## Tokyo Journal of Mathematics

### Birational Maps of Moduli Spaces of Vector Bundles on $K3$ Surfaces

#### Abstract

In this note, we construct a birational map of a moduli space of stable sheaves on a $K3$ surface induced by a reflection functor.

#### Article information

Source
Tokyo J. of Math. Volume 34, Number 2 (2011), 473-491.

Dates
First available in Project Euclid: 30 January 2012

https://projecteuclid.org/euclid.tjm/1327931397

Digital Object Identifier
doi:10.3836/tjm/1327931397

Mathematical Reviews number (MathSciNet)
MR2918917

Zentralblatt MATH identifier
1236.14013

#### Citation

KIMURA, Masanori; YOSHIOKA, Kōta. Birational Maps of Moduli Spaces of Vector Bundles on $K3$ Surfaces. Tokyo J. of Math. 34 (2011), no. 2, 473--491. doi:10.3836/tjm/1327931397. https://projecteuclid.org/euclid.tjm/1327931397.

#### References

• Ballico, E. and Chiantini, L., On some moduli spaces of rank $2$ bundles over $K3$ surfaces, Boll. Un. Mat. Ital. A (7), 7 (1993), 279–287.
• Costa, L., $K3$ surfaces: moduli spaces and Hilbert schemes. Dedicated to the memory of Fernando Serrano, Collect. Math., 49 (1998), 273–282.
• Hassett, B. and Tschinkel, Y., Abelian fibrations and rational points on symmetric products, Internat. J. Math., 11 (2000), 1163–1176.
• Huybrechts, D., Compact Hyperkähler Manifolds: Basic Results, alg-geom /9705025, Invent. Math., 135 (1999), 63–113.
• Markushevich, D., Rational Lagrangian fibrations on punctual Hilbert schemes of $K3$ surfaces, math. AG, /0509346.
• Mukai, S., Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math. J., 81 (1981), 153–175.
• Mukai, S., On the moduli space of bundles on $K3$ surfaces I, Vector bundles on Algebraic Varieties, Oxford, 1987, 341–413.
• Nakashima, T., Moduli of stable rank two bundles with ample $c\sb 1$ on $K3$ surfaces, Arch. Math. (Basel), 61 (1993), 100–104.
• Sawon, J., Lagrangian fibrations on Hilbert schemes of points on $K3$ surfaces, J. Algebraic Geom., 16 (2007), 477–497.
• Yoshioka, K., The Betti numbers of the moduli space of stable sheaves of rank 2 on ${\Bbb P}^2$, J. reine angew. Math., 453 (1994), 193–220.
• Yoshioka, K., Some examples of Mukai's reflections on $K3$ surfaces, J. reine angew. Math., 515 (1999), 97–123.
• Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces, Math. Ann., 321 (2001), 817–884, math.AG/0009001.
• Yoshioka, K., Twisted stability and Fourier-Mukai transform I, Compositio Math., 138 (2003), 261–288.
• Yoshioka, K., Twisted stability and Fourier-Mukai transform II, Manuscripta Math., 110 (2003), 433–465.
• Yoshioka, K., Stability and the Fourier-Mukai transform. II, Compos. Math., 145 (2009), 112–142.
• Kurihara, K. and Yoshioka, K., Holomorphic vector bundles on non-algebraic tori of dimension 2, Manuscripta Math., 126 (2008), 143–166.
• Zuo, K., The moduli spaces of some rank-$2$ stable vector bundles over algebraic $K3$ surfaces, Duke Math. J., 64 (1991), 403–408.