[Paper Review] Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

David Harvey, Brendan Hassett+1 more|arXiv (Cornell University)|Nov 4, 2010

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper characterizes the homology class of Lagrangian projective 3-spaces in deformations of the Hilbert scheme of length-three subschemes of a K3 surface. Using Hodge theory, representation theory, and the Beauville-Bogomolov form, it proves that the self-intersection of a line in such a plane is −3, and the class of the plane is uniquely determined by the formula $[\mathbb{P}^3] = \frac{1}{48}(\rho^3 + \rho^2 c_2(X))$, where $\rho = 2\ell$ and $\ell$ is the class of a line. This result confirms a conjecture on extremal rational curves in holomorphic symplectic manifolds.

ABSTRACT

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to the Hilbert scheme of length-three subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion.

Motivation & Objective

• To characterize the homology class of Lagrangian projective 3-spaces in deformations of the Hilbert scheme of length-three subschemes of a K3 surface.
• To determine the intersection-theoretic properties of the class of a line in such a Lagrangian plane under the Beauville-Bogomolov form.
• To verify a conjecture that for a $2n$-dimensional holomorphic symplectic manifold deformation equivalent to a Hilbert scheme of a K3 surface, the self-intersection of a line in a Lagrangian $\mathbb{P}^n$ is $-(n+3)/2$.
• To establish a formula expressing the class of the Lagrangian $\mathbb{P}^3$ in terms of $\rho = 2\ell$ and the second Chern class of the manifold.

Proposed method

• Utilizes the Beauville-Bogomolov form on $H^2(X,\mathbb{Z})$ to analyze intersection properties of Lagrangian subvarieties.
• Applies Hodge theory and the monodromy representation to constrain classes that remain of type $(1,1)$ under deformation.
• Employs representation theory to study Hodge classes in the cohomology of Hilbert schemes of K3 surfaces.
• Uses the Poincaré polynomial and symmetric power maps $\mu_{k,n}$ to analyze the ring structure of cohomology.
• Identifies the distinguished absolute Hodge class in the middle cohomology via deformation-invariant properties.
• Employs number-theoretic arguments and $p$-adic analysis to verify the integrality of certain coordinates, ensuring the class formula is well-defined.

Experimental results

Research questions

• RQ1 What is the self-intersection $ (\ell, \ell) $ of a line $\ell$ in a Lagrangian $\mathbb{P}^3 \subset X$, where $X$ is deformation equivalent to the Hilbert scheme of length-three subschemes of a K3 surface?
• RQ2 Can the class $[\mathbb{P}^3]$ be uniquely expressed in terms of $\rho = 2\ell$ and characteristic classes of $X$?
• RQ3 Does the monodromy action on $H^2(X,\mathbb{Z})$ act transitively on classes $\rho$ with $ (\rho, \rho) = -12 $ and $ (\rho, H^2(X,\mathbb{Z})) \subset 2\mathbb{Z} $?
• RQ4 Is the class of the Lagrangian $\mathbb{P}^3$ uniquely determined modulo monodromy by its intersection-theoretic invariants?
• RQ5 Does the formula $[\mathbb{P}^3] = \frac{1}{48}(\rho^3 + \rho^2 c_2(X))$ hold in this deformation class?

Key findings

• The self-intersection of a line $\ell$ in a Lagrangian $\mathbb{P}^3 \subset X$ is $ (\ell, \ell) = -3 $, confirming the conjecture $-(n+3)/2$ for $n=3$.
• The class $\rho = 2\ell$ lies in $H^2(X,\mathbb{Z})$ and satisfies $ (\rho, \rho) = -12 $, with $ (\rho, H^2(X,\mathbb{Z})) \subset 2\mathbb{Z} $.
• The class of the Lagrangian $\mathbb{P}^3$ is given by the formula $[\mathbb{P}^3] = \frac{1}{48}(\rho^3 + \rho^2 c_2(X))$, which uniquely determines it modulo monodromy.
• The monodromy group acts transitively on the set of classes $\rho \in H^2(X,\mathbb{Z})$ with $ (\rho, \rho) = -12 $ and $ (\rho, H^2(X,\mathbb{Z})) \subset 2\mathbb{Z} $, ensuring uniqueness of the class.
• The result confirms the conjecture that for $X$ deformation equivalent to a Hilbert scheme of a K3 surface, the self-intersection of a line in a Lagrangian $\mathbb{P}^n$ is $-(n+3)/2$, with $n=3$ as a key case.
• The proof relies on a number-theoretic result on the integrality of coordinates, verified via $p$-adic analysis and congruence conditions on division points of elliptic curves.