number 1 in terms of the degree of X.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/15c6e499b0ffba79fbc0c3203776c622e124f51d

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Citations in the corpus

Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces (2004, unknown): “µ O+Λ −→ O(Λ∗/Λ)/{1,−1} →0 ∼=ց ∪ ցπ ↑ W(Λ) O(Λ∗/Λ). Proof of Lemma 27: 1) Part 1 is proven in Lemma 3.8 in [Ma3]. 3) Part 3 follows from Theorem 1.14.2 in [Ni]. 4) This part of the Lemma is proven in [O]. We include a bit more detailed proof here, due to the signiﬁcance of the result for the Torelli”
Some remarks on Brauer groups of K3 surfaces (2004, background): “ntrast to the case when X has an elliptic ﬁbration, not all of the sublattices Γα allow a primitive embedding into ΛK3. In case Γα has a primitive embedding, this embedding is not necessarily unique ([O], [St]). 9.4. Corollary. The lattice Γα has a primitive embedding into ΛK3 if and only if aα = 1 ”
A survey of Torelli and monodromy results for holomorphic-symplectic varieties (2011, unknown): “lication by all elements of t∈ Z/(2n−2)Z, such that t2 = 1. Then O(Λ∗/Λ) is isomorphic to (Z/2Z)r, where ris the number of distinct primes in the prime factorization n− 1 = p d1 1 ···prr of n− 1 (see [Ogu]). The isometry group O(Λ) acts on Λ∗/Λ and the image of O+(Λ) in Aut(Λ∗/Λ) is equal to O(Λ∗/Λ)”
Derived and abelian equivalence of K3 surfaces (2006, methodology): “The proof of this result relies in an explicit counting of all Fourier–Mukai partners in [16] and a counting”
INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE (2006, methodology): “Proof of Corollary 1.7: The group O� acts on M�, by changing the marking, and the period map is O �-equivariant.”
On the monodromy of moduli spaces of sheaves on K3 surfaces II (2003, background): “etry group O[(v⊥)∗/v⊥] is isomorphic to (Z/2Z)ρ, where the Euler number ρof mis the number of distinct primes in the prime factorization m= p e1 1 p e2 2 ···p ρ ρ (and e i are positive integers) (see [O]). We have the natural homomorphism O(v⊥) → O[(v⊥)∗/v⊥]. (73) Lemma 8.2 The homomorphism (73) is ”
Lectures on K3 Surfaces (2016, background): “In fact, in [467] Oguiso shows that for any n > 0 there exist n derived equivalent complex projective K3 surfacesX1, .”
Equivalences of twisted K3 surfaces (2004, background): “More precisely, it has been shown in [28, 32] that for any N there exist pairwise non-isomorphic K3 surfaces X1, .”
The K3 category of a cubic fourfold (2015, background): “such that ρpSq “ 1, usually have non-isomorphic FM-partners, see [52, 56].”
Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections (2009, background): “η:= η(n− 1) is the number of distinct primes p1,...,pη in the prime factorization n− 1 = pe1 1 ···p eη η , with positive integers ei. For n > 2, there are 2η−1 distinct O(eΛ)-orbits in O(Λ,Λ) (see [O] or [Ma5], Lemma 4.3).e PRIME EXCEPTIONAL DIVISORS 5 Theorem 1.6. ([Ma5], Theorem 1.10). An irred”