A family of K3 surfaces $\mathscr{X}\rightarrow B$ has the \emph{Franchetta property} if the Chow group of 0-cycles on the generic fiber is cyclic. The generalized Franchetta conjecture proposed by O'Grady asserts that the universal family $\mathscr{X}_g\rightarrow \mathscr{F}_g$ of polarized K3 of degree $2g-2$ has the Franchetta property. While this is known only for small $g$ thanks to \cite{PSY}, we prove that for all $g$ there is a hypersurface in $ \mathscr{F}_g$ such that the corresponding family has the Franchetta property.
We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for $\mathscr A_{g,n}$.
Using Moriwaki's calculation of the Q-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g>2. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties.
In this paper, we study rational sections of the relative Picard variety of a linear system on a smooth projective variety. Specifically, we prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard variety come from restrictions of line bundles on the variety.
O'Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus g is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when g<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative analog, we study the Franchetta property for families of special cubic fourfolds (in the sense of Hassett) and relate it to O'Grady's conjecture for K3 surfaces. Most notably, by using special cubic fou…
We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Ma…
The generalized Franchetta conjecture for hyper-Kähler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-Kähler varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville--Donagi family of Fano varieties of lines on cubic fourfolds and its relative square, and for $0$-cycles and codimension-$2$ cycles for the Lehn--Lehn--Sorger--van Straten family of hyper-Kähler eightfolds. We also draw many…
We study relative zero cycles on the universal polarized $K3$ surface $X \to \mathcal{F}_g$ of degree $2g - 2$. It was asked by O'Grady if the restriction of any class in $\mathrm{CH}^2(X)$ to a closed fiber $X_s$ is a multiple of the Beauville-Voisin canonical class $c_{X_s} \in \mathrm{CH}_0(X_s)$. Using Mukai models, we give an affirmative answer to this question for $g \leq 10$ and $g = 12, 13, 16, 18, 20$.
Bloch's conjecture for a surface $X$ over an algebraically closed field $k$ states that every homologically trivial correspondence $Γ$ acts as 0 on the Albanese kernel $T(X_Ω)$, where $Ω$ is a universal domain containing $k$. Here we prove that, for a complex K3 surface $X$, Bloch's conjecture is equivalent to the existence of a valence for every correspondence. We also give applications of this result to the case of a correspondence associated to an automorphisms of finite order and to the existence of constant cycle curves on $X$. Finally we show that Franchetta's conjecture, as stated by K.…
O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In \cite{BL17}, this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4-th DB-cohomology group of universal oriented polarized K3 surface…
We show that Grothendieck's standard conjectures are implied by either of two other motivic conjectures: (a) by that of the existence of the motivic t-structure, and (b) by (a weak form of) Suslin's Lawson homology conjecture.
We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.
We discuss a difference between the rational and the real non-vanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem for rationally connected varieties under assuming Shokurov's ACC conjectures.
We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical $4$-folds.
We study the Lefschetz standard conjecture on a smooth complex projective variety X. In degree 2, we reduce it to a local statement concerning deformations of vector bundles on X. When X is hyperkähler, we show that the existence of nontrivial deformations of stable hyperholomorphic bundles implies the Lefschetz standard conjecture in codimension 2.
The Chisini conjecture asserts that a generic ramified covering over the complex projective plane of degree at least 5 is uniquely determined by its branch curve. We prove this for degree at least 12 using the work of Kulikov (math-AG/9803144).
After recalling the various tautological algebras of the moduli space of curves and some of its partial compactifications and stating several well-known results and conjectures concerning these algebras, we prove that the natural extension to the case of pointed curves of a 1996 conjecture of Hain and Looijenga is true if and only if two of the stated conjectures are true.
For certain product varieties, Murre's conjecture on Chow groups is investigated. In particular, it is proved that Murre's conjecture (B) is true for two kinds of four-folds. Precisely, if $C$ is a curve and $X$ is an elliptic modular threefold over $k$ (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture (B) is true for the fourfold $X\times C.$
Let C be a general curve. Mukai asked whether for every stable rank 2 vector bundle E on C with det(E) = K, the multiplication map Sym^2 H^0(C,E) --> H^0(C, Sym^2 E) is injective. We observe that this holds when h^0(E) < 7 .
Serrrano's Conjecture says that if $L$ is a strictly nef line bundle on a smooth projective variety $X$, then $K_X+tL$ is ample for $ t > dim X+1$. In this paper I will prove a few cases of this conjecture. I will also prove a generalized version of this conjecture (due to Campana, Chen and Peternell) for surfaces. In the last section, assuming the SHGH conjecture, I will give a series of examples of strictly nef non ample divisors on surfaces of arbitary Kodaira dimension.
In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \ge 2$, we give the inverse polynomial map under the condition that $JH^3=0$. We shall show that ${\operatorname{deg}}(F^{-1})\leq d^k$ if $k \le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\operatorname{deg}}(F^{-1})> d…
Let $X$ be a hyperkähler variety. Beauville has conjectured that a certain subring of the Chow ring of $X$ should inject into cohomology. This note proposes a similar conjecture for the ring of algebraic cycles on $X$ modulo algebraic equivalence: a certain subring (containing divisors and codimension $2$ cycles) should inject into cohomology. We present some evidence for this conjecture.
We study some conjectures about Chow groups of varieties of geometric genus one. Some examples are given of Calabi-Yau threefolds where these conjectures can be verified, using the theory of finite-dimensional motives.
Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\leq 6$; (ii) all known hyperkähler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi-Yau varieties. We prove that the fourth Todd…
We propose a "Bloch type" conjecture for surfaces: if the cup product map in coherent cohomology is zero, then all intersections of homologically trivial divisors should be zero in the Chow group of zero-cycles. We prove this conjecture for Sicilian surfaces.
We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the complex numbers.
We apply a degenerate version of a result due to Hirschowitz, Ramanan and Voisin to verify Green and Green-Lazarsfeld conjectures over explicit open sets inside each $d$-gonal stratum of curves $X$ with $d<[g_X/2]+2$. By the same method, we verify the Green-Lazarsfeld conjecture for any curve of odd genus and maximal gonality. The proof invokes Voisin's solution to the generic Green conjecture as a key argument.
Let $X$ be a hyperkähler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety of lines on a very general cubic fourfold.
We prove in the case of minimal Fano threefolds a conjecture stated by Dubrovin at the ICM 1998 in Berlin. The conjecture predicts that the symmetrized/alternated Euler characteristic pairing on $K_0$ of a Fano variety with an exceptional collection expressed in the basis of the classes of the exceptional objects coincides with the intersection pairing of the vanishing cycles in Dubrovin's second connection. We show that the conjecture holds for $V_{22}$, a minimal Fano threefold of anticanonical degree~22, and for $V_5$, the minimal Fano threefold of anticanonical degree~40, by applying the m…
In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of deg…
Let X be a smooth, projective variety over the field of complex numbers. On the space H of its rational cohomology of degree i we have the arithmetic filtration F^p. On the other hand, on the space of cohomology of degree i of X with complex coefficients we have the Hodge filtration G^p. It is well known that F^p is contained in the intersection of G^p with H, and that, in general, this inclusion is strict. In this paper we propose a natural substitute S^p for the Hodge filtration space G^p such that the intersection of S^p with H is the space F^p of the arithmetic filtration. In particular, S…
We discuss two conjectures by Francesco Severi and Joe Harris about the irreducibility and the dimension of the Hilbert scheme parameterizing smooth projective curves of given degree and genus.
A classical question asks whether the Abel-Jacobi map is universal among all regular homomorphisms. In this paper, we prove that we can construct a $4$-fold which gives the negative answer in codimension $3$ if the generalized Bloch conjecture holds for a $3$-fold constructed by Colliot-Thélène and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.
The Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations, and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic 0, (T) implies (S). In characteristic p an analogous result is true under stronger assumptions.
In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic zero, whose generic fiber is geometrically irreducible and of general type. If f is not birationally trivial, then there are countably many proper closed varieties { Z_i } of X such that every quasi-section of f is contained in the union of Z_i.
We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich-Zagier concerning transcendence properties of the torsors of periods of varieties over number fields. We notably establish that the Grothendieck period conjecture holds in degree 1 for products of curves, of abelian varieties, and of K3 surfaces, and that it holds in degre…
Assuming the classical Farrell-Jones conjecture we produce an explicit (commutative) group ring $R$ and a thick subcategory $\mathsf{C}$ of perfect $R$-complexes such that the Waldhausen $K$-theory space $\mathrm{K}(\mathsf{C})$ is equivalent to a rational Eilenberg-Maclane space.
We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the Betti numbers conjectures for the zeta function of the variety.
We make two observations regarding the invertibility of Keller maps. i.e., polynomial maps for which the determinant of their Jacobian matrix is identically equal to 1. In our first result, we show that if P is a n-dimensional Keller map, defined over any extension of Q, then P has a polynomial inverse if and only if the range of P contains the cartesian product of n universal Hilbert sets. In our second result, we show that if P is a 2-dimensional Keller map, defined over any algebraic number field, then P is invertible on a set that contains almost all rational integers of K.
The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.
In this note we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds.
We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it, namely that the additional factor $g$ is a power of the radical polynomial $f$. Next we show that in order to prove the Generalized Vanishing Conjecture (up to some bound on the degree of $\Lam$), we may assume that $\Lam$ is a linear combination of powers of distinct partial derivatives. At last, we show that the Generalized Vanishing Conjecture holds for produ…
We demonstrate counterexamples to Wilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic field is bounded then it must have codimension at least two. Examples are provided to show that this conclusion cannot be improved.
The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta c…
