Miguel A. Marco-Buzunáriz
We present a method to compute the Euler characteristic of an algebraic subset of $\bc^n$. This method relies on clasical tools such as Gröbner basis and primary decomposition. The existence of this method allows us to define a new invariant for such varieties. This invariant is related to the poblem of counting rational points over finite fields.
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We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.
We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.
We prove a formula that relates the Euler-Poincaré characteristic of a closed semi-algebraic set to its Lipschitz-Killing curvatures
Schanuel Integration and Euler Characteristic of Semi-algebraic Sets
该研究将Schanuel积分框架扩展到半代数集，证明其积分与基选择无关可定义欧拉特征，且与经典欧拉特征一致，其递归纤维结构能解释欧拉特征并简化经典性质证明。
The Euler characteristic of the link of a real algebraic variety is an interesting topological invariant in order to discuss local topological properties. We prove in the paper that an invariant stronger than the Euler Characteristic is well defined for the link of an algebraic variety: its virtual Poincaré polynomial.
This note studies the behavior of Euler characteristics and of intersection homology Euler characterstics under proper morphisms of algebraic (or analytic) varieties. The methods also yield, for algebraic (or analytic) varieties, formulae comparing these two kinds of Euler characteristics. The main results are direct consequences of the calculus of constructible functions and Grothendieck groups of constructible sheaves. Similar formulae for Hodge theoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in \cite{CS1, S}.
Macpherson defined Chern-Schwartz-Macpherson (CSM) classes by introducing the (local) Euler obstruction function, which is an integer valued function on the variety that is constant on each stratum of a Whitney stratification of an algebraic variety. By understanding the Euler obstruction function, one gains insights about a singular algebraic variety. It was recently shown by the author and B. Wang, how to compute these functions using maximum likelihood degrees. This paper discusses a symbolic and a numerical implementation of algorithms to compute the Euler obstruction at a point. Macaulay2…
We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.
We derive a general formula for the Euler characteristic of a fibration of projective hypersurfaces in terms of invariants of an arbitrary base variety. When the general fiber is an elliptic curve, such formulas have appeared in the physics literature in the context of calculating D-brane charge for M-/F-theory and type-IIB compactifications of string vacua. While there are various methods for computing Euler characteristics of algebraic varieties, we prove a base-independent pushforward formula which reduces the computation of the Euler characteristic of relative hypersurfaces to simple algeb…
Given a projective algebraic variety $X$, let $Π_p(X)$ denote the monoid of effective algebraic equivalence classes of effective algebraic cycles on $X$. The $p$-th Euler-Chow series of $X$ is an element in the formal monoid-ring $Z[[Π_p(X)]]$ defined in terms of Euler characteristics of the Chow varieties $\cvpd{p}α{X}$ of $X$, with $α\inΠ_p(X)$. We provide a systematic treatment of such series, and give projective bundle formulas which generalize previous results by B. Lawson and S.S.Yau and Elizondo. The techniques used involve the Chow quotients introduced by Kapranov, and this allows the …
This is an extended abstract of my talk at the Oberwolfach Workshop "Interactions between Algebraic Geometry and Noncommutative Algebra" (May 10 - 14, 2010). We present some properties of quiver Grassmannians and examples of explicit computations of their Euler characteristics.
We relate Fubini's theorem for Euler characteristics to Riemann-Hurwtiz formulae, and reprove a classical result of Iversen. The techniques used include algebraic geometry, complex geometry, and model theory. Possible applications to the study of wild ramification in finite characteristic are discussed.
We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.
We use basic algebraic topology and Ellingsrud-Stromme results on the Betti numbers of punctual Hilbert schemes of surfaces to compute a generating function for the Euler characteristic numbers of the Douady spaces of "n-points" associated with a complex surface. The projective case was first proved by L. Göttsche.
Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincaré characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincaré characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in…
We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $α$ with respect to a complex algebraic Whitney stratification of $X$. We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of $α$. Whenever $α$ is the Euler obstruction of $X$, this index formula specializes to the Seade-Tibar-Verjovsky global counterpart of the Le-Teissier formula for the local Euler obstruction.
We introduce a Grothendieck group of algebraic stacks (with affine stabilisers) analogous to the Grothendieck group of algebraic varieties. We then identify it with a certain localisation of the Grothendieck group of algebraic varieties. Several invariants of elements in this group are discussed. The most important is an extension of the Euler characteristic (of cohomology with compact support) but in characteristic zero we introduce invariants which are able to distinguish between classes with the same Euler characteristic. These invariants are actually defined on the completed localised Grot…
Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log terminal pairs. There is a natural Kawamata log terminal pair corresponding to an algebraic variety V having a regular action of a finite group G. In this situation we show that the stringy Euler number of this pair coincides with the physicists' orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture of Miles Reid on the Euler numbers of crepant desingularizations of Gorenstein quotient sing…
We study the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves and having Euler characteristic one. We determine locally free resolutions of length one for all such sheaves. We decompose the moduli space into strata which occur naturally as quotients modulo actions of certain algebraic groups. In some cases we give concrete geometric descriptions of the strata.
We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.
We study the Euler characteristic of the real Milnor fibres of a real analytic map, using a relation between complex monodromy and complex conjugation. We deduce the result of Coste and Kurdyka that the Euler characteristic of the link of an irreducible algebraic subset of a real algebraic set is generically constant modulo 4. We generalize this result to iterated links of ordered families of algebraic subsets.
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. In the space of Chern numbers there are two distinguished subspaces, one spanned by the Euler and Pontryagin numbers, the other spanned by the Hirzebruch--Todd numbers. Their intersection is the span of the …
The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.
Using the weak factorization theorem we give a simple presentation for the value group of the universal Euler characteristic with compact support for varieties of characteristic zero and describe the value group of the universal Euler characteristic of pairs. This gives a new proof for the existence of natural Euler characteristics with values in the Grothendieck group of Chow motives. A generalization of the presentation to the relative setting allows us to define duality and the six operations.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\ell$ be another prime number. O. Gabber and F. Loeser proved that for any algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf $\calF$ on $T$ the Euler characteristic $χ(\calF)$ is non-negative. We conjecture that the same result holds for any perverse sheaf $\calF$ on a reductive group $G$ over $k$ which is equivariant with respect to the adjoint action. We prove the conjecture when $\calF$ is obtained by Goresky-MacPherson extension from the set of regular semi-simple elements in $G$. From this we deduce th…
For an affine complex algebraic singular space Y, we define a global Euler obstruction Eu(Y) which extends the Euler-Poincare characteristic of a nonsingular Y. Using Lefschetz pencils, we express Eu(Y) as alternating sum of global polar invariants.
This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-…
The Euler characteristic of Chow varieties of algebraic cycles of a given degree in complex projective spaces was computed by Blaine Lawson and Stephen Yau by using holomorphic symmetries of cycles spaces. In this paper we compute this in a direct and elementary way and generalize this formula to the l-adic Euler-Poincare characteristic for Chow varieties over any algebraically closed field. Moreover, the Euler characteristic for Chow varieties with certain group action is calculated. In particular, we calculate the Euler characteristic of the space of right quaternionic cycles of a given dime…
We lift the Euler characteristic of a nearly perfect complex to a relative algebraic K-group by passing to its l-adic Euler characteristics.
Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, and $S \subset \R^{\ell+k}$ a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \geq 0, P \leq 0, P \in {\mathcal P} \cup {\mathcal Q}$. We prove that the sum of the Betti numbers of $S$ is bounded by \[ \ell^2 (O(s+\ell+m)\ell d)^{k+2m}. \] This is a common generalization of previous…
It is well known that the Euler characteristic of the cohomology of a complex algebraic variety coincides with the Euler characteristic of its cohomology with compact support. An old result of G. Laumon asserts that a relative version of this statement is true in $\ell$-adic cohomology. The purpose of this note is to extend Laumon's result to the topological setting. Some applications are also discussed.
Euler systems and the symmetric square of a Hida family
对于素数\(p\geq7\)，构建\(p\)进希达族模形式对称平方的非平凡欧拉系统，证明对偶塞尔默群的代数函数方程，并基于相关工作证明关于希达族对称平方的岩泽主猜想的可除性结果。
Let X be an algebraic projective variety in {\bf P}^n. Denote by {\cal C}_λ the space of all effective cycles on X whose homology class is λ\in H_{2p} (X,{\bf Z}). It is easy to show that {\cal C}_λ is an algebraic projective variety. Let χ({\cal C}_λ be its Euler characteristic. Define the Euler series of X by E_{p} = \sum_{λ\in{C}} \, χ({\cal C}_λ λ \in \, {\bf Z}[[C]] where {\bf Z}[[C]] is the full algebra over {\bf Z} of the monoid C of all homology classes of effective p-cyles on X. This algebra is the ring of function (with respect the convolution product) over C. Denote by {\bf Z}[C] th…
Euler characteristic of crepant resolutions of specific modular quotient singularities
本文研究正特征下特殊线性群有限子群商奇点双解析解欧拉示性数的McKay对应推广，利用有限域上wild McKay对应证明其对特定半直积群成立，并提出模情形下的猜想。
It is an observation of Kontsevich and Soibelman that generating series that produce certain (generalized) Donaldson Thomas invariants are secretly algebraic functions over the rationals. From a physical perspective this observation arises naturally for DT invariants that appear as BPS indices in theories of class S[A]: explicit algebraic equations (that completely determine these series) can be derived using (degenerate) spectral networks. In this paper, we conjecture an algebraic equation associated to DT invariants for the Kronecker 3-quiver with dimension vectors (3n,2n), n>0 in the non-tr…
The Arnold inequalities characterizing the topology of non-singular plane real algebraic curves and the generalization of these inequalities for nodal curves by Viro are extended in this paper for the curves whose singularities have non-degenerated Milnor form. This involves an invariant of such real singularities, the real linking form, which is calculated.
Euler Stratifications of Second Hypersimplices via Delta-matroids
通过Delta-拟阵与主A-行列式的非零因子对应，研究第二超单形缩放环面的欧拉示性数，证明Clarke等人(2024)关于最小ML度数的猜想。
Let $V$ be a possibly singular scheme-theoretic complete intersection subscheme of $\mathbb{P}^n$ over an algebraically closed field of characteristic zero. Using a recent result of Fullwood ("On Milnor classes via invariants of singular subschemes", Journal of Singularities) we develop an algorithm to compute the Chern-Schwartz-MacPherson class and Euler characteristic of $V$. This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern-Schwartz-MacPherson class and Euler characteristic for certain types of complete intersection subschem…
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