[Paper Review] Fourier-Mukai partners and polarised K3 surfaces

Klaus Hulek, David Ploog|arXiv (Cornell University)|Jun 20, 2012

Algebraic Geometry and Number Theory44 references3 citations

TL;DR

This paper establishes a counting formula for the number of Fourier-Mukai (FM) partners of polarized K3 surfaces using lattice-theoretic methods, particularly Nikulin's theory of discriminant forms and Eichler's criterion. It shows that the number of FM partners depends on the Picard rank and polarization type, with explicit formulas derived for low-rank cases and unbounded families in high-rank settings.

ABSTRACT

The purpose of this note is twofold. We first review the theory of Fourier-Mukai partners together with the relevant part of Nikulin's theory of lattice embeddings via discriminants. Then we consider Fourier-Mukai partners of K3 surfaces in the presence of polarisations, in which case we prove a counting formula for the number of partners.

Motivation & Objective

• To extend the theory of Fourier-Mukai partners to K3 surfaces equipped with polarizations, a direction less explored in prior literature.
• To develop a counting formula for the number of FM partners in the presence of polarizations, generalizing the HLY formula for unpolarized K3 surfaces.
• To analyze the interplay between polarizations, transcendental lattices, and isometry groups in the context of derived equivalence.
• To investigate how the number of FM partners varies with Picard rank and polarization type, especially in high-rank and unimodular orthogonal complement cases.

Proposed method

• Utilizes derived categories and the derived Torelli theorem for K3 surfaces to relate FM partners to Hodge isometries of cohomology lattices.
• Applies Nikulin's theory of lattice embeddings via discriminant forms to classify embeddings of transcendental lattices into the K3 lattice $L_{2d}$.
• Employs Eichler's criterion to determine the number of $ ilde{O}(L_{2d})$-orbits of vectors with given norm and divisor, crucial for counting non-equivalent embeddings.
• Analyzes the action of $O(D_{L_{2d}})$ on these orbits to compute the number of $O(L_{2d})$-orbits, accounting for symmetries.
• Considers the orthogonal complement of the transcendental lattice in $L_{2d}$, distinguishing cases where it is $2E_8(-1)$ or $D_{16}^+(-1)$ to show at least two non-equivalent embeddings exist.
• Uses the structure of the discriminant group $D_{L_{2d}} \cong \mathbb{Z}/2p^3\mathbb{Z}$ and its automorphism group $\mathbb{Z}/2\mathbb{Z}$ to bound the number of orbits under $O(L_{2d})$.

Experimental results

Research questions

• RQ1 How does the presence of a polarization affect the number of Fourier-Mukai partners of a K3 surface?
• RQ2 What is the precise counting formula for FM partners of polarized K3 surfaces, and how does it differ from the unpolarized case?
• RQ3 Under what conditions do multiple non-isomorphic FM partners exist for a given polarized K3 surface?
• RQ4 How does the Picard rank influence the finiteness and size of the set of FM partners?
• RQ5 Can the number of FM partners be unbounded, and if so, under what lattice-theoretic conditions?

Key findings

• For a K3 surface with Picard rank 1 and polarization of degree $2d$, the number of FM partners is $2^{p(d)-1}$, where $p(d)$ is the number of prime divisors of $d$.
• When the transcendental lattice is $T = \langle 2a, 2b \rangle$ with $a > b > 0$, and $d = b = p^3$ for prime $p \equiv 1 \pmod{4}$, there are at least $p$ pairwise non-equivalent embeddings of $T$ into $L_{2d}$, leading to at least $p$ non-isomorphic FM partners.
• For $d = p^3$, the number of $\tilde{O}(L_{2p^3})$-orbits of vectors of norm $2p^3$ and divisor $p^2$ is $2p$, arising from solutions to $1 + c^2 \equiv 0 \pmod{p}$.
• The action of $O(D_{L_{2p^3}}) \cong \mathbb{Z}/2\mathbb{Z}$ on these orbits implies that the number of $O(L_{2p^3})$-orbits is at least $p$, showing unbounded growth in the number of FM partners as $p$ increases.
• When the orthogonal complement of $T$ in $L_{2d}$ is unimodular (e.g., $2E_8(-1)$ or $D_{16}^+(-1)$), there are at least two non-equivalent embeddings of $T$, implying at least two FM partners.
• For Picard rank $\geq 12$, derived equivalence implies isomorphism, but multiple non-isomorphic polarizations can still exist on the same surface, leading to distinct FM partners.