## Duke Mathematical Journal

### Irreducible symplectic $4$-folds and Eisenbud-Popescu-Walter sextics

#### Abstract

Eisenbud, Popescu, and Walter [4] have constructed certain special sextic hypersurfaces in ${\mathbb P}^5$ as Lagrangian degeneracy loci. We prove that the natural double cover of a generic Eisenbud-Popescu-Walter (EPW) sextic is a deformation of the Hilbert square of a $K3$-surface $(K3)^{[2]}$ and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type $(1,1)$; thus we get an example similar to that (discovered by Beauville and Donagi [2]) of the Fano variety of lines on a cubic $4$-fold. Conversely, suppose that $X$ is a numerical $(K3)^{[2]}$, suppose that $H$ is an ample divisor on $X$ of square $2$ for Beauville's quadratic form, and suppose that the map $X\dashrightarrow|H|^{\vee}$ is the composition of the quotient $X\to Y$ for an antisymplectic involution on $X$ followed by an immersion $Y\hookrightarrow|H|^{\vee}$; then $Y$ is an EPW sextic, and $X\to Y$ is the natural double cover

#### Article information

Source
Duke Math. J., Volume 134, Number 1 (2006), 99-137.

Dates
First available in Project Euclid: 4 July 2006

https://projecteuclid.org/euclid.dmj/1152018505

Digital Object Identifier
doi:10.1215/S0012-7094-06-13413-0

Mathematical Reviews number (MathSciNet)
MR2239344

Zentralblatt MATH identifier
1105.14051

#### Citation

O'Grady, Kieran G. Irreducible symplectic $4$ -folds and Eisenbud-Popescu-Walter sextics. Duke Math. J. 134 (2006), no. 1, 99--137. doi:10.1215/S0012-7094-06-13413-0. https://projecteuclid.org/euclid.dmj/1152018505

#### References

• A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), 755--782.
• A. Beauville and R. Donagi, La variété des droites d'une hypersurface cubique de dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 703--706.
• G. Casnati and F. Catanese, Even sets of nodes are bundle symmetric, J. Differential Geom. 47 (1997), 237--256.
• D. Eisenbud, S. Popescu, and C. Walter, Lagrangian subbundles and codimension $3$ subcanonical subschemes, Duke Math. J. 107 (2001), 427--467.
• W. Fulton and P. Pragacz, Schubert Varieties and Degeneracy Loci, Lecture Notes in Math. 1689, Springer, Berlin, 1998.
• F. J. Gallego and B. P. Purnaprajna, Very ampleness and higher syzygies for Calabi-Yau threefolds, Math. Ann. 312 (1998), 133--149.
• D. Gieseker, Geometric invariant theory and applications to moduli problems'' in Invariant Theory (Montecatini, Italy, 1982), Lecture Notes in Math. 996, Springer, Berlin, 1983, 45--73.
• D. Huybrechts, Compact hyper-Kähler manifolds: Basic results, Invent. Math. 135 (1999), 63--113.; Erratum, Invent. Math. 152 (2003), 209--212.
• A. Iliev and K. Ranestad, Addendum to ʽʽ K$3$ surfaces of genus $8$ and varieties of sums of powers of cubic fourfolds,ʼʼ preprint, 2002.
• V. A. Iskovskih, Fano threefolds, I, Math. USSR-Izv. 11 (1977), no. 3, 485--527.
• S. Mukai, Moduli of vector bundles on $K3$ surfaces and symplectic manifolds, Sugaku Expositions 1 (1988), 139--174.
• D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
• K. G. O'Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal. 15 (2005), 1223--1274.
• —, Irreducible symplectic $4$-folds numerically equivalent to $(K3)$, preprint.
• C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Progr. Math. 3, Birkhäuser, Boston, 1980.