On the topology of algebraic surfaces and reduction modulo p

Algebraic cobordism theory attached to algebraic equivalence

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordis…

Relative exactness modulo a polynomial map and algebraic $(\mathbb{C}^p,+)$-actions

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from $\mathbb{C}^n$ to $\mathbb{C}^q$. We study the quotient ${\cal{T}}^1(F)$ of polynomial 1-forms that are exact along the fibres of $F$, by 1-forms of type $dR+\sum a_idf_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${\cal{T}}^1(F)$ is always a torsion $\mathbb{C}[t_1,...,t_q]$-module. The we determine under which conditions on $F$ we have ${\cal{T}}^1(F)=0$. As an application, we study the behaviour of a class of algebraic $(\mathbb{C}^p,+)$-actions on $\mathbb{C}^n$, and determine in particular when these actions are trivial…

Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties

We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological.

The Simplicial EHP Sequence in A1-Algebraic Topology

We give a tool for understanding simplicial desuspension in A1-algebraic topology: we show that $X \to Ω(S^1 \wedge X)\to Ω(S^1 \wedge X \wedge X)$ is a fiber sequence up to homotopy in 2-localized A1-algebraic topology for $X = (S^1)^m \wedge G_m^{\wedge q}$ with $m>1$. It follows that there is an EHP sequence spectral sequence $$Z_{(2)} \otimes π_{n+1+i}^{A1} (S^{2n+2m+1} \wedge (G_m)^{\wedge 2q}) \Rightarrow Z_{(2)} \otimes π_{i}^{A1, s} (S^{m} \wedge (G_m)^{\wedge q}).$$

Topology of Kempf-Ness sets for algebraic torus actions

In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. By making use of the recent achievements of "toric topology" we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox "geometric invariant theory" approach to toric varieties. We proceed by studying the cohomology of these "toric" Kempf-Ness sets. In the case …

The Decomposition Theorem and the topology of algebraic maps

We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples.

Topology of plane algebraic curves: the algebraic approach

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a semi-algebraic function defined on a closed semi-algebraic set X. We give formulas relating the topology of X to the indices of the critical points of the function and to the topological behavior of the function at infinity. We give applications when X is R^n and when the function is linear.

Calculabilité de la cohomologie étale modulo l

Let $X$ be an algebraic scheme over an algebraically closed field and $\ell$ a prime number invertible on $X$. According to classical results (due essentially to A. Grothendieck, M. Artin and P. Deligne), the étale cohomology groups $\mathrm{H}^i(X,\mathbb{Z}/\ell\mathbb{Z})$ are finite-dimensional. Using an $\ell$-adic variant of M. Artin's good neighborhoods and elementary results on the cohomology of pro-$\ell$ groups, we express the cohomology of $X$ as a well controlled colimit of that of toposes constructed on $BG$ where the $G$ are computable finite $\ell$-groups. From this, we deduce t…

Algebraically constructible functions: real algebra and topology

Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space to be homeomorphic to a real algebraic set.

关于实代数栈的拓扑

On the topology of real algebraic stacks

本文发展多种方法研究实Deligne-Mumford栈实轨迹的拓扑，并验证了若干情形下Smith-Thom型不等式，结合了群论、代数几何与拓扑学技术。

Algebraic structures on the topology of moduli spaces of curves and maps

We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.

Around rationality of integral cycles

In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for coefficients modulo 2. Those results have already been proved by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic different from 2.

Topology of Singularities on Algebraic Loop Spaces

We study the topology of some simple infinite dimensional singularities arising from spaces of \emph{algebraic formal loops}. We prove that in some simple cases the natural analogue of nearby cycles cohomology for a function on the loop space vanishes, and show further that a suitably renormalized version of the above cohomology produces a simple (non-zero) result.

Perverse sheaves and the topology of algebraic varieties (2015 PCMI)

Another introduction to perverse sheaves with some exercises. Expanded version of five lectures at the 2015 PCMI.

Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry

This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in th…

Representations of non-negative polynomials via critical ideals

This paper studies the representations of a non-negative polynomial $f$ on a non-compact semi-algebraic set $K$ modulo its critical ideal. Under the assumptions that the semi-algebraic set $K$ is regular and $f$ satisfies the boundary Hessian conditions (BHC) at each zero of $f$ in $K$, we show that $f$ can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if $f\ge 0$ on $K$. In particular, we focus on the polynomial ring $\mathbb R[x]$.

Some remarks about the Zariski topology of the Cremona group

For an algebraic variety $X$ we study the behavior of algebraic morphisms from an algebraic variety to the group $\bir(X)$ of birational maps of $X$ and obtain, as application, some insight about the relationship between the so-called Zariski topology of $\bir(X)$ and the algebraic structure of this group, where $X$ is a rational variety.

Projective normality of quotient varieties modulo finite groups

In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.

Topology of real algebraic curves near the non-singular tropical limit

In the 1990's, Itenberg and Haas studied the relations between combinatorial data in Viro's patchworking and the topology of the resulting non-singular real algebraic curves in the projective plane. Using recent results from Renaudineau and Shaw on real algebraic curves near the non-singular tropical limit, we continue the study of Itenberg and Haas inside any non-singular projective toric surface. We give new Haas' like criteria for patchworking non-singular real algebraic curves with prescribed number of connected components, in terms of twisted edges on a non-singular tropical curve. We the…

Topology of Injective Endomorphisms of Real Algebraic Sets

Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduce that any injective regular self-mapping of a real algebraic set is a homeomorphism. We show also a similar local property. All our results can be extended to arc-symmetric semi-algebraic sets and injec…

Topology of algebraic varieties

Let Y be a normal crossing divisor in the smooth projective algebraic variety X (defined over ${\mathbb C}$) and let U be a tubular neighbourhood of Y in X. We construct homological cycles generating $H_*(A,B)$, where (A,B) is one of the following pairs $(Y,\emptyset)$, (X,Y), (X,X-Y), $(X-Y,\emptyset)$ and $(\partial U,\emptyset)$. The construction is compatible with the weights in $H_*(A,B,{\mathbb Q})$ of Deligne's mixed Hodge structure.

Kummer面在非超奇异情况下的模2约简

Reduction of Kummer surfaces modulo 2 in the non-supersingular case

研究Kummer面在非超奇异情况下模2的良约简条件，构造了代数空间模型和方案模型的等价性。

Algebraic cycles on the Jacobian of a curve with a $g^r_d$

We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.

Affine Grassmannians in A^1-algebraic topology

Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).

代数簇上局部特征正特征局部域的有效局部微分拓扑

Effective local differential topology of algebraic varieties over local fields of positive characteristics

本文研究了代数簇上局部域的微分拓扑性质，证明了隐函数定理的类比结果，并分析了光滑测度在子浸入下的行为。

Gushel-Mukai六维流形的某些动机性质

Some Motivic Properties of Gushel-Mukai Sixfolds

本文研究了Gushel-Mukai六维流形的动机性质，证明其在代数等价意义下允许乘性Chow-Künneth分解并具有Franchetta性质，同时附带证明了双重EPW六维簇及相关立方的Franchetta性质与Chow环的消没结果。

Topology of real algebraic sets of dimension 4: necessary conditions

Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield $2^{43}-43$ independent characteristic numbers mod 2, which generalize the Akbulut-King numbers in dimension three.

Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems.

On numerical equivalence for algebraic cobordism

We define and study the notion of numerical equivalence on algebraic cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence is a finitely generated module over the Lazard ring, and it reproduces the Chow group modulo numerical equivalence. We show this theory defines an oriented Borel-Moore homology theory on schemes and oriented cohomology theory on smooth varieties. We compare it with homological equivalence and smash-equivalence for cobordism cycles. For the former, we show that homological equivalence on algebraic cobordism is strictly finer than numerical equival…

Complex monodromy and the topology of real algebraic sets

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.

最大扭绞实代数纽结的拓扑

Topology of maximally writhed real algebraic knots

针对 $RP^3$ 中 Viro 刚性同痕不变量取最大值的 $d$ 次实代数纽结，本文证明它们拓扑同痕并确定其具体纽结类型。

Topology of singular algebraic varieties

I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past 20 years have seen a series of new invariants, partly inspired by string theory, such as motivic integration and the elliptic genus of a singular variety. These theories are not defined in a topological way, but there are intriguing hints of their topological significance.

Bounds for polynomials on algebraic numbers and application to curve topology

Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $τ$, and let $V_{\mathbb{R}} (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar algebraic curve implicitly defined as the vanishing set of $P$. We give a deterministic and certified algorithm to compute the topology of $V_{\mathbb{R}} (P)$ in terms of a straight-line planar graph $\mathcal{G}$ that is isotopic to $V_{\mathbb{R}} (P)$. Our analysis yields the upper bound $\tilde O (d^5 τ+ d^6)$ on the bit complexity of our algorithm, which matches …

齐性空间及其紧化的模常数代数 Brauer 群

Groupes de Brauer algébriques modulo les constants d'espaces homogènes et leurs compactifications

本文证明连通线性代数群齐性空间的代数 Brauer 群对常数 Brauer 群的商到 $\mathrm H^1(K,\operatorname{Pic}\bar X)$ 的自然包含未必为同构，并将结论推广到光滑紧化。

On the topology of moduli spaces of real algebraic curves

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surface with boundary called the $\mathcal{A}\mathcal{B}\mathcal{C}$-complex. This complex encodes the intersection patterns of isotopy classes of essential simple arcs, boundary components, and essential simple closed curves. We show that this complex is homotopy equivalent to an inf…

Tempered subanalytic topology on algebraic varieties

On a smooth algebraic variety over $\mathbb{C}$, we build the tempered subanalytic and Stein tempered subanalytic sites. We construct the sheaf of holomorphic functions tempered at infinity over these sites and study their relations with the sheaf of regular functions, proving in particular that these sheaves are isomorphic on Zariski open subsets. We show that these data allow to define the functors of tempered and Stein tempered analytifications. We study the relations between these two functors and the usual analytification functor. We also obtain algebraization results in the non-proper ca…

Enumerative geometry meets statistics, combinatorics and topology

We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of Lorentzian polynomials. The main concept joining the mentioned fields is a linear space of matrices.

代数环与实Enriques曲面的拓扑

Algebraic cycles and topology of real Enriques surfaces

本文研究实Enriques曲面的拓扑性质，证明了同调类可由实代数曲线表示的充要条件，并刻画了Galois-Maximal与Z-Galois-Maximal曲面及Brauer群。

On discrete Zariski-dense subgroups of algebraic groups

We investigate for which linear-algebraic groups (over the complex numbers or any local field) there exists subgroups which are dense in the Zariski topology, but discrete in the Hausdorff topology. For instance, such subgroups exist for every non-solvable complex group.

Projective varieties with bad reduction at 3 only

Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p>2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of Shafarevich's Conjecture. If Y is a projective algebraic variety over the field of rational numbers with good reduction modulo all primes different from 3 and semi-stable reduction modulo 3 then for the Hodge numbers of the complexification…

拓扑、双曲性与复代数簇的沙法列夫猜想

Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties

本文探讨复代数簇的双曲性、拓扑结构及沙法列夫猜想等核心问题，通过非阿贝尔霍奇理论和基本群表示等方法，揭示这些领域间的深刻联系与最新进展。

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson-Thomas theory.…

The Markov-Zariski topology of an abelian group

According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (=totally bounded) Hausdorff group topology on G. The family of all algebraic subsets of an abelian group G forms the family of closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or verbal, topology of G. We investigate the properties of this …

Stein Domains in Banach Algebraic Geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a contribution towards the found…