We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure coreplatform@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general setup for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley–Weil theorem for stacks.
We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a polynomial on residue classes (PORC). The key ingredients in the proof are Lusztig’s Jordan decomposition of complex characters of finite reductive groups and Deriziotis’s results on their genus numbers. As a consequence of our main theorem, we obtain an expression for the E-polynomial of the character stack.
We prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.
We determine the integral Chow and cohomology rings of the moduli stack
$\mathcal {B}_{r,d}$
of rank r, degree d vector bundles on
$\mathbb {P}^1$
-bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring
$A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$
is a free
$\mathbb {Q}$
-algebra on
$2r+1$
generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring
$A^*(\mathcal {B}_{r,d})$
is torsion-free and provide multiplicative generators for
$A^*(\mathcal {B}_{r,d})$
as a subring of
$A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$
. From this description, we see that
$A^*(\mathcal {B}_{r,d})$
is not finitely generated as a
$\mathbb {Z}$
-algebra. Finally, when
$k = \mathbb {C}$
, the cohomology ring of
$\mathcal {B}_{r,d}$
is isomorphic to its Chow ring.
Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1,\,\ldots,\,d_a)$-polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1,\,\ldots,\,a$ and $a\geq 4$. We show that the fibres are in the same birational class if all the $(m,\,0)$-forms on $X_b$ are liftable to $(m,\,0)$-forms on $\mathcal {X}$, where $m=1$ and $m=a-1$. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne–Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
In 1977, Gauduchon proved that on every compact hermitian manifold $(X, \omega )$ there exists a conformally equivalent hermitian metric $\omega _\mathrm {G}$ which satisfies $\mathrm {dd}^{\mathrm {c}} \omega _\mathrm {G}^{n-1} = 0$. In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.
We study the p-rank stratification of the moduli space of cyclic degree
$\ell $
covers of the projective line in characteristic p for distinct primes p and
$\ell $
. The main result is about the intersection of the p-rank
$0$
stratum with the boundary of the moduli space of curves. When
$\ell =3$
and
$p \equiv 2 \bmod 3$
is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type
$(r,s)$
, and p-rank f satisfying the clear necessary conditions.
Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $\mathcal {E}$ be a semi-simple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal {E}_i$ of $\mathcal {E}$ have rank 2, determinant $\bar {\mathbb {Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal {E}$ at $x$ is in $\mathbb {Q}[t]\subset \mathbb {Q}_p[t]$. Then there exists a dense open subset $U\subset X$ such that $\mathcal {E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar {\mathbb {Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar {\mathbb {Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exist a dense open subset $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$.
Let $\mathbb {V}$ be a motivic variation of Hodge structure on a $K$-variety $S$, let $\mathcal {H}$ be the associated $K$-algebraic Hodge bundle, and let $\sigma \in \mathrm {Aut}(\mathbb {C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal {H}_{\mathbb {C}, s}$ above $s \in S(\mathbb {C})$ which lies inside $\mathbb {V}_{s}$, the conjugate vector $v_{\sigma } \in \mathcal {H}_{\mathbb {C}, s_{\sigma }}$ is Hodge and lies inside $\mathbb {V}_{s_{\sigma }}$. We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb {C}}$ containing $s$ whose algebraic monodromy group $\textbf {H}_{Z}$ fixes $v$. Using relationships between $\textbf {H}_{Z}$ and $\textbf {H}_{Z_{\sigma }}$ coming from the theories of complex and $\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\textbf {H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.
Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas (
$\mathrm {GW}/{\mathrm {PT}}$
) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric threefolds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.
Let A be an abelian scheme of dimension at least four over a
$\mathbb {Z}$
-finitely generated integral domain R of characteristic zero, and let L be an ample line bundle on A. We prove that the set of smooth hypersurfaces D in A representing L is finite by showing that the moduli stack of such hypersurfaces has only finitely many R-points. We accomplish this by using level structures to interpolate finiteness results between this moduli stack and the stack of canonically polarized varieties.
We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation
$A^2-DB^2=1$
, with
$A,B,D\in \mathbb {C}[t]$
and certain ramified covers
$\mathbb {P}^1\to \mathbb {P}^1$
arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials D that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann existence theorem associates to the abovementioned covers certain permutation representations: We are able to characterize the representations corresponding to ‘primitive’ solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when D has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.
We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument, we construct a dualising sheaf and trace map, in the lisse-étale topology, for families of tame twisted curves when the base stack is locally Noetherian.
Let
$f: X \to B$
be a relatively minimal fibration of maximal Albanese dimension from a variety X of dimension
$n \ge 2$
to a curve B defined over an algebraically closed field of characteristic zero. We prove that
$K_{X/B}^n \ge 2n! \chi _f$
. It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and
$\chi _f> 0$
, we prove that the general fibre F of f has to satisfy the Severi equality that
$K_F^{n-1} = 2(n-1)! \chi (F, \omega _F)$
. We also prove some sharper results of the same type under extra assumptions.
We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (nonseparated) resolutions of singularities for toric varieties. Specifically, let
$\mathcal {X}$
be a smooth Artin stack admitting a good moduli space
$\pi : \mathcal {X} \to X$
, and assume that X is a variety with log-terminal singularities,
$\pi $
induces an isomorphism over a nonempty open subset of X and the exceptional locus of
$\pi $
has codimension at least
$2$
. We conjecture a change-of-variables formula relating the motivic measure for
$\mathcal {X}$
to the Gorenstein measure for X and functions measuring the degree to which
$\pi $
is nonseparated. We also conjecture that if the stabilisers of
$\mathcal {X}$
are special groups in the sense of Serre, then almost all arcs of X lift to arcs of
$\mathcal {X}$
, and we explain how in this case (assuming a finiteness hypothesis satisfied by fantastacks) our conjectures imply a formula for the stringy Hodge numbers of X in terms of a certain motivic integral over the arcs of
$\mathcal {X}$
. We prove these conjectures in the case where
$\mathcal {X}$
is a fantastack.
We prove the generic Torelli theorem for hypersurfaces in $\mathbb {P}^{n}$ of degree $d$ dividing $n+1$, for $d$ sufficiently large. Our proof involves the higher-order study of the variation of Hodge structure along particular one-parameter families of hypersurfaces that we call ‘Schiffer variations.’ We also analyze the case of degree $4$. Combined with Donagi's generic Torelli theorem and results of Cox and Green, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.
In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.
Let
$f\colon X\to B$
be a semistable fibration where X is a smooth variety of dimension
$n\geq 2$
and B is a smooth curve. We give the structure theorem for the local system of the relative
$1$
-forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of
$f_*\omega _{X/B}$
. We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if
$B=\mathbb {P}^1$
.
In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension
$\le 1$
. In this process, we establish the decomposition of Chow groups for the cases of the Cayley trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.