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For a weight structure w on a triangulated category
$\underline {C}$
we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case
$w=w^{sph}$
(the spherical weight structure on
$SH$
), we deduce the following converse to the stable Hurewicz theorem:
$H^{sing}_{i}(M)=\{0\}$
for all
$i<0$
if and only if
$M\in SH$
is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.
The main idea is to study M that has no weights
$m,\dots ,n$
(‘in the middle’). For
$w=w^{sph}$
, this is the case if there exists a distinguished triangle
$LM\to M\to RM$
, where
$RM$
is an n-connected spectrum and
$LM$
is an
$m-1$
-skeleton (of M) in the sense of Margolis’s definition; this happens whenever
$H^{sing}_i(M)=\{0\}$
for
$m\le i\le n$
and
$H^{sing}_{m-1}(M)$
is a free abelian group. We also consider morphisms that kill weights
$m,\dots ,n$
; those ‘send n-w-skeleta into
$m-1$
-w-skeleta’.
The goal of this paper is to raise the possibility that there exists a meaningful theory of ‘motives’ associated with certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss’s
${}_2F_1$
hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show, on the one hand, how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the local action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the global action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.
We propose a variation of the classical Hilbert scheme of points, the double nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss $K$-theoretic refinements à la Nekrasov–Okounkov.
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.
We prove that any nef $b$-divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef $b$-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef $b$-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of $b$-divisors which were defined in our previous work.
We prove that torsion codimension
$2$
algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension
$2$
on supersingular abelian varieties over the algebraic closure of finite fields.
Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes
$(\mathbb {C}^{2})^{[n]}$
on
$n$
points, as
$n\rightarrow +\infty ,$
is a Gumbel distribution. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes
$((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$
that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer
$A\geq 2.$
Furthermore, if
$p_{k}(A;n)$
denotes the number of partitions of
$n$
with exactly
$k$
parts that are multiples of
$A$
, then we obtain the asymptotic
Dans cet article on caractérise la lissité du schéma Quot ponctuel emboîté d’une variété lisse—c’est-à-dire l’espace de modules paramétrant les drapeaux de quotients de dimension
$0$
d’un faisceau localement libre fixé. Nos résultats étendent la classification de Cheah dans le cadre des schémas de Hilbert ponctuels emboîtés.
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.
We prove an extension of the Kato–Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s formula for the Chow groups of
$0$
-cycles on such schemes. We identify the Chow group of
$0$
-cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasiprojective schemes over algebraically closed fields. This completes the missing p-part in the torsion theorem of Spieß and Szamuely.
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.
Consider three normalized cuspidal eigenforms of weight
$2$
and prime level p. Under the assumption that the global root number of the associated triple product L-function is
$+1$
, we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve
$X_0(p)$
is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.
In this article, we study a Gysin triangle in the category of motives with modulus (Theorem 1.2). We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives (Corollary 1.6). The corollary implies that an object in $\operatorname {\mathbf {MDM}^{\operatorname {eff}}}$ decomposes into a p-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.
A conic bundle is a contraction
$X\to Z$
between normal varieties of relative dimension
$1$
such that
$-K_X$
is relatively ample. We prove a conjecture of Shokurov that predicts that if
$X\to Z$
is a conic bundle such that X has canonical singularities and Z is
$\mathbb {Q}$
-Gorenstein, then Z is always
$\frac {1}{2}$
-lc, and the multiplicities of the fibres over codimension
$1$
points are bounded from above by
$2$
. Both values
$\frac {1}{2}$
and
$2$
are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension
$1$
with canonical singularities.
For a smooth rigid space X over a perfectoid field extension K of
$\mathbb {Q}_p$
, we investigate how the v-Picard group of the associated diamond
$X^{\diamondsuit }$
differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence
We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space
$\mathbb {A}^n$
, the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.
We offer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their six-functor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud’s approach to rigid analytic geometry.
In this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.
We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves.
Let X be a smooth proper variety over a field k and suppose that the degree map ${\mathrm {CH}}_0(X \otimes _k K) \to \mathbb {Z}$ is isomorphic for any field extension $K/k$. We show that $G(\operatorname {Spec} k) \to G(X)$ is an isomorphism for any $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers G. This generalises a result of Binda, Rülling and Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge–Witt cohomology is a $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers.