Let $K$ be a field of characteristic 0. Fix integers $r,d$ coprime with $r \geq 2$. Let $X_K$ be a smooth, projective, geometrically connected curve of genus $g \geq 2$ defined over K. Assume there exists a line bundle $L_K$ on $X_K$ of degree $d$. In this article we prove the existence of a stable locally free sheaf on $X_K$ with rank $r$ and determinant $L_K$. This trivially proves the $C_1$ conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.
Using the theory of mixed perverse sheaves, we extend arguments on the Hodge conjecture initiated by Lefschetz and Griffiths to the case of the Tate conjecture, and show that the Tate conjecture for divisors is closely related to the de Rham conjecture for nonproper varieties, finiteness of the Tate-Shafarevich groups, and also to some conjectures in the analytic number theory.
Let $f(\bf z,\bar{\bf z})$ be a strongly mixed homogeneous polynomial of 3 variables $\bf z=(z_1,z_2,z_3)$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space $\mathbb P^2$. The fundamental group of the complement $\mathbb P^2\setminus C$ is cyclic group of order $q$ if $f$ is homogeneous polynomial without $\bar{\bf z}$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.
In this note we prove the semiampleness conjecture for klt Calabi--Yau surface pairs over an excellent base ring. As applications we deduce that generalised abundance and Serrano's conjecture hold for surfaces. Finally, we study the semiampleness conjecture for Calabi--Yau threefolds over a mixed characteristic DVR.
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).
We prove a version of weakly functorial big Cohen-Macaulay algebras that suffices to establish Hochster-Huneke's vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot's theorem that direct summands of regular rings are pseudo-rational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by André and Bhatt.
We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the André-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.
We give a simple geometric characterization of the locus where the inscribed Banach--Colmez Tangent Spaces of moduli of mixed characteristic local shtukas with one leg and fixed determinant are connected. We conjecture that the structure morphism for the underlying diamond is cohomologically smooth over this locus and, applying the Fargues--Scholze Jacobian criterion, we prove this conjecture in the case of EL infinite level Rapoport--Zink spaces, generalizing a result of Ivanov--Weinstein in the basic case.
The Poincaré torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincaré torsor, but other, more geometric, approache…
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let $A$ be a mixed characteristic DVR, $G$ a reductive group scheme over $A$, $X$ an irreducible smooth projective $A$-scheme, $\mathcal G$ a principal $G$-bundle over $X$. Suppose $\mathcal G$ is generically trivial. We prove that in this case $\mathcal G$ is Zariski locally trivial. This result confirms the conjecture.
Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split.
We strengthen a conjecture by the author. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of mixed tilt-stable complexes on fibred threefolds. We extend it from complexes of mixed tilt-slope zero to arbitrary relative tilt-slope. We show that this stronger conjecture implies the support property of Bridgeland stability conditions, and the existence of explicit stability conditions. We prove our conjecture for ruled threefolds, hence improving a previous result by the author.
The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic …
We use plumbing calculus to prove the homotopy commutativity assertion of the Geometric P=W conjecture in all Painlevé cases. We discuss the resulting Mixed Hodge structures on Dolbeault and Betti moduli spaces.
Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with $r$ Laurent polynomials $f_1,...,f_r$.We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of $f_1,...,f_r$. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.
We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
In this paper, we prove that, if Deligne's "petites camarades conjecture" holds, then a Langlands type correspondence holds also for $p$-adic coefficients on a smooth curve over a finite field. We also prove that any overconvergent $F$-isocrystal of rank less than or equal to 2 on a smooth variety is $ι$-mixed.
Building on his reduction to the case of an unramified complete regular local ring R of mixed characteristic, we propose a proof in the framework of P. Scholze's perfectoid theory. The main ingredients are the perfectoid "Abhyankar lemma" and an analysis of Kummer extensions of R by a thickening technique.
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theo…
We construct smooth families of elliptic surface pairs with terminal singularities over a DVR of positive or mixed characteristic $(X,B)\to \mathrm{Spec}R$, such that $P_m(X_k,B_k)>P_m(X_K,B_K)$ for all sufficiently divisible $m>0$. In particular, this shows that invariance of all sufficiently divisible plurigenera does not follow from the MMP and Abundance Conjectures.
We study Deligne's conjecture on the monodromy weight filtration on the nearby cycles in the mixed characteristic case, and reduce it to the nondegeneracy of certain pairings in the semistable case. We also prove a related conjecture of Rapoport and Zink which uses only the image of the Cech restriction morphism, if Deligne's conjecture holds for a general hyperplane section. In general we show that Deligne's conjecture is true if the standard conjectures hold.
We propose a definition of ``nonabelian mixed Hodge structure'' together with a construction associating to a smooth projective variety $X$ and to a nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted $H=Hom(X_M, V)$. We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where $V$ has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with $\pp …
The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late sixties, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p. Recently, perfectoid techniques coming from p-adic Hodge theory have allowed us to get rid of any base field; this solves the direct summand conjecture and establishes the existence and weak functoriality of big Cohen-Macaulay algebras, which solve i…
We define etale cohomology of the moduli spaces of mixed characteristic local shtukas so that it gives smooth representations including the case where the relevant elements of the Kottwitz set are both non-basic. Then we relate the etale cohomology of different moduli spaces of mixed characteristic local shtukas using convolution morphisms, duality morphisms and twist morphisms. As an application, we show the Kottwitz conjecture in some new cases including the cases for all inner forms of $\mathrm{GL}_3$ and minuscule cocharacters. We study also some non-minuscule cases and show that the Kottw…
This is the second installment in a sequence of articles devoted to "explicit Chabauty-Kim theory" for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit equation whose halting will be conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via $p$-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm…
We prove that the category of mixed Tate motives over $\Z$ is spanned by the motivic fundamental group of $\Pro^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\Q$-linear combination of $ζ(n_1,..., n_r)$ where $n_i\in \{2,3\}$.
This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type C+ and D, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer group into secondary K-theory; construct a noncommutative motivic Gysin triangle; compute the localizing …
We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties,…
We construct the t-motivic mixed Carlitz zeta category over $\F_q(t)$ and show that it contains all the (mixed) t-motives with Carlitz-Thakur multi-zeta values as periods constructed by Anderson and Thakur. Our construction is canonical and our category is Tannakian and neutral and every object is equipped with a weight filtration whose graded pieces are Carlitz motives over $\F_q(t)$. For any finite separable extension L/\F_q(t) we show that existence of a similar category over $L$ is a consequence of a version of a conjecture of L. Taelman. Along the way we also prove the existence of the ca…
The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' $S$ there exists a motivic $t$-structure for the category of Voevodsky's $S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very reasonable' (for example, of finite type over a field) then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characteri…
The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.
We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.
In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over Noetherian normal domains, which is fundamental in the study of Euler system theory over normal domains and Iwasawa main conjectures.
We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures. In other words, every non-zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.
