Selecta Mathematica

We apply the barcodes of persistent homology theory to the c Chekanov–Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov–Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level. As an application we show that it is not possible to $$C^0$$ -approximate a stabilized Legendrian by a Legendrian that admits an augmentation.

We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $$2^{g-1}$$ even theta characteristics and $$2^{g-1}$$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.

We study diophantine properties of a typical point with respect to measures on $\mathbb{R}^n .$ Namely, we identify geometric conditions on a measure μ on $\mathbb{R}^n $ guaranteeing that μ-almost every ${\bf y}\,\in\,\mathbb{R}^n $ is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called ‘friendly’. Examples include smooth measures on nondegenerate manifolds; thus this paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.

We obtain sharp inequalities between the large scale asymptotic of the J functional with respect to the $$d_1$$ metric on the space of Kähler metrics. Applications regarding the initial value problem for geodesic rays are presented.

We define various height functions for motives over number fields. We compare these height functions with classical height functions on algebraic varieties, and also with analogous height functions for variations of Hodge structures on curves over $${\mathbb C}$$ . These comparisons provide new questions on motives over number fields.

In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to $${{\mathbb {C}}} \times {\mathcal {H}}$$ for $$l\ge 3$$ . In the present paper we define a new notion of norm, which distinguishes $$\{D^b(K(l)) \}_{l\ge 2}$$ . More precisely, to a triangulated category $${\mathcal {T}}$$ which has property of a phase gap we attach a non-negative real number $$\left\| {\mathcal {T}}\right\| ^{\varepsilon }$$ . Natural assumptions on a SOD $${\mathcal {T}} =\langle {\mathcal {T}}_1,{\mathcal {T}}_2\rangle $$ imply $$\left\| \langle {\mathcal {T}}_1,{\mathcal {T}}_2\rangle \right\| ^{\varepsilon }\le {\mathrm{min}}\{\left\| {\mathcal {T}}_1 \right\| ^{\varepsilon },\left\| {\mathcal {T}}_2\right\| ^{\varepsilon } \}$$ . Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that $$[D^b(point)] \in \mathrm{Cl}([D^b(S)])$$ . Categories in a neighborhood of $$D^b(K(l))$$ have the property that $$D^b(K(l))$$ is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi–Yau curve-counting, where the entities we count are a Calabi–Yau modification of $$D^b(K(l))$$ . In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.

We propose an alternative definition for families of stable pairs (X, D) over an arbitrary (possibly non-reduced) base in the case in which D is reduced, by replacing (X, D) with an appropriate orbifold pair $$(\mathcal {X},\mathcal {D})$$ . This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Adjunction for $$(\mathcal {X},\mathcal {D})$$ holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial morphisms from $$(n + 1)$$ dimensional stable pairs to n dimensional polarized orbispaces. As an application, we study the deformation theory of some surface pairs.