Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincaré-Hopf when the ambient variety is a smooth manifold. One has the Schwartz index, the local Euler obstruction, the GSV-index, the virtual index and the homological index (and probably other indices too). In this article we give a brief overview of all these indices and the relations amongst them. We also indicate their relations with the various generalisations to complex analytic singular varieties of the Chern classes of complex manifolds.
This paper gives the all possible global indices of log Calabi-Yau 3-folds with standard coefficients on the boundaries and having lc, non-klt singularities. This follows easily from the discussion in the paper: The indices of log canonical singularities by Fujino (AG/9909035).
We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their computation.
This article is about 1-forms on complex analytic varieties and it is particularly relevant when the variety has non-isolated singularities. We first show how the radial extension technique of M.-H. Schwartz can be adapted to 1-forms, allowing us to define the Schwartz index of 1-forms with isolated singularities on singular varieties. Then we see how MacPherson's local Euler obstruction, adapted to 1-forms in general, relates to the Schwartz index, thus obtaining a proportionality theorem for these indices analogous to the one for vector fields. We also extend the definition of the GSV-index …
To each subvariety $X$ in projective $n$-space of codimension $m$ we associate an integer sequence of length $m + 1$ from $1$ to the degree of $X$ recording the maximal cardinalities of finite, reduced intersections of $X$ with linear subvarieties. We call this the sequence of secant indices of $X$. Similar numbers have been studied independently with the aim of classifying subvarieties with extremal secant spaces. Our focus in this note is the study of the combinatorial properties that the secant indices satisfy collectively. We show these sequences are strictly increasing for nondegenerate s…
By extending the notion of grid classes to include infinite grids, we establish a structural characterisation of the simple permutations in Av(4231, 35142, 42513, 351624), a pattern class which has three different connections with algebraic geometry, including the specification of indices of Schubert varieties defined by inclusions. This characterisation leads to the enumeration of the class.
For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of the group. We show that on a germ of a smooth complex-analytic G-variety these indices coincide. This permits to consider the difference between them as a version of the equivariant Milnor number of a germ a G-variety with an isolated singular point.
We discuss the notions of indices of vector fields and 1-forms and their generalizations to singular varieties and varieties with actions of finite groups, as well as indices of collections of vector fields and 1-forms.
This note was prepared after the workshop on cdh descent and algebraic $K$-theory in Hara-mura, Japan (1 - 5 Sept. 2016), to complement the author's talk on the history and applications of pro cdh descent. In the final section pro cdh descent is used to define $K$-groups of rigid analytic varieties. This note was revised and released in light of Kerz, Strunk, and Tamme establishing pro cdh descent in general (arXiv:1611.08466).
An overview of the geometry of Kottwitz-Viehmann varieties
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Morningside Center of Mathematics（数学研究所（晨兴中心））；State Key Laboratory of Mathematical Sciences（数学科学学院国家重点实验室）
We record an observation about the Witten indices in two families of gauged linear sigma models: the U(2) model for linear sections of Grassmannians, and the U(1) model for quadric complete intersections. We describe how the Witten indices are related to the Euler characteristics of the singular skew-symmetric or symmetric determinantal varieties featuring in the analysis of their opposite phases, and we discuss the extent to which these relationships can be reconciled with standard Born-Oppenheimer arguments.
In this paper we give top most and least indices of non vanishing cohomology modules of line bundles in the Kac Moody setting in terms of Weyl group combinatorics. We also prove a surjection Theorem.
Let G be a semisimple algebraic group over a field k. We introduce the higher Tits indices of G as the set of all Tits indices of G over all field extensions K/k. In the context of quadratic forms this notion coincides with the notion of the higher Witt indices introduced by M. Knebusch and classified by N. Karpenko and A. Vishik. We classify the higher Tits indices for exceptional algebraic groups. Our main tools involve the Chow groups and the Chow motives of projective homogeneous varieties, Steenrod operations, and the notion of the J-invariant of algebraic groups.
The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.
This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$. For a reduced word $\bf i$ of the longest element in the Weyl group of $G$ and a dominant integral weight $λ$, one can construct the string polytope $Δ_{\bf i}(λ)$, whose lattice points encode the character of the irreducible representation $V_λ$. The string polytope $Δ_{\bf i}(λ)$ is singular in general and combinatorics of string polytopes heavily depends on the choice of $\mathbf i$. In this paper, we study combinatorics of string polytopes when $G = SL_{n+1}(\mathbb{C})$, and present a sufficient condition on $\mathbf i$ such that …
Homological index of a holomorphic 1-form on a complex analytic variety with an isolated singular point is an analogue of the usual index of a 1-form on a non-singular manifold. One can say that it corresponds to the top Chern number of a manifold. We offer a definition of homological indices for collections of 1-forms on a (purely dimensional) complex analytic variety with an isolated singular point corresponding to other Chern numbers. We also define new invariants of germs of complex analytic varieties with isolated singular points related to "vanishing Chern numbers" at them.
This paper gives an overview of the various approaches towards F_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar's F_1-schemes, Toën and Vaquié's F_1-schemes, Haran's F-schemes, Durov's generalized schemes, Soulé's varieties over F_1 as well as his and Connes-Consani's variations of this theory, Connes and Consani's F_1-schemes, the author's torified varieties and Borger's Lambda-schemes. In a second part, we will tie up these different theories by describing functors between the different F_1-geometries, which partly rely on the work of other…
Given a scheme X over a field k, a generalized jet scheme parametrizes maps from Spec(A) to X, where A is a finite-dimensional, local algebra over k. We give an overview of known results concerning the dimensions of these schemes when A has embedding dimension one, when they are related to invariants of singularities in birational geometry. We end with a discussion of more general jet schemes.
We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.
In this expository note, we offer an overview of the relationship between Hodge-theoretic and geometric compactifications of moduli spaces of algebraic varieties.
Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behaviour of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as we…
We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.
There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, …
We give an overview of the theory of local G-shtukas and their moduli spaces that were introduced in joint work of U.~Hartl and the author, and in the past years studied by many people. We also discuss relations to moduli of global G-shtukas, properties of their special fiber through affine Deligne-Lusztig varieties and of their generic fiber, such as the period map.
