scholarly journals GENERALIZED CALABI–YAU STRUCTURES, K3 SURFACES, AND B-FIELDS

2005 ◽  
Vol 16 (01) ◽  
pp. 13-36 ◽  
Author(s):  
DANIEL HUYBRECHTS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Weak Form ◽  
K3 Surfaces ◽  
Standard Theory ◽  
General Notion ◽  
Symplectic Structures ◽  
Special Cases ◽  
Torelli Theorem

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

scholarly journals K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

2021 ◽  
pp. 1-41
Author(s):  
KENNETH ASCHER ◽  
KRISTIN DEVLEMING ◽  
YUCHEN LIU
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
Quadric Surface ◽  
Intersection Curves

Abstract We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

scholarly journals On Some Generalized Rapoport–Zink Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1111-1187
Author(s):  
Xu Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Connected Components ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Mixed Characteristic ◽  
Special Fibers ◽  
Type Spaces ◽  
Local Invariants

AbstractWe enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

A Torelli theorem for moduli spaces of principal bundles on curves defined over ℝ

2017 ◽  
Vol 28 (06) ◽  
pp. 1750049
Author(s):  
Indranil Biswas ◽  
Olivier Serman
Keyword(s):  
Algebraic Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Isomorphism Class ◽  
Projective Curve ◽  
Real Numbers ◽  
Principal Bundles ◽  
Smooth Projective Curve ◽  
Simple Factor ◽  
Torelli Theorem

Let [Formula: see text] be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let [Formula: see text] be a connected reductive affine algebraic group, defined over [Formula: see text], such that [Formula: see text] is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal [Formula: see text]-bundles on [Formula: see text] determine uniquely the isomorphism class of [Formula: see text].

The Euler number of certain primitive Calabi–Yau threefolds

2000 ◽  
Vol 128 (1) ◽  
pp. 79-86
Author(s):  
MEI-CHU CHANG ◽  
HOIL KIM
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Three Dimensional ◽  
Building Blocks ◽  
K3 Surfaces ◽  
Physical Models ◽  
Ample Divisor ◽  
K3 Surface ◽  
Superstring Theory ◽  
Two Parameters

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define Mh,d to be {(X, H)[mid ]H3 = h, c2(X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know that Ng is of dimension 19 for every g and is irreducible but we do not know the dimension of Mh,d and whether or not Mh,d is irreducible. In fact, the dimension of Mh,d = h1(X, TX), where (X, H) ∈ Mh,d. Furthermore, it is well known that χ(X) = 2 (rank of Pic (X) − h1(X, TX)), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an important role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c3 of Calabi–Yau threefolds with Picard rank 1.

K3 Surfaces

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Moduli Spaces ◽  
Special Class ◽  
K3 Surfaces ◽  
Derived Categories ◽  
K3 Surface ◽  
Hodge Structures ◽  
Surface Theory ◽  
Torelli Theorem ◽  
Stable Sheaves ◽  
Moduli Theory

After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.

scholarly journals Singularities on the 2-Dimensional Moduli Spaces of Stable Sheaves on K3 Surfaces

2003 ◽  
Vol 14 (08) ◽  
pp. 837-864 ◽  
Author(s):  
Nobuaki Onishi ◽  
Kōta Yoshioka
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Double Points ◽  
Exceptional Locus ◽  
Stable Sheaves

We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on k3 surfaces. We show that the moduli space is normal, in particular the siguralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.

scholarly journals Integrable Systems and Torelli Theorems for the Moduli Spaces of Parabolic Bundles and Parabolic Higgs Bundles

2016 ◽  
Vol 68 (3) ◽  
pp. 504-520
Author(s):  
Indranil Biswas ◽  
Tomás L. Gómez ◽  
Marina Logares
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Integrable Systems ◽  
Algebraic Curve ◽  
Higgs Bundles ◽  
Parabolic Bundles ◽  
Smooth Complex ◽  
Torelli Theorem ◽  
Complex Projective

AbstractWe prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

scholarly journals Zero cycles on the moduli space of curves

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Rahul Pandharipande ◽  
Johannes Schmitt
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Rational Surface ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
K3 Surface ◽  
Moduli Space Of Curves ◽  
Moduli Spaces Of Curves ◽  
Open Questions ◽  
Witten Theory

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed. Comment: Published version

scholarly journals AUTOMORPHISMS OF MODULI SPACES OF SYMPLECTIC BUNDLES

2012 ◽  
Vol 23 (05) ◽  
pp. 1250052 ◽  
Author(s):  
INDRANIL BISWAS ◽  
TOMAS L. GÓMEZ ◽  
VICENTE MUÑOZ
Keyword(s):  
Automorphism Group ◽  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Projective Curve ◽  
Smooth Complex ◽  
Complex Projective ◽  
Complex Projective Curve

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let MSp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of MSp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where [Formula: see text], together with the automorphisms induced by automorphisms of X.

scholarly journals On K3 Surface Quotients of K3 or Abelian Surfaces

2017 ◽  
Vol 69 (02) ◽  
pp. 338-372
Author(s):  
Alice Garbagnati
Keyword(s):  
Abelian Group ◽  
Moduli Space ◽  
Minimal Model ◽  
K3 Surfaces ◽  
Rational Curves ◽  
The Other ◽  
Abelian Surface ◽  
K3 Surface ◽  
Abelian Surfaces ◽  
Special Cases

Abstract The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group G (respectively of a K3 surface by an Abelian group G) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces that are (rationally) G-covered by Abelian or K3 surfaces (in the latter case G is an Abelian group). When G has order 2 or G is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases. Moreover, we prove that a K3 surface XG is the minimal model of the quotient of an Abelian surface by a group G if and only if a certain configuration of rational curves is present on XG . Again, this result was known only in some special cases, in particular, if G has order 2 or 3.

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