Manuscripta Mathematica最新文献

We compute the integral Grothendieck rings of the moduli stacks, ({{mathcal {M}}}_2), (overline{{{mathcal {M}}}}_2) of smooth and stable curves of genus two respectively. We compute ({{,textrm{K},}}_0({{mathcal {M}}}_2)) by using the presentation of ({{mathcal {M}}}_2) as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring ({{,textrm{K},}}_0(overline{{{mathcal {M}}}}_2)) we decompose (overline{{{mathcal {M}}}}_2) as (Delta _1) and its complement (overline{{{mathcal {M}}}}_2 setminus Delta _1) and use their presentations as quotient stacks given by Larson (Algebr Geom 8 (3):286–318, 2021) to compute the Grothendieck rings. We show that they are torsion-free and this, together with the Riemann–Roch isomorphism allows us to ultimately give a presentation for the integral Grothendieck ring ({{,textrm{K},}}_0(overline{{{mathcal {M}}}}_2)).

We show that shrinking Kähler-Ricci solitons over a compact Kähler manifold are gradient shrinking Kähler-Ricci solitons. The proof relies on a remarkable identity on the kernels of a real and a complex elliptic operator proved in our solution of the variational stability problem for gradient shrinking Kähler-Ricci solitons in Pali (Complex Manifolds 3(1):41–144, 2016).

For an elliptic surface (pi :Xrightarrow mathbb {P}^1) defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of (pi ). When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of (mathbb {P}^1) for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.

Let (mathcal {I}_{d,g,r}) be the union of irreducible components of the Hilbert scheme whose general points represent smooth, irreducible, non-degenerate curves of degree d and genus g in (mathbb {P}^r). Using a family of curves found on ruled surfaces over smooth curves of genus (gamma ), we show that for (gamma ge 7) and (g ge 6 gamma + 5), the scheme (mathcal {I}_{2g-4gamma + 1, g, g - 3gamma + 1}) acquires a non-reduced component (mathcal {D}^{prime }) such that ({text {dim}}T_{[X^{prime }]} mathcal {D}^{prime } = {text {dim}}mathcal {D}^{prime } + 1) for a general point ([X^{prime }] in mathcal {D}^{prime }).

In this paper, we describe the structure of the negative part of a Zariski decomposition of (K_X+K_{{{mathcal {F}}}}) for a relatively minimal foliation ((X,{{mathcal {F}}})) whenever (K_X+K_{{{mathcal {F}}}}) is pseudoeffective.

Let (ell ) be a prime number different from the residue characteristic of a non-archimedean local field F. We give formulations of (ell )-adic local Langlands correspondences for connected reductive algebraic groups over F, which we conjecture to be independent of a choice of an isomorphism between the (ell )-adic coefficient field and the complex number field.

We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case (textrm{U} (1)times textrm{U} (2)).

Given a scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field, the Greenberg transform produces a new scheme over the residue field thicker than the special fiber. In this paper, we will generalize this transform to the case of imperfect residue field. We will then construct a certain kind of cycle class map defined on this generalized Greenberg transform applied to the Néron model of a semi-abelian variety, which takes values in the relatively perfect nearby cycle functor defined by Kato and the second author.

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension 2 closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension 2 closed subset in an affine smooth complete intersection of low degree.

For an effective Cartier divisor D on a scheme X we may form an ({n}^{text {th}}) root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is (2n)-periodic. For (n=2) this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For (n > 2) we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (N-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.