30 pages; v5: added the proof of the Lefschetz type formula conjectured by Frenkel and Ngo; v6: added comparison with Mukai pairing, revised the section on applications to matrix factorizations
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 V. Lunts 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. Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity w(x). We apply this formula to get some restrictions on the Betti numbers of a Z/2-equivariant module over k[[x_1,...,x_n]]/(w) in the case when w(-x)=w(x).
We consider the bounded derived category of $S_k$-equivariant coherent sheaves on $(\mathbb{P}^n)^k$. The goal of this paper is to construct in this category a rectangular Lefschetz exceptional collection when this is possible, or a minimal Lefschetz exceptional collection when a rectangular one does not exist. The main results of the paper include the construction of a rectangular Lefschetz exceptional collection in the case $k=3$ and in the case $n=1$ when $\mathrm{gcd}(n+1,k)=1$. We also construct minimal Lefschetz exceptional collection for $n=1$ and even $k$, and for $n=2$ and $k=3$.
We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian $\text{G}(k,n)$ we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for $k = p$, a prime number, modulo completeness of Fonarev's collection (and for $p = 3$ we check this completeness).
This is the first article in an upcoming series of papers. They have arisen through an attempt to answer open questions of Deligne proposed in "Le determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf. "Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm. Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary results on virtual categories which are the categories where the Deligne…
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 particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. Rössler.
We reprove the Lefschetz trace formula for Lubin-Tate spaces, based on the locally finite cell decompositions of these spaces obtained by Fargues, and Mieda's theorem of Lefschetz trace formula for certain open adic spaces (\cite{Mi1} theorem 3.13). This proof is rather different from those of Strauch in \cite{St} (theorem 3.3.1) and of Mieda in \cite{Mi1} (example 4.21), and is quite hopeful to generalized to some other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, we proved a Lefschetz trace formula for some unitary Rapoport-Zink spaces in \cite{Sh} b…
In this article, we discuss the Lefschetz trace formula for an adic space which is separated smooth of finite type but not necessarily proper over an algebraically closed non-archimedean field. Under a certain condition on the absence of set-theoretical fixed points on the boundary, we obtain a fixed point formula. As an application, we can establish a trace formula for some formal schemes, which is applicable to the Rapoport-Zink tower for GSp(4). A partial generalization of Fujiwara's trace formula for contracting morphisms is also given.
The goal of this paper is to generalize a theorem of Fujiwara (formerly Deligne's conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is more elementary than the original one and stays in the realm of ordinary algebraic geometry, that is, does not use rigid geometry. We also include proof of the Lefschetz-Verdier trace formula and of the additivity of filtered trace maps, thus making the paper essentially self-contained.
In this paper we prove a concentration theorem for arithmetic $K_0$-theory, this theorem can be viewed as an analog of R. Thomason's result in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. Köhler and D. Roessler and they also gave a correct route of its proof. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.
We study Lefschetz fixed point formulas for constructible sheaves with higher-dimensional fixed point sets. Under fairly weak assumptions, we prove that the local contributions from them are expressed by some constructible functions associated to hyperbolic localizations. This gives an affirmative answer to a conjecture of Goresky-MacPherson in particular for smooth fixed point components. In the course of the proof, the new Lagrangian cycles introduced in our previous paper will be effectively used. Moreover we show various examples for which local contributions can be explicitly determined b…
We continue our study of the genus-$0$ permutation-equivariant quantum K-theory of the target $X=pt$, and completely determine the "big J-function" of this theory. The computation is based on the application of Lefschetz' fixed point formula to the action of $S_n$ on $\overline{M}_{0,n+1}$. It is an instance of the general "adelic characterization" (which we state at the end with reference to arXiv:1106.3136) of quantum K-theory for any target $X$ in terms of quantum cohomology theory. Yet, some simplifications of non-conceptual nature occur in this example, making it a lucid illustration to t…
We use Lefschetz pencil methods to derive structural results about Fukaya categories of Calabi-Yau hypersurfaces; in particular, concerning their dependence on the Novikov parameter. Warning to the reader: the arguments in this paper cite two preprints (papers 6.5 and 7 in the series) which do not at present exist. Until that is remedied, the proofs cannot be considered complete, and this paper should be regarded as a preliminary research announcement.
Laszlo and Olsson constructed Grothendieck's six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. In this article we give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed in arXiv:1211.1877. As an application, we prove a Lefschetz-Verdie…
A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration.
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. We produce virtual cycles on both the projective completion and projectivisation and show the ones on the former push down to Jiang-Thomas cycles and the one on the latter 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.
We introduce new Lagrangian cycles which encode local contributions of Lefschetz numbers of constructible sheaves into geometric objects. We study their functorial properties and apply them to Lefschetz fixed point formulas with higher-dimensional fixed point sets.
In this follow-up to arXiv:2007.11642, our main result is a tropical Lefschetz-Hopf trace formula for matroidal automorphisms. We show that both sides of the formula are equal to the (generalized) beta invariant of the lattice of fixed flats.
Deligne's conjecture is the Lefschetz trace formula for correspondences defined over a finite field. In this paper, we prove an analogous statement of Deligne's conjecture with respect to $p^n$-torsion étale cohomology under certain conditions, where $p$ is the characteristic of the base field.
This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut-Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.
Let $Y$ be a complete intersection in an affine variety $X$, with action by a complex reductive group $G$. Let $T \subset G$ be a maximal torus. A character $θ$ of $G$ defines GIT quotients $Y//_θG$ and $X//_θT$. We prove formulas relating the small quasimap I-function of $Y//_θG$ to that of $X//_θT$. When $X$ is a vector space, this provides a completely explicit formula for the small $I$-function of $Y//_θG$.
Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.
We use Noether-Lefschetz theory to study the reduced Gromov--Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov-Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov-Witten techniques we also determine the generating series of Noether-Lefschetz numbers of a general pencil of Debarre-Voisin varieties. This reproves and extends a result of Debarre, Han, O…
We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.
In this article we prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $p>0$ degenerate at $E_1$. We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.
We investigate the alternating sum of the l-adic cohomology of the Rapoport-Zink tower for GSp(4) by the Lefschetz trace formula. Under some assumptions on L-packets of GSp(4) and its inner form, we observe that the local Jacquet-Langlands correspondence appears in the cohomology.
We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milno…
There exists a well-known Lefschetz formula for the number of fixed points in algebraic topology. In algebraic geometry, there exist cohomologies of coherent sheaves. It is natural to consider the same alternated sum of traces as in Lefschetz formula for cohomologies of sheaves. In the present paper, we consider the case of the action of a one-dimensional torus on a smooth projectve variety. Using theory of adelic complexes it becomes possible to rewrite coherent Lefschetz formula in a new compact form and then to prove it.
V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.
We study the Lefschetz fixed point formula for constructible sheaves with higher-dimensional fixed point sets. We give another proof to the explicit description of Lefschetz cycles in our previous paper. For this purpose, we introduce a new notion of shrinking subbundles and describe Lefschetz cycles by using hyperbolic localization with respect to shrinking subbundles.
We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bézout's theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation the…
In this paper we investigate the representations of reductive groups over a finite field, introduced in 1987 by D.Kazhdan and G.Laumon. We show that generically these representations are irreducible and that their character is equal to the function obtained by traces of Frobenius morphism on the stalks of the corresponding Lusztig's character sheaf. This, together with the corresponding result by G.Lusztig, implies that generic Kazhdan-Laumon representations are isomorphic to the corresponding Deligne-Lusztig representations. In the second half of the paper we give a direct geometric construct…
We give a new proof of a slightly weaker form of a theorem of P. Colmez. This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over $\bf Q$ is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in the first paper of the series. Our proof is intrinsic in the sense that it does not rely on the computation of the periods of any particular abelian variety.
Let $G$ be a compact Lie group acting on a compact complex manifold $M$. We prove a trace density formula for the $G$-Lefschetz number of a differential operator on $M$. We generalize Engeli and Felder's recent results to orbifolds.
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz …
The Lefschetz Type Theorem For Fundamental Group Schemes
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education（同济大学数学科学学院，智能计算与应用教育部重点实验室）
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport \cite{LR87} of deriving the Kottwitz's formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands-Rapoport conjecture), which replaces their Galois gerb theoretic arguments by more geometric …
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported étale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Es…
We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the "Euler characteristic integral" of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are étale, we compute this integral in terms of Morel's identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck-Witt ring. In particular, we show that the Euler characteristic of an étale algebra corresponds to the class of its trace form in the Grothendieck…
We introduce a notion of Hochschild Lefschetz class for a good coherent D-module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann-Roch formula for a D-module on a compact complex orbifold.
Let ${\mathcal E}$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{\mathbb C}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $ψ_{2n}$ of the algebra ${\rm Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{{\mathcal E},ψ_{2n}}(\mathcal D)$ from any holomorphic differential operator ${\mathcal D}$ on ${\mathcal E}$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that $\int_X f_{{\mathcal E},ψ_{2n}}({\mathcal D})$ gives the Lefschetz number of $\mathcal D$ upto a…
We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.
A formula for calculating the Lefschetz number of an automorphism acting on a crepant resolution for a quotient of a Kahler manifold derived from an equivariant version of McKay correspondence. The latter is proven in some cases. As an application the Lefschetz numbers of of involutions acting on Calabi-Yau threefolds and their mirrors are compared.
The Lefschetz formula for the action of a Hecke correspondence on the weighted cohomology of a locally symmetric space is derived. It is also proven that each Hecke correspondence on the reductive Borel-Serre compactification of the locally symmetric space is weakly hyperbolic.
We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and étale cohomology spaces of projective varieties. The gen…
We reprove the Lefschetz trace formula for stacks (in the context of derived categories and the six operations for stacks developed by Laszlo and Olsson), and give the meromorphic continuation of L-series (in particular, zeta functions) of Artin stacks over a finite field. We also give an upper bound for the weights of the cohomology groups of stacks, and an "independence of l" result for a certain class of quotient stacks.
Given a symmetric monoidal $(\infty,2)$-category $\mathscr E$ we promote the trace construction to a functor. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).
