I have finalized my old (1979) results about enumeration of connected components of moduli of real polarized K3 surfaces.
As an application, using recent results of math.AG/0312396, the complete classification of real polarized K3 surfaces which are deformations of real hyper-elliptically polarized K3 surfaces is obtained. This could be important in some questions, because real hyper-elliptically polarized K3 surfaces can be constructed explicitly.
In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that…
A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial, on the smooth projective polarized surface (S,L), is constructed. Families of locally free sheaves on the surface S are completed by locally free sheaves on schemes which are modifications of S. Gieseker -- Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs polarized scheme -- vector bundle with moduli space of such type.
In this note it is shown that associating to a principally polarized abelian variety its de Rham cohomology defines a faithfully flat morphism of the moduli space of principally polarized abelian varieties in positive characteristics to the moduli space of symplectic $F$-Zips as defined in \cite{MW}.
We study the global structure of moduli spaces of quasi-isogenies of polarized p-divisible groups introduced by Rapoport and Zink. Using the corresponding results for non-polarized p-divisible groups from a previous paper, we determine their dimensions and their sets of connected components and of irreducible components.
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.
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.
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.
We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface corresponding to twisted ideal sheaves. We use Fourier-Mukai techniques to extend the ideas of Arcara and Bertram to express wall-crossings as Mukai flops and show that the moduli spaces are projective.
We generalize the construction of moduli spaces of parabolic sheaves given by Maruyama and Yokogawa in [MY92] to the case of a projective fine saturated log scheme with a fixed global chart. Furthermore we construct moduli spaces of parabolic sheaves without fixing the weights.
We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.
Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.
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.
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmativ…
In this paper we introduce the stack of polarized twisted conics and we use it to give a new point of view on $\overline{\mathcal{M}}_2$. In particular, we present a new and independent approach to the computation of the integral Chow ring of $\overline{\mathcal{M}}_2$, previously determined by Eric Larson.
We prove that the moduli space X(1,7) of (1,7)-polarized abelian surfaces with canonical level structure is birational to the Fano 3-fold V22 of polar hexagons of the Klein quartic X(7). In particular X(1,7) is rational and the birational map to P^3 is defined over Q. As a byproduct we obtain explicitely the equations of the (1,7)-very-ample polarized abelian surfaces in P^6.
Around 1970 Griffiths introduced the moduli of polarized Hodge structures/the period domain $D$ and described a dream to enlarge $D$ to a moduli space of degenerating polarized Hodge structures. Since in general $D$ is not a Hermitian symmetric domain, he asked for the existence of a certain automorphic cohomology theory for $D$, generalizing the usual notion of automorphic forms on symmetric Hermitian domains. Since then there have been many efforts in the first part of Griffith's dream but the second part still lives in darkness. The objective of the present text is two-folded. First, we giv…
We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.
Let $(X,D)$ be a smooth log pair over $\mathbb{C}$ such that the complement $U := X \setminus D$ carries a maximally varied family of polarized manifolds. We prove a version of second main theorem on $(X,D)$ by using the Viehweg-Zuo construction of the family and McQuillan's tautological inequality. As an application, we generalize a classical result of Nadel about the distribution of entire curves in the (compactified) base space of polarized families.
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.
This paper investigates the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit real valued function f(x) is obtained such that if $X$ is a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0 such that $K_X^2 <f(p)$, then $X$ is unirational and the order of its algebraic fundamental group is at most two. As a con…
In this note we define moduli functors of (primitively) polarized K3 spaces. We show that they are representable by Deligne-Mumford stacks over Spec(Z). Further, we look at K3 spaces with a level structure. Our main result is that the moduli functors of K3 spaces with a primitive polarization of degree 2d and a level structure are representable by smooth algebraic spaces over open parts of Spec(Z). To do this we use ideas of Grothendieck, Deligne, Mumford, Artin and others. These results are the starting point for the theory of complex multiplication for K3 surfaces and the definition of Kuga-…
We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational but not rational, and there are no Gamma_{11}-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A_{9}^{lev}.
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.
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.
By means of analytic methods the quasi-projectivity of the moduli space of algebraically polarized varieties with a not necessarily reduced complex structure is proven including the case of non-uniruled polarized varieties.
In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degeneration of Calabi-Yau threefolds is given.
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.
In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and we have natural GIT constructions for the corresponding moduli spaces and we show that the K3 surfaces with at worst ADE singularities are GIT stable. We give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. We compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga's…
We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of…
In this paper we consider the following problem: For a given natural number $d$ and a prime $p$ determine all Newton polygons of polarized K3 surfaces of degree 2d over fields of characteristic $p$. This is an analogue of the Manin problem for Newton polygons of abelian varieties. This question is equivalent to determining the non-empty height strata of the moduli stack $\M_{2d}\otimes \F_p$ of K3 surfaces with a polarization of degree 2d over $\F_p$. We prove here that if $d$ is large enough and prime to $p$, then the height strata of $\M_{2d}\otimes \F_p$ are non-empty.
We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A-infinity-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a version of Hilbert stability for non-commutative projective schemes.
We announce the construction of toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We include an application to the analyticity of zero loci of some functions.
We consider the natural action of a finite group on the moduli space of polarized K3 surfaces which induces a duality defined by Mukai for surfaces of this type. We show that the group permutes polarized Fourier-Mukai partners of polarized K3 surfaces and we study the divisors in the fixed loci of the elements of this finite group.
Let $X$ be a smooth polarized algebraic surface over the compex number field. We discuss the invariants obtained from the moduli stacks of semistable sheaves of arbitrary ranks on $X$. For that purpose, we construct the virtual fundamental classes of some moduli stacks, and we show the transition formula of the integrals over the moduli stacks of the $δ$-stable Bradlow pairs for the variation of the parameter $δ$. Then, we study the relation among the invariants. In the case $p_g>0$, we show that the invariants are independent of the choice of a polarization of $X$. We also show that the invar…
For a smooth canonically embedded curve $C$ of genus $9$ together with a pencil $|L|$ of degree $6$, we study the relative canonical resolution of $C\subset X\subset \mathbb{P}^8$, where $X$ is the scroll swept out by the pencil $|L|$. We show that the second syzygy bundle in this resolution of $C\subset X$ is unbalanced. The proof reveals a new geometric connection between the universal Brill--Noether variety $\mathcal{W}^1_{9,6}$ and a moduli space $\mathcal{F}^\mathfrak{h}$ of lattice polarized $K3$ surfaces (for a certain rank $3$ lattice $\mathfrak{h}$). As a by-product we prove the unira…
Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.
We prove that, for any $g \geq 2$, the étale double cover $ρ_g:\mathcal{E}_{g} \to \widehat{\mathcal{E}}_{g}$ from the moduli space $\mathcal{E}_{g}$ of complex polarized genus $g$ Enriques surfaces to the moduli space $\widehat{\mathcal{E}}_{g}$ of numerically polarized genus $g$ Enriques surfaces is disconnected precisely over irreducible components of $\widehat{\mathcal{E}}_{g}$ parametrizing $2$-divisible classes, answering a question of Gritsenko and Hulek. We characterize all irreducible components of $\mathcal{E}_{g}$ in terms of a new invariant of line bundles on Enriques surfaces that…
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a non-constant map to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration facto…
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 show that every smooth toric variety (and many other algebraic spaces as well) can be realized as a moduli space for smooth, projective, polarized varieties. Some of these are not quasi--projective. This contradicts a recent paper (Quasi--projectivity of moduli spaces of polarized varieties, Ann. of Math. 159 (2004) 597--639.).
We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the Generalized Riemann Hypothesis is true, there are no further examples of such curves. More generally, we prove under the Generalized Riemann Hypothesis that there exist exactly 46 genus-2 curves over $\overline{\mathbb{Q}}$ with field of moduli $\mathbb{Q}$ whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal o…
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…
Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav of dimension 6 is a Prym-Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge-Hurwitz ei…
Let A_2 be the moduli stack of principally polarized abelian surfaces and V a smooth l-adic sheaf on A_2 associated to an irreducible rational finite dimensional representation of Sp(4). We give an explicit expression for the cohomology of V in any degree in terms of Tate type classes and Galois representations attached to elliptic and Siegel cusp forms. This confirms a conjecture of Faber and van der Geer. As an application we prove a dimension formula for vector-valued Siegel cusp forms for Sp(4,Z) of weight three, which had been conjectured by Ibukiyama.
As a result of our study of the hyperbolicity of the moduli space of polarized manifold, we give a general big Picard theorem for a holomorphic curve on a log-smooth pair $(X,D)$ such that $W=X\setminus D$ admits a Finsler pseudometric that is strongly negatively curved when pulled back to the curve. We show, by some refinements of the classical Viehweg-Zuo construction, that this latter condition holds for the base space $W$, if nonsingular, of any algebraic family of polarized complex projective manifolds with semi-ample canonical bundles whose induced moduli map $ϕ$ to the moduli space of s…
