Rigidity for orientable functors

For a natural class of cohomology theories with support (including \'etale or pro-\'etale cohomology with suitable coefficients), we prove a moving lemma for cohomology classes with support on smooth quasi-projective k-varieties that admit a smooth projective compactification (e.g. if char(k)=0). This has the following consequences for such k-varieties and cohomology theories: a local and global generalization of the effacement theorem of Quillen, Bloch--Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, and a generalization of the injectivity property and the codimension 1 purity theorem for \'etale cohomology. Our results imply that the refined unramified cohomology groups from [Sch23] are motivic.Comment: 50 pages, to appear in EPIGA (special volume in honour of Claire Voisin)

There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e., for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the proétale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the proétale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating Milnor K-theory to Galois cohomology. The latter conjecture is known to be true in weight 2 for all primes [M-S] and in all weights for the prime 2 [V 3].

Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress toward a long-standing question. Along this line we prove that Chern numbers on nef vector bundles obey reverse dominance ordering, which improves upon some classical and recent results. We propose a simultaneous positivity question on (signed) Chern numbers of compact complex or Kähler manifolds whose (co)tangent bundles are semipositive in various senses, and show that it holds true for compact homogeneous complex manifolds.

The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g. to Hasse principles and the computation of motivic cohomology, and discuss some open questions.

For first-order topological semimetals, non-Hermitian perturbations can drive the Weyl nodes into Weyl exceptional rings having multiple topological structures and no Hermitian counterparts. Recently, it was discovered that higher-order Weyl semimetals, as a novel class of higher-order topological phases, can uniquely exhibit coexisting surface and hinge Fermi arcs. However, non-Hermitian higher-order topological semimetals have not yet been explored. Here, we identify a new type of topological semimetal, i.e., a higher-order topological semimetal with Weyl exceptional rings. In such a semimetal, these rings are characterized by both a spectral winding number and a Chern number. Moreover, the higher-order Weyl-exceptional-ring semimetal supports both surface and hinge Fermi-arc states, which are bounded by the projection of the Weyl exceptional rings onto the surface and hinge, respectively. Noticeably, the dissipative terms can cause the coupling of two exceptional rings with opposite topological charges, so as to induce topological phase transitions. Our studies open new avenues for exploring novel higher-order topological semimetals in non-Hermitian systems.

We establish the existence of an “Atiyah-Hirzebruch-like” spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.

Invariants, torsion indices and oriented cohomology of complete flags

This study adopts Hodge theory as a rigorous mathematical framework to construct a quantitative mapping system between the high-dimensional topological invariants of CalabiYau (CY) manifolds and nuclear physics parameters, thereby establishing a strict logical closure from algebraic geometry to low-energy nuclear physics. First, based on the Hodge decomposition theorem of complex manifolds and the geometric properties of compact Kähler manifolds, the study develops a topological characterization system for 3-dimensional complex CY manifolds, clarifying the algebraic structures of Hodge numbers, Chern classes, and Hodge classes, as well as their corresponding physical interpretations. Second, the proton/neutron distributions of nuclides are abstracted as algebraic cycles in Hodge theory (protons correspond to 1-codimensional cycles 𝒵1𝒵2cl: 𝒵𝑝 → ℋ𝑝ℙ3 , and neutrons correspond to 2-codimensional cycles). Using the surjectivity of the cohomology mapping (Lefschetz (1,1) theorem and its high-dimensional generalization), the quantitative relationship between “Hodge numbers and nucleon numbers” is derived.

The lectures which provided the source for these notes covered several different topics which are related to each other but which do not in any reasonable sense form a coherent whole. As a result, this text is a collection of four parts which refer to each other but otherwise are independent. In the first part we introduce the motivic homotopy category and connect it with the motivic cohomology theory discussed in [7]. The exposition is a little unusual because we wanted to avoid any references to model structures and still prove the main theorem 7. We were able to do it modulo 12 where we had to refer to the next part.

An analogue of the Dyer-Lashof algebra and an analogue of the Steenrod algebra are defined for generalized homology and cohomology theories. It is shown that if there is an -multiplicative structure on a spectrum , then on the corresponding generalized cohomology of a topological space there is an action of the Steenrod algebra, while if the space is an -space, then on the generalized homology there is an action of the Dyer-Lashof algebra. These actions are computed for cobordism of topological spaces. A connection is established between the Steenrod operations and the Landweber-Novikov operations.

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie bialgebra. This result is extended in Liu et al. (1998) to a classification of the Poisson homogeneous spaces of a Poisson groupoid (G⇉P, πG) via a special class of Dirac structures in the Courant algebroid defined by the corresponding Lie bialgebroid. It is hence natural to ask what corresponds via these classifications, or extensions of them, to arbitrary Dirac structures in the Drinfel'd double or the Courant algebroid associated to a Poisson group(oid). We show in this thesis that there is a bigger class of Dirac structures that fits in this correspondence. They correspond naturally to Dirac homogeneous spaces of the Poisson group(oid). Dirac Lie groups and Dirac groupoids have been defined by Ortiz (2009) as a generalization of Poisson Lie groups, Poisson groupoids and presymplectic groupoids. We prove in an alternative manner the existence of a natural Lie bialgebra associated to a Dirac Lie group, and we find good candidates for the infinitesimal data of a Dirac groupoid; a square of morphisms of Lie algebroids associated to the multiplicative Dirac structure. These objects generalize simultaneously the Lie bialgebroid of Poisson groupoids and the IM-2-forms associated to presymplectic groupoids (Bursztyn et al. (2004)). We show also the existence of a Courant algebroid associated to these algebroids. This Courant algebroid structure is induced in a natural way by the ambient standard Courant algebroid TG ×G T*G and is isomorphic to the Courant algebroid AG ×P A*G in the Poisson case. We show that Dirac homogeneous spaces of Dirac Lie groups (respectively Dirac groupoids) correspond to certain classes of Dirac structures in the Drinfel'd double of the Lie bialgebra (respectively in the Courant algebroid). These results generalize the classification theorems in Drinfel'd (1993) and Liu et al. (1998). Along the way, we study involutive multiplicative foliations on Lie groupoids and show when and how there is a natural groupoid structure defined on the leaf spaces in the set of objects and the set of units of the Lie groupoid. This is a fact that is always true in the Lie group case, where a multiplicative foliation is automatically the left and right invariant image of an ideal in the Lie algebra, i.e., the leaves are the cosets of a normal subgroup of the Lie group.

The quantitative nature of reduced Floer theory

. In this article, we consider the robust estimation for a class of varying coefficient partially linear EV models with a more general measurement error structure, which contains some classical measurement error structures such as the additive structure and the multiplicative structure. By combining the instrumental variable adjustment technique and the composite quantile regression method, we propose a robust empirical likelihood estimation procedure for parametric and non parametric components in such models. Under some regularity conditions, the resulting estimator of the parametric component is proved to be asymptotically normal, and the convergence rate of the estimator for non parametric component is also derived. Some simulations and a real data analysis are conducted for further illustrating the performance of the proposed method, and simulation results show that the proposed method is workable.