K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

We study the local structure of the singularity in the moduli space of sheaves on a K3 surface which has been resolved by K. O’Grady in his construction of new examples of hyperkaehler manifolds. In particular, we identify the singularity with the closure of a certain nilpotent orbit in the coadjoint representation of the group Sp(4). We also prove that the moduli spaces for some other sets of numerical parameters do not admit a smooth symplectic resolution of singularities. 2000 Math. Subj. Class. 14D20.

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Local structure of hyperkaehler singularities in O'Grady's examples

2-group symmetries arise when 1-form symmetries and 0-form symmetries of a theory mix with each other under group multiplication. We discover the existence of 2-group symmetries in 5d N=1 abelian gauge theories arising on the (non-extended) Coulomb branch of 5d superconformal field theories (SCFTs), leading us to argue that the UV 5d SCFT itself admits a 2-group symmetry. Furthermore, our analysis determines the global forms of the 0-form flavor symmetry groups of 5d SCFTs, irrespective of whether or not the 5d SCFT admits a 1-form symmetry. As a concrete application of our method, we analyze 2-group symmetries of all 5d SCFTs, which reduce in the IR, after performing mass deformations, to 5d N=1 non-abelian gauge theories with simple, simply connected gauge groups. For rank-1 Seiberg theories, we check that our predictions for the flavor symmetry groups match with the superconformal and ray indices available in the literature. We also comment on the mixed 't Hooft anomaly between 1-form and 0-form symmetries arising in 5d N=1 non-abelian gauge theories and its relation to the 2-groups, discussing various scenarios for the UV SCFT completions of these gauge theories. In particular, we provide one possible scenario that consists of an anomaly cancellation mechanism by an additional (topological) sector.

The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs

We prove that the moduli space M0;0(N;2) of stable maps of degree 2 to the moduli space N of rank 2 stable bundles of flxed odd determinant OX(ix) over a smooth projective curve X of genus g ‚ 3 has two irreducible components which intersect transversely. One of them is Kirwan's partial desingularization e MX of the moduli space MX of rank 2 semistable bundles with determinant isomorphic to OX(yix) for some y 2 X. The other component is the partial desingularization of the GIT quotient PHom(sl(2) _ ;W)==PGL(2) for a vector bundle W = R 1 …⁄L i2 (ix) of rank g over the Jacobian of X. We also show that the Hilbert scheme H, the Chow scheme C of conics in N and M0;0(N;2) are related by explicit contractions.

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Hecke correspondence, stable maps, and the Kirwan desingularization

2-group symmetries arise when 1-form symmetries and 0-form symmetries of a theory mix with each other under group multiplication. We discover the existence of 2-group symmetries in 5d N=1 abelian gauge theories arising on the (non-extended) Coulomb branch of 5d superconformal field theories (SCFTs), leading us to argue that the UV 5d SCFT itself admits a 2-group symmetry. Furthermore, our analysis determines the global forms of the 0-form flavor symmetry groups of 5d SCFTs, irrespective of whether or not the 5d SCFT admits a 1-form symmetry. As a concrete application of our method, we analyze 2-group symmetries of all 5d SCFTs, which reduce in the IR, after performing mass deformations, to 5d N=1 non-abelian gauge theories with simple, simply connected gauge groups. For rank-1 Seiberg theories, we check that our predictions for the flavor symmetry groups match with the superconformal and ray indices available in the literature. We also comment on the mixed 't Hooft anomaly between 1-form and 0-form symmetries arising in 5d N=1 non-abelian gauge theories and its relation to the 2-groups. 

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Let J be an abelian surface with a generic ample line bundle Open image in new window. For n≥1, the moduli space MJ(2,0,2n) of Open image in new window(1)-semistable sheaves F of rank 2 with Chern classes c1(F)=0, c2(F)=2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of MJ(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of MJ(2,0,2n) for n≥2.

On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Cohomology of the moduli space of Hecke cycles

Let $M_c=M(2,0,c)$ be the moduli space of O(1)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$ where O(1) is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a symplectic structure on the smooth locus. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This implies that there is no symplectic desingularization.

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Jaeyoo Choy

Papers

On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Cohomology of the moduli space of Hecke cycles

Nonexistence of a crepant resolution of some moduli spaces of sheaves on a k3 surface

Nonexistence of a crepant resolution of some moduli spaces of sheaves on a K3 surface

Geometry of Uhlenbeck partial compactification of orthogonal instanton spaces and the K-theoretic Nekrasov partition functions