Manuscripta Mathematica最新文献

We present a complete classification of invariant generalised Killing spinors on three-dimensional Lie groups. We show that, in this context, the existence of a non-trivial invariant generalised Killing spinor implies that all invariant spinors are generalised Killing with the same endomorphism. Notably, this classification is independent of the choice of left-invariant metric. To illustrate the computational methods underlying this classification, we also provide the first known examples of homogeneous manifolds admitting invariant generalised Killing spinors with n distinct eigenvalues for each $n>4$ .

Theta functions were defined for varieties with effective anticanonical divisor [11] and are related to certain punctured Gromov-Witten invariants [2]. We show that in the case of a log Calabi-Yau surface (X, D) with smooth very ample anticanonical divisor we can relate theta functions and their multiplicative structure to certain 2-marked log Gromov-Witten invariants. This is a natural extension of the correspondence between wall functions and 1-marked log Gromov-Witten invariants [8]. It gives an enumerative interpretation for the intrinsic mirror construction of [17] and will be related to the open mirror map of outer Aganagic-Vafa branes in [9].

We compute the quadratic Euler characteristic of the symmetric powers of a smooth, projective curve over any field k that is not of characteristic two, using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an application, we show that the power structure on the Grothendieck-Witt ring introduced by Pajwani-Pál computes the compactly supported $A^1$ -Euler characteristic of symmetric powers for all curves.

We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem [22] for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $C^1$ . With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $\mathrm{RCD}$ -splitting theorem [30, 31]. Along the way, we establish a Bochner-Weitzenböck identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in [62]. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see [62]) between distributional Ricci curvature bounds and $\mathrm{RCD}$ -type bounds, using in part the stability of the variable $\mathrm{CD}$ -condition under suitable limits [47].

Let X be a complex smooth Fano variety of dimension n. In this paper, we give a classification of such X when the pseudoindex is equal to (dfrac{dim X+1}{2}) and the Picard number greater than one.

We investigate the reduced unramified Witt group of the product of two smooth projective conics (X_1), (X_2) over a field. In some particular cases, this group denoted as (W(X_1,X_2)) turns out to be very small (zero or ({mathbb {Z}}/2{mathbb {Z}})). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of (W(X_1,X_2)) in terms of 2-fold Pfister forms (pi _1), (pi _2) associated with the conics. These conditions and constructions of the corresponding nonzero elements in (W(X_1,X_2)) depend on ({text {ind}}(pi _1+pi _2)). Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.

We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the k-th eigenvalue (lambda _{k}) as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of (lambda _{k})-extremal Kähler metric. We deduce a condition for a Kähler metric to be (lambda _{k})-extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.

We construct log canonical pairs (X, B) with B a nonzero reduced divisor and (K_X+B) ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in dimension 2, by Liu and Shokurov. The examples are weighted projective hypersurfaces that are not quasi-smooth. We also develop an example for a related extremal problem. Esser constructed a klt Calabi–Yau variety which conjecturally has the smallest mld in each dimension (for example, mld 1/13 in dimension 2 and 1/311 in dimension 3). However, the example was only worked out completely in dimensions at most 18. We now prove the desired properties of Esser’s example in all dimensions (in particular, determining its mld).

We construct some integral elements in the motivic cohomology of the Hesse cubic curves and express their regulators in terms of generalized hypergeometric functions and Kampé de Fériet hypergeometric functions. By using these hypergeometric expressions, we obtain numerical examples of the Bloch-Beilinson conjecture on special values of L-functions.

Let X be an orthogonal Shimura variety, and let (mathcal {C}^{textrm{ort}}_{r}(X)) be the cone generated by the cohomology classes of orthogonal Shimura subvarieties in X of dimension r. We investigate the asymptotic properties of the generating rays of (mathcal {C}^{textrm{ort}}_{r}(X)) for large values of r. They accumulate towards rays generated by wedge products of the Kähler class of X and the fundamental class of an orthogonal Shimura subvariety. We also compare (mathcal {C}^{textrm{ort}}_{r}(X)) with the cone generated by the special cycles of dimension r. The main ingredient to achieve the results above is the equidistribution of orthogonal Shimura subvarieties.