For an algebraic K3 surface with complex multiplication (CM), algebraic fibres of the associated twistor space away from the equator are again of CM type. In this paper, we show that algebraic fibres corresponding to points at the same altitude of the twistor base $S^2\simeq \mathbb{P}^1_\mathbb{C}$ share the same CM endomorphism field. Moreover, we determine all the admissible Picard numbers of the twistor fibres.
We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction i…
We prove that there exist smooth surfaces of degree d in projective 3-space such that the group of rational equivalence classes of decomposable 0-cycles has rank at least the integer part of (d-1)/3.
We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P^3 whose function field has level 2 is dense in the set of those that have no real points.
We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective toric variety, with V\subset X. Under some assumptions, we prove that the cohomological class in H^{k,k}(X) associated to V remains of type (k,k) under an infinitesimal deformation if and only if V remains algebraic. Actually we prove that locally the Noether-Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 …
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
Let $ϕ:X\rightarrow \mathbb{P}^n$ be a morphism of varieties. Given a hyperplane $H$ in $\mathbb{P}^n$, there is a Gysin map from the compactly supported cohomology of $ϕ^{-1}(H)$ to that of $X$. We give conditions on the degree of the cohomology under which this map is an isomorphism for all but a low-dimensional set of hyperplanes, generalizing results due to Skorobogatov, Benoist, and Poonen-Slavov. Our argument is based on Beilinson's theory of singular supports for étale sheaves.
Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda (such as $\mathbb P^3$, the quintic threefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class then we show it has very negative square.
On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems
Wuhan University（武汉大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Johannes Gutenberg-Universität Mainz（美因茨约翰内斯古腾堡大学）
The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $\mathbb P^3$ of degree $d \geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d \geq 4$, a very general surface in a simplicial toric threefold $\mathbb P_Σ$ (with orbifold singularities) has the same Picard number as $\mathbb P_Σ$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $\mathbb P_Σ$ in a linear system of a Cartier ample divisor with respect to a (-1)…
In this manuscript we sharpen the lower bound on the codimension of the irreducible components of the Noether-Lefschetz locus of surfaces in projective toric threefolds given in [BG17]. We also provide a simpler proof of Theorem 4.11 in [BG17], which allows one to avoid some technical assumptions.
We take a sum $C_1+r C_2,\ r\in\mathbb Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside a smooth surface of degree $8$ and with $C_1\cap C_2=\emptyset$. We gather evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+ r C_2$ are distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a conjectural descri…
We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensi…
Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for O(2,19) lattices to determine the Noether-Lefschetz degrees in classical families of K3 surfaces of degrees 2, 4, 6 and 8. For the quartic K3 surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coeffi…
We study BPS jumping loci, or the subloci in moduli spaces of supersymmetric string vacua where BPS states come into existence discontinuously. This phenomenon is distinct from wall-crossing. We argue that these loci should be thought of as special cycles in the sense of Noether-Lefschetz loci or special Shimura subvarieties, which are indeed examples of BPS jumping loci for certain string compactifications. We use the Hodge-elliptic genus as an informative tool, suggesting that our work can be extended to understand the jumping behavior of motivic Donaldson-Thomas invariants.
Let $f:X\to Y$ be a morphism of complex manifolds. Suppose that $X$ is a Kähler manifold. Let $(\mathcal{T},\mathcal{S})$ be a regular polarized pure twistor $\mathcal{D}$-module of weight $w$ on $X$ whose support is proper over $Y$. We prove the Hard Lefschetz Theorem for the push-forward of $(\mathcal{T},\mathcal{S})$ by $f$. As one of the key steps, we obtain the twistor version of a theorem of Kashiwara and Kawai about the Hodge structure on the intersection complex of polarized variation of Hodge structure.
We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler's relation in genus 1, we construct a universal decomposition of the diagonal in Chow in the third fiber product of the universal K3 surface. The decomposition has terms supported on Noether-Lefschetz loci which are not visible in the Beauville-Voisin decomposition for a fixed K3 surface. As a result of our universal decomposition, we prove the conjecture of Marian-Oprea-Pandharipande: the full tautological ring of the moduli space of K3 surfaces…
As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a {\em generic} chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in $Compl$ where a fixed second cohomology class stays of Hodge type (1…
Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide a closed formul…
A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $π$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $\mathbb{CP}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in $\mathbb{CP}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest t…
We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed. …
