Lefschetz Formula Research Articles

Lefschetz Formula for Nonlocal Elliptic Problems Associated with a Bundle

Lefschetz Formula for Nonlocal Elliptic Problems Associated with a Bundle

A categorification of the quantum Lefschetz principle

The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus 0, regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the ∞-categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to recover the classical Quantum Lefschetz formula when its hypotheses are satisfied.

Virtual cycles on projective completions and quantum Lefschetz formula

For a compact quasi-smooth derived scheme M with (−1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas [17]. We produce virtual cycles on both the projective completion N‾:=P(N⊕OM) and projectivisation P(N) and show the ones on N‾ push down to Jiang-Thomas cycles and the one on P(N) computes the difference.Using similar ideas we give an expression for the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan [15].

A geometric Jacquet\u2013Langlands correspondence for paramodular Siegel threefolds

We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between {text {GSp}}_4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of {text {GSp}}_4.

Building lattices and zeta functions

AbstractWe give a Lefschetz formula for tree lattices and apply it to geometric zeta functions. We further generalize Bass’s approach to Ihara zeta functions to the higher-dimensional case of a building.

Polynomial Factorization Statistics and Point Configurations in ℝ3

Abstract We use combinatorial methods to relate the expected values of polynomial factorization statistics over $\mathbb{F}_q$ to the cohomology of ordered configurations in $\mathbb{R}^3$ as a representation of the symmetric group. Our method gives a new proof of the twisted Grothendieck–Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.

Geometric zeta functions for higher rank $p$-adic groups

The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions

Lefschetz type formulas for dg-categories

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230–256, 2012) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngo (Bull Math Sci 1(1):129–199, 2011). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity $${\varvec{w}}$$ . We apply this formula to get some restrictions on the Betti numbers of a $${\mathbb Z}/2$$ -equivariant module over $$k[[x_1,\ldots ,x_n]]/({\varvec{w}})$$ in the case when $${\varvec{w}}(-x)={\varvec{w}}(x)$$ .

A Brouwer fixed-point theorem for graph endomorphisms

We prove a Lefschetz formula for graph endomorphisms , where G is a general finite simple graph and ℱ is the set of simplices fixed by T. The degree of T at the simplex x is defined as , a graded sign of the permutation of T restricted to the simplex. The Lefschetz number is defined similarly as in the continuum as , where is the map induced on the k th cohomology group of G. The theorem can be seen as a generalization of the Nowakowski-Rival fixed-edge theorem (Nowakowski and Rival in J. Graph Theory 3:339-350, 1979). A special case is the identity map T, where the formula reduces to the Euler-Poincare formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if is nonzero, then T has a fixed clique. A special case is the discrete Brouwer fixed-point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. If is the automorphism group of a graph, we look at the average Lefschetz number . We prove that this is the Euler characteristic of the chain and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function is a product of two dynamical zeta functions and, therefore, has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits. MSC:58J20, 47H10, 37C25, 05C80, 05C82, 05C10, 90B15, 57M15, 55M20.

Degenerate flag varieties of type A: Frobenius splitting and BW theorem

Let \(\mathcal F ^a_\lambda \) be the PBW degeneration of the flag varieties of type \(A_{n-1}\). These varieties are singular and are acted upon with the degenerate Lie group \(SL_n^a\). We prove that \(\mathcal F ^a_\lambda \) have rational singularities, are normal and locally complete intersections, and construct a desingularization \(R_\lambda \) of \(\mathcal F ^a_\lambda \). The varieties \(R_\lambda \) can be viewed as towers of successive \(\mathbb{P }^1\)-fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties \(R_\lambda \) are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for \(\mathcal F ^a_\lambda \). Using the Atiyah–Bott–Lefschetz formula for \(R_\lambda \), we compute the \(q\)-characters of the highest weight \(\mathfrak sl _n\)-modules.

On the minimum dilatation of braids on punctured discs

We find the minimum dilatation of pseudo-Anosov braids on n-punctured discs for 3 <= n <= 8. This covers the results of Song-Ko-Los (n=4) and Ham-Song (n=5). The proof is elementary, and uses the Lefschetz formula.

Erratum to: A higher rank Lefschetz formula

For Proposition 1.4.1 to be correct, one has to insist that the open set C be regular in the sense that C is the open interior of its closure. In this way the boundary of C is also the boundary of the complement. Furthermore, in order to apply Proposition 1.4.1 to the construction of the test function f in Section 4.3, the condition ∂F (C) ⊂ F (∂C) needs to be satisfied. As it stands (page 35), this condition is not satisfied. One can, however, replace the set N in the definition of the set C on page 35 with an open subset U of N which has compact support. This suffices for the application and then the proof of the property ∂F (C) ⊂ F (∂C) runs as follows: Let (kj , nj ,mj , aj) be a sequence in C such that F (kj , nj ,mj , aj) = kjnjajmjn −1 j k −1 j converges to some g ∈ G. As K is compact, we can assume that kj converges to some k ∈ K. Replacing g with k−1gk we reduce to k = 1, so the sequence njajmjn −1 j converges to g. As nj varies in a relatively compact set, we can assume that nj converges to some n ∈ N . We have njajmjn −1 j = aj }{{} ∈A mj }{{} ∈M n ajmj j n −1 j } {{ } ∈N .

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes' C ∗ -algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.

A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres

In this article, we consider singular equivariant arithmetic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason in [31]. In particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. Rössler in [25].

A higher rank Lefschetz formula

Since the early seventies flows with the Lefschetz property were studied by several authors. In this paper a Lefschetz formula is proved for the geodesic flow of a compact locally symmetric space. The flow is described in terms of actions of split tori of various dimensions and the geometric side of the Lefschetz formula is a sum over closed geodesics which correspond to a given torus. The cohomological side is given in terms of Lie algebra cohomology.

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ℓ-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost–Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.

A discrete Lefschetz formula

A Lefschetz formula is given that relates loops in a regular finite graph to traces of a certain representation. As an application the vanishing orders of the Ihara/Bass zeta function are expressed as dimensions of global section spaces of locally constant sheaves.

On a holomorphic Lefschetz formula in strictly pseudoconvex subdomains of complex manifolds

The classical Lefschetz formula expresses the number of fixed points of a continuous map in terms of the transformation induced by on the cohomology of . In 1966, Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they obtained a holomorphic Lefschetz formula on compact complex manifolds without boundary. Brenner and Shubin (1981, 1991) extended the Atiyah-Bott theory to compact manifolds with boundary. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, therefore the Atiyah-Bott theory is not applicable. Bypassing difficulties related to the boundary behaviour of Dolbeault cohomology, Donnelly and Fefferman (1986) obtained a formula for the number of fixed points in terms of the Bergman metric. The aim of this paper is to obtain a Lefschetz formula on relatively compact strictly pseudoconvex subdomains of complex manifolds with smooth boundary, that is, to find the total Lefschetz number for a holomorphic endomorphism of the Dolbeault complex and to express it in terms of local invariants of the fixed points of .

A Prime geodesic theorem for higher rank spaces

A prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula for higher rank torus actions.

A higher Lefschetz formula for flat bundles

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.