arXiv subjects
Publications and source records attributed to Paolo Aluffi.
We introduce trim resolutions of complex algebraic varieties, a strengthening of the notion of small resolution. We prove that the characteristic cycle of the intersection cohomology sheaf of a variety admitting a trim resolution is irreducible and that for such varieties the stringy and Chern-Mather classes coincide.
We obtain explicit expressions for the class in the Grothendieck group of varieties of the moduli space of genus 0 stable curves with n marked points. This information is equivalent to the Poincar\'e polynomial; it implies explicit expressions for the Betti numbers of the moduli space in terms of Stirling numbers or, alternatively, Bernoulli numbers. The expressions are obtained by solving a differential equation characterizing the generating function for the Grothendieck class as shown in work of Yuri Manin from the 1990s. This differential equation is equivalent to S. Keel's recursion for the Betti numbers of these moduli spaces. Our proof reduces the solution to two combinatorial identities which follow from applications of Lagrange series. We also study generating functions for the individual Betti numbers. In previous work it had been shown that these functions are determined by a set of polynomials with positive rational coefficients, which are conjecturally log-concave. We verify this conjecture for many infinite families of these polynomials, corresponding to the generating functions for the $2k$-Betti numbers of the moduli spaces for all $k\le 100$. Further, studying these polynomials allows us to prove that the generating function for the Grothendieck class of the moduli spaces may be written as a series of rational functions in the Lefschetz motive and the principal branch of the Lambert W-function. We include an interpretation of the main result in terms of Stirling matrices and a discussion of the Euler characteristic of the moduli space.
Using a known recursive formula for the Grothendieck classes of the moduli spaces $\overline{\mathcal M}_{0,n}$, we prove that they satisfy an asymptotic form of ultra-log-concavity as polynomials in the Lefschetz class. We also observe that these polynomials are $γ$-positive. Both properties, along with numerical evidence, support the conjecture that these polynomials only have real zeros. This conjecture may be viewed as a particular case of a possible extension of a conjecture of Ferroni-Schröter and Huh on Hilbert series of Chow rings of matroids. We prove asymptotic ultra-log-concavity by studying differential equations obtained from the recursion, whose solutions are the generating functions of the individual betti numbers of $\overline{\mathcal M}_{0,n}$. We obtain a rather complete description of these generating functions, determining their asymptotic behavior; their dominant term is controlled by the coefficients of the Lambert W function. The $γ$-positivity property follows directly from the recursion, extending the argument of Ferroni et al. proving $γ$-positivity for the Hilbert series of the Chow ring of matroids.
We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are `sectional log-concave', that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences. We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with Segre classes of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation. Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the adjoint polynomial of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomial. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact dually Lorentzian, that is, Lorentzian after a certain change of variables.
The equivariant motivic Chern class of a Schubert cell in a `complete' flag manifold $X=G/B$ is an element in the equivariant K theory ring of $X$ to which one adjoins a formal parameter $y$. In this paper we prove several `folklore results' about the motivic Chern classes, including finding specializations at $y=-1$ and $y=0$; the coefficient of the top power of $y$; how to obtain Chern-Schwartz-MacPherson (CSM) classes as leading terms of motivic classes; divisibility properties of the Schubert expansion of motivic Chern classes. We collect several conjectures about the positivity, unimodality, and log concavity of CSM and motivic Chern classes of Schubert cells, including a conjectural positivity of structure constants of the multiplication of Poincar\'e duals of CSM classes. In addition, we prove a `star duality' for the motivic Chern classes. We utilize the motivic Chern transformation to define two equivariant variants of the Hirzebruch transformation, which appear naturally in the Grothendieck-Hirzebruch-Riemann-Roch formalism. We utilize the Demazure-Lusztig recursions from the motivic Chern class theory to find similar recursions giving the Hirzebruch classes of Schubert cells, their Poincar{\'e} duals, and their Segre versions. We explain the functoriality properties needed to extend the results to `partial' flag manifolds $G/P$.
Segre classes encode essential intersection-theoretic information concerning vector bundles and embeddings of schemes. In this paper we survey a range of applications of Segre classes to the definition and study of invariants of singular spaces. We will focus on several numerical invariants, on different notions of characteristic classes for singular varieties, and on classes of Le cycles. We precede the main discussion with a review of relevant background notions in algebraic geometry and intersection theory.
We express the Segre class of a monomial scheme -- or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections -- in terms of an integral computed over an associated body in euclidean space. The formula is in the spirit of the classical Bernstein-Kouchnirenko theorem computing intersection numbers of equivariant divisors in a torus in terms of mixed volumes, but deals with the more refined intersection-theoretic invariants given by Segre classes, and holds in the less restrictive context of `r.c. monomial schemes'.
We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-$4$ internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure, in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space $\mathcal M_{0,4}$. We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.
Motivic Chern classes are elements in the K-theory of an algebraic variety $X$, depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds $G/B$. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of $G$, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Feh\'er, Rim\'anyi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman's basis.
Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on $X$ with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Feh\'er and Rim\'anyi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.
Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its singular locus and, in the hypersurface case, Milnor numbers. We obtain simple proofs of these formulas, recovering along the way several classically known results.
Given a homogeneous ideal in a polynomial ring over C, we adapt the construction of Newton-Okounkov bodies to obtain a convex subset of Euclidean space such that a suitable integral over this set computes the Segre zeta function of the ideal. That is, we extract the numerical information of the Segre class of a subscheme of projective space from an associated (unbounded) Newton-Okounkov convex set. The result generalizes to arbitrary subschemes of projective space the numerical form of a previously known result for monomial schemes.
There is an explicit formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety (in characteristic $0$) in terms of the Segre class of its jacobian subscheme; this has been known for a number of years. We generalize this formula to arbitrary embeddable schemes: for every subscheme $X$ of a nonsingular variety $V$, we define an associated subscheme $Y$ of a projective bundle over $V$ and provide an explicit formula for the Chern-Schwartz-MacPherson class of $X$ in terms of the Segre class of $Y$. If $X$ is a local complete intersection, a version of the result yields a direct expression for the Milnor class of $X$.
We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X.
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex projective manifold $X$, the homogenized, torus equivariant CSM class of a constructible function $\varphi$ is the restriction of the characteristic cycle of $\varphi$ via the zero section of the cotangent bundle of $X$. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize $X$ to be a (generalized) flag manifold $G/B$. In this case CSM classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke orthogonality' of CSM classes, determined by the DL operator and its Poincar{\'e} adjoint. We further use the theory of holonomic $\mathcal{D}_X$-modules to show that the characteristic cycle of a Verma module, restricted to the zero section, gives the CSM class of the corresponding Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov's stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rim{\'a}nyi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the expansion in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds $G/P$.
We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a `Segre-Bertini' theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general hyperplane sections. These results interpolate between an observation of Samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideals. The Segre-Bertini theorem has applications to characteristic classes of singular varieties. The main results are motivated by the problem of computing Segre classes explicitly and applications of Segre classes to enumerative geometry.
We define a power series associated with a homogeneous ideal in a polynomial ring, encoding information on the Segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension. We prove that this power series is rational, with poles corresponding to generators of the ideal, and with numerator of bounded degree and with nonnegative coefficients. We also prove that this `Segre zeta function' only depends on the integral closure of the ideal. The results follow from good functoriality properties of the `shadows' of rational equivalence classes of projective bundles. More precise results can be given if all homogeneous generators have the same degree, and for monomial ideals. In certain cases, the general description of the Segre zeta function given here leads to substantial improvements in the speed of algorithms for the computation of Segre classes. We also compute the projective ranks of a nonsingular variety in terms of the corresponding zeta function, and we discuss the Segre zeta function of a local complete intersection of low codimension in projective space.
We study a class obtained from the Segre class $s(Z,Y)$ of an embedding of schemes by incorporating the datum of a line bundle on $Z$. This class satisfies basic properties analogous to the ordinary Segre class, but leads to remarkably simple formulas in standard intersection-theoretic situations such as excess or residual intersections. We prove a formula for the behavior of this class under linear joins, and use this formula to prove that a `Segre zeta function' associated with ideals generated by forms of the same degree is a rational function.