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Abstract

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy (psi ^1) of a symplectic manifold ((M, omega )) fixing a relatively exact Lagrangian L setwise must act trivially on (R_*(L)) , where (R_*) is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over ({mathbb {Z}}/2) and over ({mathbb {Z}}) under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, (psi ^1|_L) is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that (psi ^1|_L) acts trivially on (R_*({mathcal {L}}L)) , where ({mathcal {L}}L) is the free loop space of L. From this we deduce that when L is a surface or a (K(pi , 1)) , (psi ^1|_L) is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over ({mathbb {Z}}/2) .

Abstract

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type.

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra (textrm{R}) by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over (textrm{R}) equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over (textrm{R}[t]/(t^2)). Using these results together with results of Geiss, Leclerc and Schröer we give, when (textrm{k}) is algebraically closed, a classification of pairs ((Q,textrm{R})) such that the set of isomorphism classes of indecomposable locally free representations of Q over (textrm{R}) is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra (mathbb {F}_q[t]/(t^r)). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.

Let (ell ) be a prime number. We classify the subgroups G of ({text {Sp}}_4({mathbb {F}}_ell )) and ({text {GSp}}_4({mathbb {F}}_ell )) that act irreducibly on ({mathbb {F}}_ell ^4), but such that every element of G fixes an ({mathbb {F}}_ell )-vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree (ell ) between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and (ell ) is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes (ell ) for which some abelian surface (A/{mathbb {Q}}) fails the local-global principle for isogenies of degree (ell ).

We consider closed subschemes in the affine grassmannian obtained by degenerating e-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For (G= {text {GL}}_2), and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of G. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton–Gee stack classifying p-adic representations of the Galois group of a finite extension of ({mathbb {Q}}_p). As an application we prove new cases of the Breuil–Mézard conjecture in dimension two.

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi–Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional hypersurfaces of degree (dge lceil 5(n+3)/6rceil ) are not birational to Jacobian fibrations of dimension one. A key part of the argument is to resolve singularities of general (mu _{p})-covers in mixed characteristic p.

Let ( {{textbf{k}}}) be a field and (Qin {{textbf{k}}}[x_1, ldots , x_s]) a form (homogeneous polynomial) of degree (d>1.) The ({{textbf{k}}})-Schmidt rank (text {rk}_{{textbf{k}}}(Q)) of Q is the minimal r such that (Q= sum _{i=1}^r R_iS_i) with (R_i, S_i in {{textbf{k}}}[x_1, ldots , x_s]) forms of degree (<d). When ( {{textbf{k}}}) is algebraically closed and ( text {char}({{textbf{k}}})) doesn’t divide d, this rank is closely related to ( text {codim}_{{mathbb {A}}^s} (nabla Q(x) = 0)) - also known as the Birch rank of Q. When ( {{textbf{k}}}) is a number field, a finite field or a function field, we give polynomial bounds for ( text {rk}_{{textbf{k}}}(Q) ) in terms of ( text {rk}_{{bar{{{textbf{k}}}}}} (Q) ) where ( {bar{{{textbf{k}}}}} ) is the algebraic closure of ( {{textbf{k}}}. ) Prior to this work no such bound (even ineffective) was known for (d>4). This result has immediate consequences for counting integer points (when ( {{textbf{k}}}) is a number field) or prime points (when ( {{textbf{k}}}= {mathbb {Q}})) of the variety ( (Q=0) ) assuming ( text {rk}_{{{textbf{k}}}} (Q) ) is large.

We prove that for a Baire-generic Riemannian metric on a closed smooth manifold, the union of the images of all stationary geodesic nets forms a dense set.

Let k be a perfect field of characteristic (p ge 3), and let K be a finite totally ramified extension of (K_0 = W(k)[p^{-1}]). Let (L_0) be a complete discrete valuation field over (K_0) whose residue field has a finite p-basis, and let (L = L_0otimes _{K_0} K). For (0 le r le p-2), we classify (textbf{Z}_p)-lattices of semistable representations of (textrm{Gal}(overline{L}/L)) with Hodge–Tate weights in [0, r] by strongly divisible lattices. This generalizes the result of Liu (Compos Math 144:61–88, 2008). Moreover, if (mathcal {X}) is a proper smooth formal scheme over (mathcal {O}_L), we give a cohomological description of the strongly divisible lattice associated to (H^i_{acute{text {e}}text {t}}(mathcal {X}_{overline{L}}, textbf{Z}_p)) for (i le p-2), under the assumption that the crystalline cohomology of the special fiber of (mathcal {X}) is torsion-free in degrees i and (i+1). This generalizes a result in Cais and Liu (Trans Am Math Soc 371:1199–1230, 2019).