Stable Ulrich bundles on cubic fourfolds

TL;DR

This paper characterizes when Ulrich bundles exist on cubic fourfolds, constructs large families of such bundles, and explores related geometric structures including holomorphic symplectic manifolds and special surfaces.

Contribution

It provides necessary and sufficient conditions for Ulrich bundles on cubic fourfolds, constructs explicit examples, and links these bundles to new geometric structures.

Findings

Existence conditions for Ulrich bundles of any rank on cubic fourfolds.

Construction of a 19-dimensional family of holomorphic symplectic manifolds.

Identification of special surfaces in cubic fourfolds not arising as intersections.

Abstract

In this paper, we give necessary and sufficient conditions for the existence of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we show that for every integer $r\ge 2$ there exists a family of indecomposable rank $r$ Ulrich bundles on the certain cubic fourfolds, depending roughly on $r$ parameters, and in particular they are of wild representation type; special surfaces on the cubic fourfolds are explicitly constructed by Macaulay2; a new $19$-dimensional family of projective ten-dimensional irreducible holomorphic symplectic manifolds associated to a certain cubic fourfold is constructed; and for certain cubic fourfold $X$, there exist arithmetically Cohen-Macaulay smooth surface $Y \subset X$ which are not an intersection $X \cap T$ for a codimension two subvariety $T \subset \Bbb P^5$.