Boris Khesin, Alexei Rosly
For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue on it.
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On rational homology projective planes with quotient singularities of small indices
本文研究拓扑与光滑阻碍对带小指数商奇点的有理同调复射影平面存在性的影响，完成了指数≤3且光滑轨迹一阶整同调平凡的这类曲面的商奇点类型分类。
We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under suitable assumptions. We also provide examples which demonstrate sharpness of the results. The ambient surface is singular in these examples.
We construct rational projective 4-dimensional varieties with the property that certain Lawson homology groups tensored with Q are infinite dimensional Q-vector spaces. More generally, each pair of integers p and k, with k\geq 0, p>0, we find a projective variety Y, such that L_pH_{2p+k}(Y) is infinitely generated. We also construct two singular rational projective 3-dimensional varieties Y and Y' with the same homeomorphism type but different Lawson homology groups, specifically L_1H_3(Y) is not isomorphic to L_1H_3(Y')even up to torsion.
In this paper we classify all the compactifications of affine homology $3$-cells into the blow-ups of the projective $3$-space along smooth curves such that the log canonical divisors are linearly trivial. As a result, we prove that each embedded affine $3$-fold is isomorphic to the affine $3$-space except one example.
A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $\mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. So far all known examples have at most 4 singular points. In this paper, we prove that a rational homology projective plane $S$ with quotient singularities such that $K_S$ is nef has at most 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth ra…
We present the complete list of all singularity types on Gorenstein $\mathbb{Q}$-homology projective planes, i.e., normal projective surfaces of second Betti number one with at worst rational double points. The list consists of $58$ possible singularity types, each except two types supported by an example.
A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $CP^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2, d \le 4$ and $n=3=d$ are calculated.
We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.
We give a complete classification of complex Q-homology projective planes with isolated rational double point singularities and numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all moduli curves are rational and defined over Q, and we determine all families explicitly using extremal rational elliptic surfaces and Enriques involutions of base change type.
In this paper we develop homology and cohomology theories which play the same role for real projective varieties that Lawson homology and morphic cohomology play for projective varieties respectively. They have nice properties such as the existence of long exact sequences, the homotopy invariance, the Lawson suspension property, the homotopy property for bundle projection, the splitting principle, the cup product, the slant product and the natural transformations to singular theories. The Friedlander-Lawson moving lemma is used to prove a duality theorem between these two theories. This dualit…
Chiral homology of the projective line and extensions of vertex algebra modules
该研究建立了顶点代数两模的扩张向量空间与对应手性代数系数下一维射影直线的一阶手形同调间的同构关系，推进了手形同调与顶点代数模扩张的关联研究。
Compactification of homology cells, Fujita's conjectures and the complex projective space
该研究证明了在特定条件下，紧致凯勒流形必须为复射影空间，并回答了Fujita的猜想。
A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line
本文提出一种半数值算法，用于计算复射影直线上椭圆曲面的同调有效基并进行周期积分，从而启发式恢复Néron-Severi格、超越格及Mordell-Weil群与格等代数不变量，并提供SageMath实现。
Lawson Homology for projective varieties with C^*-action
本文针对带C^*-作用的射影簇，建立了相容的整系数劳森同调分解，证明了相关提升的性质，推导了特定基维数下的比较映射性质，计算了对称多项式并给出多类应用。
Effective homology and periods of complex projective hypersurfaces
本文提出了一种基于Picard-Lefschetz理论的新算法，用于高效计算复射影超曲面的周期，通过结合同调基计算与monodromy作用分析，实现了高精度的周期计算。
In this paper, the Lawson homology and morphic cohomology are defined on the Chow motives. We also define the rational coefficient Lawson homology and morphic cohomology of the Chow motives of finite quotient projective varieties. As a consequence, we obtain a formula for the Hilbert scheme of points on a smooth complex projective surface. Further discussion concerning generic finite maps is given. As a result, we give examples of self-product of smooth projective curves with nontrivial Griffiths groups by using a result of Ceresa.
In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.
Real topological Hochschild homology of schemes
该研究证明带对合概形的实拓扑Hochschild同调THR满足基变换与Z/2等变平展拓扑下降性质，并给出射影直线及高维射影空间的相关计算，为相关同调理论研究提供了新结果。
Computing the Homology of Real Projective Sets
本文提出一种数值算法，用于计算实射影簇的同调群，包括贝蒂数和 torsion 系数，算法在数值稳定性上有效，计算复杂度与变量数和输入条件有关。
We characterize integral homology classes of the product of two projective planes which are representable by a subvariety.
The homology growth for finite abelian covers of smooth quasi-projective varieties
本文研究了复光滑拟射影簇在有限生成阿贝尔群满射下奇异同调挠子群阶与 Betti 数的渐近行为，并在 $ν$ 为 orbifold effective 时给出了 1 次 $L^2$-型不变量的显式公式，推广了先前关于 $H\cong \Z$ 的结论。
New birational invariants for a projective manifold are defined by using Lawson homology. These invariants can be highly nontrivial even for projective threefolds. Our techniques involve the weak factorization theorem of Wlodarczyk and tools developed by Friedlander, Lawson, Lima-Filho and others. A blowup formula for Lawson homology is given in a separate section. As an application, we show that for each n\geq 5, there is a smooth rational variety X of dimension n such that the Griffiths groups Griff}_p(X) are infinitely generated even modulo torsion for all p with 2\leq p\leq n-3.
We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to the Hilbert scheme of length-three subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion.
We describe polar homology groups for complex manifolds. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue on it. The polar homology groups may be regarded as holomorphic analogues of the homology groups in topology. We also describe the polar homology groups for quasi-projective one-dimensional varieties (affine curves). These groups obey the Mayer--Vietoris property. A complex counterpart of the Gauss linking number of two curves in a three-fold and various gau…
The comparison of the 0-th Suslin homology and Chow groups of 0-cycles with modulus
本文证明了在正特征域上，射影正规曲面及任意维数射影光滑概形中，带模零循环周群与0次Suslin同调之间的同构关系。
We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus 0. Generalizations apply in the case of multiple singular points.
Homology fiber bundles of varieties, that are not topological fiber bundles
构造平坦射影态射，它是\(\mathbb{Z}\)-同调纤维丛，全空间光滑但非正则，反驳了第二作者等人的一个猜想。
We construct an Abel-Jacobi type map on the homologically trivial part of Lawson homology groups. It generalizes the Abel-Jacobi map constructed by Griffiths. By using a result of H. Clemens, we give some examples of smooth projective manifolds with infinite generated Lawson homology groups L_pH_k(X,Q) when k>2p.
Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any resolution $X_r\to X$. This paper uses uniform (and non-uniform) expressions to define what is believed to be the usual intersection homology (and its local-global variant) of a convex polytope (or a projective toric variety). Such expressions are generated by the facets, and so may lead to necessary numerical conditions on the flag vector. Most of the conc…
Lawson homology groups of Chow varieties
本文计算了柴可夫变种的有理恰氏同调群，并证明了其与奇异同调群的同构性，同时建立了柴可夫变种在自然嵌入和代数悬垂映射下的稳定性。
Let X be a smooth projective variety of dimension n on which rational and homological equivalence coincide for algebraic p-cycles in the range 0\leq p\leq s. We show that the homologically trivial sector of rational Lawson homology L_pH_k(X,Q)_{hom} vanishes for 0\leq n-p\leq s+2. This is an analogue of a theorem of C. Peters in "dual dimensions". Together with Peters' theorem we get that the natural transformation L_pH_k(X,Q)--> H_k(X,Q) is injective for all p and k when X is a smooth projective variety of dimension 4 and Ch_0(X) =Z.
Natural transformations from constructible functions to homology
对于复射影簇，所有从可构造函数到同调（模挠）的自然变换都是MacPherson-Schwartz-Chern类的线性组合。(The作者愿意邮寄论文的硬拷贝。
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric varieties.
A Secondary Multiplicative Structure on Lawson Homology
本文在光滑复射影簇的Lawson同调上引入一种次级乘法运算，细化交积并产生非平凡的整系数Massey括号，其值在奇异同调中消失，且在射影丛等映射下保持，给出形式性障碍。
We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L-classes generalizing results of Hirzebruch-Zagier and Moonen. Our methods also apply to the study of Todd classes of…
Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each projective R/I-module V has an R-projective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_1 --> P_0 for all n, if d = 1. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if d = 2, we compute the homology of the complex N Sym^2 K(P.) where K and N denote the…
We prove the vanishing of the subgroup of Bloch's cubical higher Chow groups $CH^2(\text{Spec}(\MF_p),3)$, $p=2,3$, generated by the images of corresponding projective Grassmannian homology groups ${}^PGH_1^2(\MF_p)$ using computer calculations.
The topology of spaces of holomorphic maps to projective space
本文证明了从光滑复射影簇到射影空间的全纯映射与连续映射在依赖次数的范围内同调相同。
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in particular, to homology planes arising from arrangements of lines. This permits to determine the infinitude of the fundamental group of homology planes of log-general type that may be the first examples of this type.
We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring of infinite adeles of K, provides the right theory to obtain, using the lambda-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.
We show that for a smooth projective variety $X$ over a field $k$ and a reduced effective Cartier divisor $D \subset X$, the Chow group of 0-cycles with modulus $\mathrm{CH}_0(X|D)$ coincides with the Suslin homology $H^S_0(X \setminus D)$ under some necessary conditions on $k$ and $D$. We derive several consequences, and we answer to a question of Barbieri-Viale and Kahn.
Algebraic Montgomery-Yang problem and smooth obstructions
本文利用 Donaldson 对角化定理和 Heegaard Floer 校正项等光滑 4-流形工具，排除特定四奇点情形并识别满足 orbifold BMY 不等式但被光滑条件阻碍的无穷多奇点类型，推进了代数 Montgomery-Yang 问题的研究。
Degenerations of the complex projective plane with only rational singularities
在瓦尔猜想成立的前提下，对仅具有理奇点的复射影平面正规退化进行分类，包括马内蒂和哈金 - 普罗霍罗夫分类的由马尔可夫方程控制的曲面，以及含四个非对数典范奇点的六个新退化。
Using the Bialynicki-Birula method, we determine the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (5, 1, 4), (7, 2, 6), respectively, (0, 5, 19). We compute the Hodge numbers of these moduli spaces.
Galois Action on Homology of the Heisenberg Curve
本文研究海森堡曲线的基本群与同调结构，构造了阿廷辫子群在其上的作用，并用特征0域上海森堡群的不可约表示描述其同调。
The tropical (p, q)-homology groups of Itenberg, Katzarkov, Mikhalkin and Zharkov are the tropical analogues of the Hodge decomposition of the cohomology of complex algebraic varieties. There is a well-defined intersection pairing on tropical (1, 1)-classes of a compact non-singular tropical surface. Here we compute directly the (1, 1)-homology of a non-singular floor decomposed tropical surface in tropical projective space, along with the intersection form.
A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.
Given any connected topological space $X$, assume that there exists an epimorphism $ϕ: π_1(X) \to \mathbb{Z}$. The deck transformation group $\mathbb{Z}$ acts on the associated infinite cyclic cover $X^ϕ$ of $X$, hence on the homology group $H_i(X^ϕ, \mathbb{C})$. This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring $\mathbb{C}[t,t^{-1}]$, which is a finite dimensional $\mathbb{C}$-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When $X$ is a compact Kähler manifold, we show that all the Jordan …
This is a written-up version of eight introductory lectures to the Hodge theory of projective manifolds. The table of contents should be self-explanatory. The only exception is section 8 where I discuss, in a simple example, a technique for studying the class map for homology classes on the fibers of a map and one for approximating a certain kind of primitive vectors.
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
Cyclic homology of categories of matrix factorizations
本文证明光滑拟射影簇上矩阵分解范畴的周期循环同调与带符号扭转单值性的消没上同调一致，Hochschild同调与相应超上同调一致，并将Hodge猜想推广到该框架。
In this paper we compute Lawson homology groups and semi-topological K-theory for some threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of a technique of Bloch and Srinavas, of the Bloch-Kato conjecture and of the spectral sequence relating morphic cohomology and semi-topological K-theory.
We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sa…
Let X be the complex Fermat variety of dimension n=2d and degree m>2. We investigate the submodule of the middle homology group of X with integer coefficients generated by the classes of standard d-dimensional subspaces contained in X, and give an algebraic (or rather combinatorial) criterion for the primitivity of this submodule.
We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are $K$-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.
In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.
Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.
We show that the chow group of $p$-cycles with rational coefficients are isomorphic to the corresponding rational homology groups for smooth complex projective varieties carrying a holomorphic vector field with an isolated zero locus. As applications, we obtain Chow groups and Lawson homology groups with rational coefficients and verify the Friedlander-Mazur conjecture and the Generalized Hodge conjecture for those varieties.
Line bundles of rational degree are defined using Perfectoid spaces, and their co-homology computed via standard Čech complex along with Kunneth formula. A new concept of `braided dimension' is introduced, which helps convert the curse of infinite dimensionality into a boon, which is then used to do Bezout type computations, define euler characters, describe ampleness and link integer partitions with geometry. This new concept of 'Braided dimension' gives a space within a space within a space an infinite tower of spaces, all intricately braided into each other. Finally, the concept of Blow Up …
Weighted projective lines and Hochschild cohomology
本文研究了加权射影曲线的Hochschild上同调维数，发现其仅与曲线的亏格和例外点数量相关，并给出了加权射影线作为商堆栈的实现方法。
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