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Register a new userUlrich bundles on K3 surfaces
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Authors
Daniele Faenzi
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We show that any polarized K3 surface supports special Ulrich bundles of rank 2.
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We assume that $mathcal{E}$ is a rank $r$ Ulrich bundle for $(P^n, mathcal{O}(d))$. The main result of this paper is that $mathcal{E}(i)otimes Omega^{j}(j)$ has natural cohomology for any integers $i in mathbb{Z}$ and $0 leq j leq n$, and every Ulrich bundle $mathcal{E}$ has a resolution in terms of $n$ of the trivial bundle over $P^n$. As a corollary, we can give a necessary and sufficient condition for Ulrich bundles if $n leq 3$, which can be used to find some new examples, i.e., rank $2$ bundles for $(P^3, mathcal{O}(2))$ and rank $3$ bundles for $(P^2, mathcal{O}(3))$.
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.
Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $mathcal{E}$ on $Xtimes M$ can be understood as a complete flat family of stable vector bundles on $M$ parametrized by $X$, which identifies $X$ with a smooth connected component of some moduli space of stable sheaves on $M$.
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating $lambda_g$ integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibered rational surfaces. New formulas relating reduced virtual classes on K3 surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds. Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix A, we answer a number of questions about the relationship between the Kiem-Li approach, traditional virtual cycles, and symmetric obstruction theories. The interplay between the boundary geometry of the moduli spaces of curves, K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.
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