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In this note, we prove that the moduli stack of vector bundles on a curve with a fixed determinant is
${\mathbb A}^1$
-connected. We obtain this result by classifying vector bundles on a curve up to
${\mathbb A}^1$
-concordance. Consequently, we classify
${\mathbb P}^n$
-bundles on a curve up to
${\mathbb A}^1$
-weak equivalence, extending a result in [3] of Asok-Morel. We also give an explicit example of a variety which is
${\mathbb A}^1$
-h-cobordant to a projective bundle over
${\mathbb P}^2$
but does not have the structure of a projective bundle over
${\mathbb P}^2$
, thus answering a question of Asok-Kebekus-Wendt [2].
We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure.
We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.
We categorify the commutation of Nakajima’s Heisenberg operators
$P_{\pm 1}$
and their infinitely many counterparts in the quantum toroidal algebra
$U_{q_1,q_2}(\ddot {gl_1})$
acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical
$U_{q_1,q_2}(\ddot {gl_1})$
action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.
We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack
$\mathcal {X}$
, which specializes to the Batyrev–Manin conjecture when
$\mathcal {X}$
is a scheme and to Malle’s conjecture when
$\mathcal {X}$
is the classifying stack of a finite group.
We consider the family
$\mathrm {MC}_d$
of monic centered polynomials of one complex variable with degree
$d \geq 2$
, and study the map
$\widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$
which maps each
$f \in \mathrm {MC}_d$
to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber
$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$
for every
$\bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$
except when the fiber
$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$
contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math.322 (2017), 132–185] which gave a rather long algorithm with some induction processes.
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point
$x \in X$
to show that there exists an embedding from the Grassmannian variety
$\mathbb{G}(E_x,m)$
into the moduli space of torsion-free sheaves
$\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$
which induces an injective morphism from
$X \times M_{X,H}(n;\,c_1,c_2)$
to
$Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$
.
The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus 4 with unique trigonal structure.
An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group
${\mathbb {K}^{*}}$
commuting with the G-action. We show that X is determined by the
${\mathbb {K}^{*}}$
-variety
$X^U$
of fixed points under a maximal unipotent subgroup
$U \subset G$
. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient
$X /\!\!/ G$
.
If G is of type
${\mathsf {A}_n}$
(
$n\geq 2$
),
${\mathsf {C}_{n}}$
,
${\mathsf {E}_{6}}$
,
${\mathsf {E}_{7}}$
, or
${\mathsf {E}_{8}}$
, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If
$n \geq 5$
, every smooth affine
$\operatorname {\mathrm {SL}}_n$
-variety of dimension
$< 2n-2$
is an
$\operatorname {\mathrm {SL}}_n$
-vector bundle over the smooth quotient
$X /\!\!/ \operatorname {\mathrm {SL}}_n$
, with fiber isomorphic to the natural representation or its dual.
Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.
For any odd integer $d$, we give a presentation for the integral Chow ring of the stack $\mathcal {M}_{0}(\mathbb {P}^r, d)$, as a quotient of the polynomial ring $\mathbb {Z}[c_1,c_2]$. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form. The paper concludes with explicit computations of some examples for low values of $d$ and $r$, and a conjecture for a minimal set of generators.
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.
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 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 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 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 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.
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.
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.