We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds.
We show that, in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to strata. For non-generic strata similar results can be shown by a case-by-case inspection.
We also propose to study a notion of 'linear manifold' that comprises Teichmueller curves, Hilbert modular surfaces and the ball quotients of Deligne and Mostow. Moreover, we give an explanation for the difference between Hilbert modular surfaces and Hilbert modular threefolds with respect to this notion of linearity.
A Finsler metric, whose holomorphic curvature is bounded from above by a negative constant, is constructed on the moduli stack of canonically polarized manifolds including singularities. Demailly's version of Ahlfors' lemma yields hyperbolicity of the moduli stack.
By using Oprea's Bialynicki-Birula decomposition for the stack of genus zero stable maps to flag manifolds. We calculate the Poincaré polynomial of the moduli space in degree one and degree two.
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
These are notes of my lectures given at the school on intersection theory and moduli at the ICTP, Trieste. We construct moduli spaces of K3 surfaces and higherdimensional hyperkaehler manifolds, including moduli spaces of (2,2)-conformal field theories associated to hyperkaehler manifolds, and describe their period domains. Recent results on these manifolds are discussed from the moduli space point of view. The second goal is to present a detailed account of mirror symmetry for K3 surfaces. Arguments due to Aspinwall and Morrison allow one to explain various versions of mirror symmetry (e.g. f…
In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperkähler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperkähler manifolds with $n\leq 2$. Secondly, we prove the…
We show that the moduli space of polarized irreducible symplectic manifolds of $K3^{[n]}$-type, of fixed polarization type, is not always connected. This can be derived as a consequence of Eyal Markman's characterization of polarized parallel-transport operators of $K3^{[n]}$-type.
Moduli spaces of polarised (with fixed polarisation type) irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on $K3$ surfaces are not connected in general and A. Apostolov counted the number of their components. We extend Apostolov's results to the case of manifolds deformation equivalent to generalised Kummer varieties.
On moduli spaces of vector bundles on $K3^{[n]}$-type IHS manifolds
SISSA（的里雅斯特国际高等研究学校）；Dipartimento di Matematica (DIMA)（都灵大学数学系）；Facultat de Matemàtiques i Informàtica（巴塞罗那大学数学与计算机学院）；Università degli Studi di Roma Tor Vergata（罗马二大）；Université Paris-Saclay（巴黎萨克雷大学）
In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, i.e. the Torelli space, is a complex affine manifold. As applications we prove that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the completion space is a bounded domain of holomorphy with a complete Kähler-Einstein metric. As a corollary we show that the period map from the moduli space of polarized Calabi-Yau manifolds with level $m$ structure is also injective.
Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the case $e=14$), we prove that the moduli space of polarized K3 surfaces of degree $2e$ contains complete curves for all $e\geq 62$ and for some sporadic lower values of $e$ (starting at $14$). We also construct complete curves in the moduli spaces of polarized hyper-Kähler manifolds of $\mathrm{K3}^{[n]}$-type or $\mathrm{Kum}_n$-type for all $n\ge 1$ and polarizations of various degrees and divisibilities.
In this paper we construct the analogue of Dedekind eta-function on the moduli space of polarized CY manifolds. We prove that the L-two norm of eta is the regularized determinants of the Laplacians of the CY metric on (0,1) forms. We construct the analogue of the Baily-Borel Compactification of the moduli space of polarized CY and prove that it has the same properties as the Baily-Borel compactification of the locally symmetric Hermitian spaces. We proved that the compactification constructed in the paper is the minimal.
In this paper we first study the moduli spaces related to Calabi-Yau manifolds. We then apply the results to the following problem. Let $C$ be a fixed Riemann surface with fixed finite number of points on it. Given a CY manifold with fixed topological type, we consider the set of all families of CY manifolds of the fixed topological type over $C$ with degenerate fibres over the fixed points up to isomorphism. This set is called Shafarevich set. The analogue of Shafarevich conjecture for CY manifolds is for which topological types of CY the Shafarevich set is finite. It is well-known that the a…
In this note, we introduce the notion of a singular principal G-bundle, associated to a reductive algebraic group G over the complex numbers by means of a faithful representation $\varrho^\p\colon G\lra \SL(V)$. This concept is meant to provide an analogon to the notion of a torsion free sheaf as a generalization of the notion of a vector bundle. We will construct moduli spaces for these singular principal bundles which compactify the moduli spaces of stable principal bundles.
We extend some of the results obtained for subvarieties of the moduli stack of canonically polarized manifolds in "Base spaces of non-isotrivial families of smooth minimal models" (math.AG/0103122) to moduli of polarized minimal models of Kodaira dimension zero. To this aim we first show that for those manifolds there are no non isotrivial families of minimal models, which are birationally isotrivial. In the last chapter, we discuss the rigidity of curves in the moduli stack of canonically polarized manifolds and of polarized minimal models of Kodaira dimension zero.
This article describes some complex-analytic aspects of the moduli space of the finite-dimensional complex representations of a finite quiver, which are stable with respect to a fixed rational weight. We construct a natural structure of a complex manifold on this moduli space, and a Kähler metric on the complex manifold. We then define a Hermitian holomorphic line bundle on the moduli space, and show that its curvature is a rational multiple of the Kähler form.
We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large anti-canonical volume. Our arguments are based on recent progress about the geometry of metric tangent cones and on related ideas about the algebro-geometric study of singularities of K-stable Fano varieties.
We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12.
In this paper it is proved that the volumes of the moduli spaces of polarized CY manifolds with respect to the Weil-Petersson metrics are finite and they are rational numbers.
We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.
We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a semipositive symplectic manifold of dimension 4, when GW([point],...,[point]) is enumerative. In particular, we show that the intersection theoretic property depends on the existence of a critical point on a stable map.
We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give complex analytic charts on the topological space consisting of Kähler-Ricci solitons, by studying differential geometric aspects of this infinite dimensional moment map. Some stacky words and arguments on Gromov-Hausdorff convergence help to glue them together in the holomorphic mann…
Consider the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h, and let M_h be the corresponding coarse quasi-projective moduli scheme. We show that M_h is Brody hyperbolic in the following sense: Assume that for some quasi-projective variety U there exists a morphism U --> M_h, quasi-finite over its image, which is induced by a family f: V --> U, belonging to the moduli problem. Then all holomorphic maps from the complex plane to U are constant.
We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama moduli spaces with respect to two different chosen polarisations are related via Thaddeus-flips through other "multi-Gieseker"-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding D…
Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.
Let $X$ be a closed $6-$dimensional manifold with a half-closed $SU(3)-$structure. On the product manifold $X\times S^{1}$, with respect to the product $G_{2}-$structure and on a pullback vector bundle from $X$, we show that any $G_{2}-$instanton is equivalent to a Hermitian Yang-Mills connection on $X$ via a "broken gauge". This result reveals the topological type of the moduli of $G_{2}-$instantons on $X\times S^{1}$. In dimension $8$, similar result holds for moduli of $Spin(7)-$instantons. A generalization and an example are given.
We show that the moduli space of all Calabi-Yau manifolds that can be realized as hypersurfaces described by a transverse polynomial in a four dimensional weighted projective space, is connected. This is achieved by exploiting techniques of toric geometry and the construction of Batyrev that relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the previously known fact that the moduli space of all CICY's is connected, and is moreover connected to the moduli space of the present class of Calabi-Yau manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface …
In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.
We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-Kähler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves.
Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic symplectic manifolds by deformation type. This paper studies a natural Lagrangian fibration on these moduli spaces to determine that they are not Kähler or simply connected, ruling out most of the known deformation types of holomorphic symplectic manifolds.
Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold $V$. When the associated moduli map $V\to P_h$ from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold $V$: for any projective compactification $Y$ of $V$, any holomorphic map $f:Δ-\{0\}\to V$ from the punctured unit disk to $V$ extends to a holomorphic map of the unit disk $Δ$ into $Y$. This result generalizes our previous work on the Brody hyperbolicity of $V$ (i.e. there are no entire cu…
We study the moduli space of rank stable based instantons over a connected sum of q copies of CP^2. For c_2=1 we give the homotopy type of the moduli space. For c_2=2 we compute the cohomology of the moduli space.
We show that a family of minimal surfaces of general type with p_g = 0, K^2=7, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family. The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue-type manifolds: these are obtained as quotients \hat{X} / G, where \hat{X} is an ample divisor in a K(Γ, 1) projective manifold Z, and G is a finite group acting freely on \hat{X} . For these type of manif…
We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is quasi-projective we prove that F is actually a Kaehler form.
In this paper, we construct a completion of the moduli space for polarized Calabi-Yau manifolds by using Ricci-flat Kähler-Einstein metrics and the Gromov-Hausdorff topology, which parameterizes certain Calabi-Yau varieties. We then study the algebro-geometric perperties and the Weil-Petersson geometry of such completion. We show that the completion can be exhausted by sequences of quasi-projective varieties, and new points added have finite Weil-Petersson distance to the interior.
We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the tau function of a semiclassical limit of the r-th Gelfand-Dickey (or KdV_r) hierarchy. Additionally, we prove that tensor products of the Frobenius manifolds associated to such hierarchies admit a geometric interpretation in terms of moduli spaces of higher spin structures. We also elaborate upon the analogy to Gromov-Witten invariants of a smooth, projective variety.
The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.
In this paper we obtain an ordinary differential equation ${\sf H}$ from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic $n$-form. We work on a moduli space ${\sf T }$ constructed from a Calabi-Yau $n$-fold $W$ together with a basis of the middle complex de Rham cohomology of $W$. We verify the existence of a unique vector field ${\sf H}$ on ${\sf T }$ such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by ${\sf H}$ is a generalization of differential equations introduced by Darboux, Halphen an…
The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil's programme was that of the geometric type of the moduli spaces of polar…
We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces $BK$ where $K$ is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form.…
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth pr…
We review the role played by tau functions of special type - called {\it Bergman} tau functions in various areas: theory of isomonodromic deformations, solutions of Einstein's equations, theory of Dubrovin-Frobenius manifolds, geometry of moduli spaces and spectral theory of Riemann surfaces. These tau functions are natural generalizations of Dedekind's eta-function to higher genus. Study of their properties allows to get an explicit form of Einstein's metrics, obtain new relations in Picard groups of various moduli spaces and derive holomorphic factorization formulas of determinants of Laplac…
We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of symplectic forms modulo diffeomorphisms on such a manifold has at least n connected components.
In this paper we first show that on projective manifolds (M, ω), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern class on a compact Kahler manifold Z (with Todd genus non-zero) and a choice of a geometric quantization of (M, ω). Next we further study the generalization of the vortex equations on Kahler 4-manifold which has been studied earlier by Bradlow. We show that when the Kahler 4-manifol…
Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $ω$ on \lq twisted' moduli spaces of representations of the fundamental group $π$ of a 2-manifold $Σ$ (the smooth analogues of ${\rm Hom} (π_1(Σ), G)/G$) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian $G$-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment ma…
We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear sigma-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kahler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kahler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the unio…
