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We provide a new Hodge theoretical characterization of the set of complex numbers that arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over ${\mathbb {P}}^1$ with four singular fibers. The characterization is approached via a detailed analysis of the periodicity of the uniformizing Higgs bundle attached to ${\mathbb {P}}^1$ minus four points over the field of complex numbers.

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.

Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and $e_2 \in T$ are negatively correlated for any distinct edges $e_1$ and $e_2$. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events $e \in B$, where $B$ is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of $k$-element independent sets of a matroid forms an ultra-log-concave sequence in $k$., 16 pages. Supersedes arXiv:1804.03071

To a one-parameter family of Calabi-Yau threefolds, we can associate the extended period map by the log Hodge theory of Kato and Usui. In the present paper, we study the image of a maximally unipotent monodromy point under the extended period map. As an application, we prove the generic Torelli theorem for a large class of one-parameter families of Calabi-Yau threefolds.

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models., Comment: 47 pages, updated to match the published version, to avoid confusion. Following referee's suggestion, we reorganized the paper so that all matrix factorization material appears in its own separate section

In this paper, we describe recent work towards the mirror $$\mathrm P=\mathrm W$$ conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi–Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [1] and de Cataldo and Migliorini [2]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror $$\mathrm P=\mathrm W$$ conjecture provides a possible bridge between the mirror $$\mathrm P=\mathrm W$$ conjecture and the well-known $$\mathrm P=\mathrm W$$ conjecture in non-Abelian Hodge theory.

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space., Comment: v1: 9 pages, comments welcome; v2: 8 pages, final and published version

Let p>3 be a prime number and K a finite extension of Q_p. We consider a proper and smooth surface XK over K, with a semistable model X over the ring of integers OK of K. In this thesis, we give a criterion for the good reduction of XK for the case of K3 surfaces, in terms of the monodromy operator in the second De Rham cohomology group. We don't use trascendental methods nor p-adic Hodge Theory as in other works concerning this problem. Instead, we first get a p-adic version of the Clemens-Schmid exact sequence and use it to study the degree of nilpotency of the monodromy operator N on the log-crystalline cohomology group of the special fiber Xs of the semistable model X. By the work of Nakkajima, we can assume that Xs is a combinatorial K3 surface. Then, we prove that Xs is of type I iff N=0; Xs if of type II iff N is not zero and N^2=0; Xs is of type III iff N^2 is not zero. In particular, this implies that XK has good reduction if and only if the monodromy operator on its second De Rham cohomology is zero. Finally, we also give some ideas on how to address the same problem for the case of Enriques surfaces. In particular, we prove that we are reduced to the case of K3 surfaces.

We study the nonnegativity of stringy Hodge numbers of a projective variety with Gorenstein canonical singularities, which was conjectured by Batyrev. We prove that the $(p,1)$-stringy Hodge numbers are nonnegative, and for threefolds we obtain new results about the stringy Hodge diamond, which hold even when the stringy $E$-function is not a polynomial. We also use the Decomposition Theorem and mixed Hodge theory to prove Batyrev's conjecture for a class of fourfolds., Comment: 34 pages

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kahler manifolds, which generalize Enoki’s original injectivity theorem.

Let K be a finite extension of ℚ p and G K the absolute Galois group. Then G K acts on the fundamental curve X of p -adic Hodge theory and we may consider the abelian category ℳ ( G K ) of coherent O X -modules equipped with a continuous and semilinear action of G K . An almost ℂ p -representation of G K is a p -adic Banach space V equipped with a linear and continuous action of G K such that there exists d ∈ ℕ , two G K -stable finite dimensional sub- ℚ p -vector spaces U + of V , U − of ℂ p d , and a G K -equivariant isomorphism V ∕ U + → ℂ p d ∕ U − . These representations form an abelian category C ( G K ) . The main purpose of this paper is to prove that C ( G K ) can be recovered from ℳ ( G K ) by a simple construction (and vice-versa) inducing, in particular, an equivalence of triangulated categories D b ( ℳ ( G K ) ) → D b ( C ( G K ) ) .

We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory., Comment: 18 pages; comments are welcome!

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of $L^2$-harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of $L^2$-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near $t = 0$., 48 pages. 1 figure

We present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2,C)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincare metric with Sl(2,R) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture., Comment: 57 pages, 2 figures, v2: minor clarifications, typos corrected, references added

We use the Decomposition Theorem to derive several generalizations of the Clemens–Schmid sequence, relating asymptotic Hodge theory of a degeneration to the mixed Hodge theory of its singular fiber(s).

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincar\'e duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well., Comment: 42 pages, 1 figure

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

Let $k $ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow and prove that the category of periodic Higgs-de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the \'{e}tale fundamental group $\pi_1(X_K)$ of the generic fiber of $X$, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber $X_k$ of $X$ of rank $\leq p$ initiates a semistable Higgs-de Rham flow and thus those of rank $\leq p-1$ with trivial Chern classes induce $k$-representations of $\pi_1(X_K)$. A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic $p$, it was constructed by Ogus-Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus-Vologodsky correspondence of Shiho., Comment: 60 pages (Updated Version). Subsume the manuscript with title "Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case", arXiv:1210.8280. Accepted for publication in "Journal of European Mathematical Society"

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme-precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis., 33 pages; Final version. Fixed typos, minor expository changes. Changed code repository link

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the Hodge theory. Our method is based on the theory of harmonic integrals and the L2-method for the dbar-equation, and it enables us to generalize the injectivity theorem to the complex analytic setting., Comment: 26pages, comments welcome

In this article, we consider $$L^{2}$$ harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form $$\omega $$ . The main result is that if $$(X,\omega )$$ is a complete $$G_{2}$$ - (or $$\textit{Spin}(7)$$ -) manifold with a d(linear) $$G_{2}$$ - (or $$\textit{Spin}(7)$$ -) structure form $$\omega $$ , then the $$L^{2}$$ harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on $$(X,\omega )$$ only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over $$(X,\omega )$$ .

We extend the Ax–Schanuel theorem recently proven for Shimura varieties by Mok–Pila–Tsimerman to all varieties supporting a pure polarizable integral variation of Hodge structures. In fact, Hodge theory provides a number of conceptual simplifications to the argument. The essential new ingredient is a volume bound for Griffiths transverse subvarieties of period domains.

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singularities, then any vector bundle on the resolution that appears to come from X numerically, does indeed come from X. Furthermore and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established., Final version. To appear in Compositio Mathematica

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.

The purpose of this paper is to work out the theoretical aspects of the vibration problem of rubber wiper blade on convex windshield. Over the past 20 years, some $2$-dimensional spring-mass models were presented in engineering science to simulate the vibration of rubber wiper blade on windshield. In this paper, we will consider the elasticity perspective on this $3$-dimensional vibration problem. Our theoretical analysis suggests that there should exist two classes of vibration frequencies corresponding to “$\ast$-exact deformations (Class I)” and “$\ast$-closed deformations (Class II)”. We prove mathematical theorems on the characterization of deformations of Class I. We also explain how elementary deformations of Class II can be constructed. We then deduce two mathematical formulas, for the vibration problem of rubber wiper blade on convex windshield, from our theoretical analysis. Our theoretical predictions are in almost perfect agreement with experimental data. One of the crucial steps of our analysis is a decomposition theorem motivated by the de Rham Cohomology and the Hodge Theory.

In p-adic Hodge theory, it is known that if a Galois representation is de Rham, then it becomes semistable after extension of the base field. Liu and Zhu asked whether a corresponding result holds in the relative setting: given an \'etale local system on a quasi-compact rigid analytic variety (for example, a projective scheme) over a p-adic field, does it become semistable after a finite extension of the base field? We give a counterexample., Comment: 1 page (!)

International audience; Using a relation due to Katz linking up additive and multiplicative convolutions, we make explicit the behaviour of some Hodge invariants by middle multiplicative convolution, following [DS13] and [Mar18a] in the additive case. Moreover, the main theorem gives a new proof of a result of Fedorov computing the Hodge invariants of hypergeometric equations.

This talk will explain the structure of Khovanov homology for knots and links and will present a quantum algorithm for its computation that is based on the use of combinatorial Hodge theory. By using combinatorial Hodge theory we can formulate the problem of computing the homology in terms of the determination of the dimension of the eigenvalue one subspace of a unitary operator that is associated with the boundary and coboundary operators of the Khovanov complex. We then use the quantum phase estimation algorithm to find the dimension of this subspace. This is a first foray into the problems of producing a quantum algorithm for the significant knot and link invariant Khovanov homology. There are other deeper problems related to the structure of the chain complex and we shall discuss these difficulties. This paper represents joint work with Sam Lomonaco and Nadya Shirakova.

We give explicit harmonic representatives of Dolbeault cohomology of Oeljeklaus-Toma manifolds and show that they are geometrically Dolbeault formal. We also give explicit harmonic representatives of Bott-Chern cohomology of Oeljeklaus-Toma manifolds of type $(s,1)$ and study the Angella-Tomassini inequality., Comment: 10 pages To appear in Proc. Amer. Math. Soc

The notion of Higgs-de Rham flows was introduced by Lan-Sheng-Zuo, as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory. In this short note we investigate a small part of this theory, and study those Higgs-de Rham flows which are of level zero. We improve the original definition of level-zero Higgs-de Rham flows (which works for general levels), and establish a Hitchin-Simpson-type correspondence between such objects and certain representations of fundamental groups in positive characteristic, which generalizes the classical results of Katz. We compare the deformation theories of two sides in the correspondence, and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side., Comment: 29 pages

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line., Comment: 35 pages. Comments are welcome

The cohomology rings of smooth complex projective algebraic varieties satisfy a “package” of properties: Poincare duality, weak Lefschetz, hard Lefschetz, and the Hodge–Riemann bilinear relations. These properties impose conditions on the structure of the cohomology ring, and give rise to interesting linear operators and bilinear forms on the ring. In several contexts this “package” yields important results about geometry, representation theory, or combinatorics. We discuss the above properties purely in terms of linear algebra, so that some of the techniques and intuition from geometry can be brought to bear on more general problems. Our focus is on the techniques which appear in the proof of Soergel’s conjecture, but there have been successes in other fields as well (e.g. in the combinatorics of matroids and polytopes), and it seems there will be more to come.

The ring of Witt vectors over a perfect valuation ring of characteristic p, often denoted \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), plays a pivotal role in p-adic Hodge theory; for instance, Bhatt–Morrow–Scholze have recently reinterpreted and refined the crystalline comparison isomorphism by relating it to a certain \(\mathbf {A}_{{{\mathrm{inf}\,}}}\)-valued cohomology theory. We address some basic ring-theoretic questions about \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), motivated by analogies with two-dimensional regular local rings. For example, we show that in most cases \(\mathbf {A}_{{{\mathrm{inf}\,}}}\), which is manifestly not noetherian, is also not coherent. On the other hand, it does have the property that vector bundles over the complement of the closed point in \({{\,\mathrm{Spec}\,}}\mathbf {A}_{{{\mathrm{inf}\,}}}\) do extend uniquely over the puncture; moreover, a similar statement holds in Huber’s category of adic spaces.

In this note we study the Bochner formula on smooth metric measure spaces. We introduce weighted curvature conditions that imply vanishing of all Betti numbers.

Functional transcendence results have in the last decade found a number of important applications to the algebraic and arithmetic geometry of varieties X admitting flat or hyperbolic uniformizations: Pila and Zannier’s new proof of the Manin–Mumford conjecture, the proof of the Andre–Oort conjecture for Ag, and the generic Shafarevich conjecture for hypersurfaces of Lawrence–Venkatesh, to name a few. The key insight (originally stemming from work of Pila and Zannier) is the use of o-minimality to pass between the geometry of X and that of its uniformizing space.

In the late ’80s, Faltings established an integral p-adic Hodge theory with coefficients, in which he generalized Fontaine–Laffaille theory of crystalline \(\mathbb Z_p\)-representations of the absolute Galois group of a p-adic field to the fundamental group of a non-singular algebraic variety over a p-adic field with good reduction. In this paper, we study the theory of coefficients above in the framework of integral p-adic Hodge theory via \(A_{\text {inf}}\)-cohomology recently introduced by Bhatt, Morrow, and Scholze. We give a local theory (i.e. a theory on an affine open) of \(A_{\inf }\)-cohomology for a p-torsion free crystalline \(\mathbb Z_p\)-representation of the fundamental group by constructing the associated \(A_{\inf }\)-representation with Frobenius, which is a variant of the construction by N. Wach of the \((\varphi ,\varGamma )\)-module associated to a crystalline \(\mathbb Z_p\)-representation of the absolute Galois group.

We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent hyper-K\"{a}hler varieties with $b_2(X_i)>6$ and if there exists a Hodge isometry $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, then the Andr\'e motives of $X_1$ and $X_2$ are isomorphic after a finite extension of $K$, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the \'{e}tale cohomology of $X_1$ and $X_2$ are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-K\"{a}hler varieties for which the Mumford--Tate conjecture is true., Comment: added Section 5; accepted for publication in Math. Zeit

In the previous chapter, an outline of the proof of Hodge theory for Soergel bimodules was given. A key issue is the absence of the weak Lefschetz theorem. In the proof of Elias and Williamson, a substitute for this theorem is provided by the homological algebra of Rouquier complexes—remarkable complexes of Soergel bimodules which categorify the braid group. In this chapter, we briefly describe the underlying idea, review some necessary bits of homological algebra, and introduce Rouquier complexes formally. We also discuss the “diagonal miracle,” which is a key step in establishing the hard Lefschetz theorem for Soergel bimodules.

We study a cohomology theory $$H^{\bullet }_{\varphi }$$ , which we call the $${\mathcal {L}}_B$$ -cohomology, on compact torsion-free $$\mathrm {G}_2$$ manifolds. We show that $$H^k_{\varphi } \cong H^k_{\mathrm {dR}}$$ for $$k \ne 3, 4$$ , but that $$H^k_{\varphi }$$ is infinite-dimensional for $$k = 3,4$$ . Nevertheless, there is a canonical injection $$H^k_{\mathrm {dR}} \rightarrow H^k_{\varphi }$$ . The $${\mathcal {L}}_B$$ -cohomology also satisfies a Poincare duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $$\mathrm {d}$$ and the derivation $${\mathcal {L}}_B$$ and uses both Hodge theory and the special properties of $$\mathrm {G}_2$$ -structures in an essential way. As an application of our results, we prove that compact torsion-free $$\mathrm {G}_2$$ manifolds are ‘almost formal’ in the sense that most of the Massey triple products necessarily must vanish.

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

We introduce the notion of effective resistance for a simplicial network ( X , R ) where X is a simplicial complex and R is a set of resistances for the top simplices, and prove two formulas generalizing previous results concerning effective resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a current generator σ, an extra top-dimensional simplex to be attached to X. We will show that the harmonic class gives rise to the current I σ and the voltage V σ for X ∪ σ , satisfying Thomson's energy-minimizing principle and Ohm's law for simplicial networks. The effective resistance R σ of a current generator σ shall be defined as a ratio of the σ-components of V σ and I σ . By introducing potential for voltage vectors, we present a formula for R σ via the inverse of the weighted combinatorial Laplacian of X in codimension one. We also derive a formula for R σ via weighted high-dimensional tree-numbers for X, providing a combinatorial interpretation for R σ . As an application, we generalize Foster's Theorem, and discuss various high-dimensional examples.

This note comes out of the author's lecture presented at the workshop Primitive forms and related subjects, Feb 10-14 2014.