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It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form
$f=f^1\cdots f^{k_0}$
is uniquely determined by the Newton boundaries of
$f^1,\ldots , f^{k_0}$
if
$\{f^{k_1}=\cdots =f^{k_m}=0\}$
is a nondegenerate complete intersection variety for any
$k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$
.
A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.
We propose a conjectural list of Fano manifolds of Picard number
$1$
with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number
$1$
are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number
$1$
.
Topological complexity naturally appears in the motion planning in robotics. In this paper we consider the problem of finding topological complexity of real Grassmann manifolds $G_k(\mathbb {R}^{n})$. We use cohomology methods to give estimates on the zero-divisor cup-length of $G_k(\mathbb {R}^{n})$ for various $2\leqslant k< n$, which in turn give us lower bounds on topological complexity. Our results correct and improve several results from Pavešić (Proc. Roy. Soc. Edinb. A151 (2021), 2013–2029).
We explicate the combinatorial/geometric ingredients of Arthur’s proof of the convergence and polynomiality, in a truncation parameter, of his noninvariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthur’s results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthur’s work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence–Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii–Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.
We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally
$3$
-Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam [18]. We show that our categorification can be realised as a full extension closed subcategory of Jensen–King–Su’s Grassmannian cluster category [28], in a way compatible with their bijection between rank
$1$
modules and Plücker coordinates.
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.
The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.
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 study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks (with possibly nontrivial generic stabilisers K and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. An Aganagic-Vafa brane in this paper is a possibly ineffective
$C^\infty $
orbifold that admits a presentation
$[(S^1\times \mathbb {R} ^2)/G_\tau ]$
, where
$G_\tau $
is a finite abelian group containing K and
$G_\tau /K \cong \boldsymbol {\mu }_{\mathfrak {m}}$
is cyclic of some order
$\mathfrak {m}\in \mathbb {Z} _{>0}$
.
1. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack
$\mathcal {X}$
with boundaries mapped into an Aganagic-Vafa brane
$\mathcal {L}$
. All genus open-closed Gromov-Witten invariants of
$\mathcal {X}$
relative to
$\mathcal {L}$
are defined by torus localisation and depend on the choice of a framing
$f\in \mathbb {Z} $
of
$\mathcal {L}$
.
2. We provide another definition of all genus open-closed Gromov-Witten invariants in (1) based on algebraic relative orbifold Gromov-Witten theory, which agrees with the definition in (1) up to a sign depending on the choice of orientation on moduli of maps in (1). This generalises the definition in [57] for smooth toric Calabi-Yau 3-folds and specifies an orientation on moduli of maps in (1) compatible with the canonical orientation on moduli of relative stable maps determined by the complex structure.
3. When
$\mathcal {X}$
is a toric Calabi-Yau 3-orbifold (i.e., when the generic stabiliser K is trivial), so that
$G_\tau =\boldsymbol {\mu }_{\mathfrak {m}}$
, we define generating functions
$F_{g,h}^{\mathcal {X},(\mathcal {L},f)}$
of open-closed Gromov-Witten invariants of arbitrary genus g and number h of boundary circles; it takes values in
$H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )^{\otimes h}$
, where
$H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )\cong \mathbb {C} ^{\mathfrak {m}}$
is the Chen-Ruan orbifold cohomology of the classifying space
$\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}$
of
$\boldsymbol {\mu }_{\mathfrak {m}}$
.
4. We prove an open mirror theorem that relates the generating function
$F_{0,1}^{\mathcal {X},(\mathcal {L},f)}$
of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of
$\mathcal {X}$
. This generalises a conjecture by Aganagic-Vafa [6] and Aganagic-Klemm-Vafa [5] (proved in full generality by the first and the second authors in [33]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.
Let G be a simple complex algebraic group, and let
$K \subset G$
be a reductive subgroup such that the coordinate ring of
$G/K$
is a multiplicity-free G-module. We consider the G-algebra structure of
$\mathbb C[G/K]$
and study the decomposition into irreducible summands of the product of irreducible G-submodules in
$\mathbb C[G/K]$
. When the spherical roots of
$G/K$
generate a root system of type
$\mathsf A$
, we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of
$G/K$
is a direct sum of subsystems of rank 1.
We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups
${\textrm {GL}}_{2}$
and
${\textrm {GL}}_{3}$
.
We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.
Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text {der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb {F}_p$-sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb {F}_p$-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$-regular, and hence they are $F$-rational and have pseudo-rational singularities.
Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of $\operatorname{GL}(N-1,{\mathbb {C}}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $\operatorname{GL}_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
We consider G, a linear algebraic group defined over
$\Bbbk $
, an algebraically closed field (ACF). By considering
$\Bbbk $
as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space
$S^\mu _G(\Bbbk )$
consisting of
$\mu $
-types on G. We show that for each
$p_\mu \in S^\mu _G(\Bbbk )$
,
$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$
is a solvable infinite algebraic group when
$p_\mu $
is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of
$\mathrm {Stab}\left (p_\mu \right )$
in terms of the dimension of p.
Moduli spaces of bounded local G-shtukas are a group-theoretic generalisation of the function field analogue of Rapoport and Zink’s moduli spaces of p-divisible groups. In this article we generalise some very prominent concepts in the theory of Rapoport-Zink spaces to our setting. More precisely, we define period spaces, as well as the period map from a moduli space of bounded local G-shtukas to the corresponding period space, and we determine the image of the period map. Furthermore, we define a tower of coverings of the generic fibre of the moduli space, which is equipped with a Hecke action and an action of a suitable automorphism group. Finally, we consider the
$\ell $
-adic cohomology of these towers.
Les espaces de modules de G-chtoucas locaux bornés sont une généralisation des espaces de modules de groupes p-divisibles de Rapoport-Zink, au cas d’un corps de fonctions local, pour des groupes plus généraux et des copoids pas nécessairement minuscules. Dans cet article nous définissons les espaces de périodes et l’application de périodes associés à un tel espace, et nous calculons son image. Nous étudions la tour au-dessus de la fibre générique de l’espace de modules, équipée d’une action de Hecke ainsi que d’une action d’un groupe d’automorphismes. Enfin, nous définissons la cohomologie
$\ell $
-adique de ces tours.
We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra (dga) we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are $\hbox{M}$-varieties.
We explain how to form a novel dataset of Calabi–Yau threefolds via the Gross–Siebert algorithm. We expect these to degenerate to Calabi–Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities. In particular, we explain how to ‘smooth the boundary’ of a class of four-dimensional reflexive polytopes to obtain polarised tropical manifolds. We compute topological invariants of a compactified torus fibration over each such tropical manifold, expected to be homeomorphic to the general fibre of the Gross–Siebert smoothing. We consider a family of examples related to products of reflexive polygons. Among these we find $14$ topological types with $b_2=1$ that do not appear in existing lists of known rank-one Calabi–Yau threefolds.