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In this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field
$K$
. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to
$\mathbb{Z}[1/p]$
-coefficients.
In this note, we prove the logarithmic
$p$
-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally
$K(\unicode[STIX]{x1D70B},1)$
(in a certain sense) with respect to
$\mathbb{F}_{p}$
-local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.
We describe generators of disguised residual intersections in any commutative Noetherian ring. It is shown that, over Cohen–Macaulay rings, the disguised residual intersections and algebraic residual intersections are the same, for ideals with sliding depth. This coincidence provides structural results for algebraic residual intersections in a quite general setting. It is shown how the DG-algebra structure of Koszul homologies affects the determination of generators of residual intersections. It is shown that the Buchsbaum–Eisenbud family of complexes can be derived from the Koszul–Čech spectral sequence. This interpretation of Buchsbaum–Eisenbud families has a crucial rule to establish the above results.
We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is
$F$
-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly
$F$
-regular varieties over arbitrary fields.
We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball
$\mathbb{B}^{n}$
using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice
$\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$
and the associated uniformization map
$\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$
. Given an algebraic subset
$S\,\subset \,\mathbb{B}^{n}$
and writing
$Z$
for the Zariski closure of
$\unicode[STIX]{x1D70B}(S)$
in
$X_{\unicode[STIX]{x1D6E4}}$
(which is equipped with a canonical quasi-projective structure), in some precise sense we realize
$Z$
as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component
$\widetilde{Z}$
of
$\unicode[STIX]{x1D70B}^{-1}(Z)$
as
$\widetilde{Z}$
exits the boundary
$\unicode[STIX]{x2202}\mathbb{B}^{n}$
by exploiting the strict pseudoconvexity of
$\mathbb{B}^{n}$
, culminating in the proof that
$\widetilde{Z}\,\subset \,\mathbb{B}^{n}$
is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of
$\text{ Aut}(\unicode[STIX]{x1D6FA})$
for (possibly reducible) bounded symmetric domains
$\unicode[STIX]{x1D6FA}$
.
The Chern–Schwartz–MacPherson class of a hypersurface in a nonsingular variety may be computed directly from the Segre class of the Jacobian subscheme of the hypersurface; this has been known for a number of years. We generalize this fact to arbitrary embeddable schemes: for every subscheme
$X$
of a nonsingular variety
$V$
, we define an associated subscheme
$\mathscr{Y}$
of a projective bundle
$\mathscr{V}$
over
$V$
and provide an explicit formula for the Chern–Schwartz–MacPherson class of
$X$
in terms of the Segre class of
$\mathscr{Y}$
in
$\mathscr{V}$
. If
$X$
is a local complete intersection, a version of the result yields a direct expression for the Milnor class of
$X$
.
For
$V=\mathbb{P}^{n}$
, we also obtain expressions for the Chern–Schwartz–MacPherson class of
$X$
in terms of the ‘Segre zeta function’ of
$\mathscr{Y}$
.
We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including
$\overline{\mathbb{Q}}$
and
$\overline{\mathbb{F}_{p}}$
, we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.
In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives
$\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$
generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over
$k$
. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.
Let
$G/H$
be a homogeneous variety and let
$X$
be a
$G$
-equivariant embedding of
$G/H$
such that the number of
$G$
-orbits in
$X$
is finite. We show that the equivariant Borel–Moore homology of
$X$
has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the
$G$
-orbits. If
$T$
is a maximal torus of
$G$
such that each
$G$
-orbit has a
$T$
-fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel–Moore homology of
$X$
. We apply our findings to certain wonderful compactifications as well as to double flag varieties.
We study the singularity at the origin of
$\mathbb{C}^{n+1}$
of an arbitrary homogeneous polynomial in
$n+1$
variables with complex coefficients, by investigating the monodromy characteristic polynomials
$\unicode[STIX]{x1D6E5}_{l}(t)$
as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case
$n=2$
, we give a description of
$\unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$
in terms of the multiplier ideal.
In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for
$k$
a perfect field of characteristic
$p$
, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over
$k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$
lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with
$\mathbb{Q}_{p}$
-coefficients.
We study
$F$
-signature under proper birational morphisms
$\unicode[STIX]{x1D70B}:Y\rightarrow X$
, showing that
$F$
-signature strictly increases for small morphisms or if
$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$
. In certain cases, we can even show that the
$F$
-signature of
$Y$
is at least twice as that of
$X$
. We also provide examples of
$F$
-signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses.
This note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.
We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.
Let
$X$
be a normal complex projective variety,
$T\subseteq X$
a subvariety of dimension
$m$
(possibly
$T=X$
) and
$a:X\rightarrow A$
a morphism to an abelian variety such that
$\text{Pic}^{0}(A)$
injects into
$\text{Pic}^{0}(T)$
; let
$L$
be a line bundle on
$X$
and
$\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$
a general element.
We introduce two new ingredients for the study of linear systems on
$X$
. First of all, we show the existence of a factorization of the map
$a$
, called the eventual map of
$L$
on
$T$
, which controls the behavior of the linear systems
$|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$
, asymptotically with respect to the pullbacks to the connected étale covers
$X^{(d)}\rightarrow X$
induced by the
$d$
-th multiplication map of
$A$
.
Second, we define the so-called continuous rank function
$x\mapsto h_{a}^{0}(X_{|T},L+xM)$
, where
$M$
is the pullback of an ample divisor of
$A$
. This function extends to a continuous function of
$x\in \mathbb{R}$
, which is differentiable except possibly at countably many points; when
$X=T$
we compute the left derivative explicitly.
As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form
We prove that a Küchle fourfold
$X$
of type d3 has a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of
$X$
.
In this paper, we construct Chern classes from the relative
$K$
-theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.
The aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold
$X$
by a non-symplectic involution
$\unicode[STIX]{x1D6FC}$
. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where
$X$
is the Hilbert scheme of two points on a K3 surface
$S$
, and the involution
$\unicode[STIX]{x1D6FC}$
is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold
$Y_{S}$
, which is the crepant resolution of
$X/\unicode[STIX]{x1D6FC}$
, with the Calabi–Yau 4-fold
$Z_{S}$
, constructed from
$S$
through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational
$2:1$
map from
$Z_{S}$
to
$Y_{S}$
.
We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo
$p$
prevents the existence of an action without fixed points of certain finite
$p$
-groups. The case of base fields of characteristic
$p$
is included. Counterexamples are systematically provided to test the sharpness of our results.