Foliations on Projective Complete Intersection K3 Surfaces

TL;DR

The paper extends the Campillo-Olivares finding that a foliation on projective space is determined by its singular scheme to complete intersection K3 surfaces. It computes the spaces of foliations with tangent $\mathcal{L}_d = i^{\ast}\mathcal{O}_{\mathbb{P}^n}(1-d)$ on smooth K3 CI surfaces and determines existence thresholds ($d\ge 3$) and explicit dimension formulas depending on the CI type. Using Koszul resolutions and cohomology computations, it then establishes vanishing ranges $H^1(X, \Theta_X \otimes i^{\ast}\mathcal{O}_{\mathbb{P}^n}(1-d))=0$ in specific degrees, yielding that foliations are uniquely determined by their singular schemes for quartics with $d\ge 6$, $(2,3)$ intersections with $d\ge 5$, and $(2,2,2)$ intersections with $d\ge 4$. This advances Poincaré-type questions on foliations to K3 complete intersections and provides a framework potentially useful for foliation classification on such surfaces.

Abstract

We study foliations $\mathscr{F}$ on projective complete intersection K3 surfaces $X \hookrightarrow \mathbb{P}^n$, where $\mathscr{F}$ has isolated singularities and it is the restriction of a foliation of degree $d$ on $\mathbb{P}^n$ that leaves $X$ invariant. We compute the values of the degrees $d$ for which $\mathscr{F}$ is uniquely determined by its singular scheme.

Theorems & Definitions (20)

• Theorem
• Lemma 1.1
• Lemma 1.2: Huybrechts
• Corollary 1.3
• Proposition 1.4
• Proposition 2.1
• proof
• Lemma 2.2
• Theorem 2.3
• proof