Chad Schoen
Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds.
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We determine the Betti numbers of the Springer fibers in type A. To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers.
A lower bound on the top Betti number of the Milnor fiber
本文给出了解析函数Milnor纤维顶部非零Betti数的下界，该下界由零维Lê数和临界轨迹复链接上消失循环的内部单值化决定。
We study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of $C^n$ by using the theory of motivic Milnor fibers. The results will be explicitly described by the mixed volumes of the faces of Newton polyhedrons.
Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps
本文研究了带有 lagrangian 纤维的不可约 holomorphically symplectic 流形的抛物自同构，证明了相关 Betti 映射具有最大秩，从而得出诱导平移为有限阶的纤维集合是稠密的。
For a Weyl group $W$ of classical type, we present a formula to calculate the restriction of (graded) Springer representations of $W$ to a maximal parabolic subgroup $W'$ where the types of $W$ and $W'$ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.
Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work…
Starting from a homogeneous affine Springer fiber $Fl_ψ$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $Fl_ψ$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $Fl_ψ$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.
We discuss combinatorial invariance of the betti numbers of the Milnor fiber for arrangements of lines with points of multiplicity at most three and describe a link between this problem and enumeration of solutions of the Catalan equation over function field in the case when its coefficients are products of linear forms and the equation defines an elliptic curve.
We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. We indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.
Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of a complex hyperplane arrangement has the homotopy type of a CW complex in which the number of p-cells equals the p-th betti number.
Lê modules and hypersurfaces with one-dimensional singular sets
本文利用Lê模块和上界定理，研究具有单维奇异性集和小Lê数的超曲面的Milnor纤维上同调及消失环局部系统。
Let X --> S be a smooth projective family of surfaces over a smooth curve S such that the generic fiber is a surface with Weil H^2 spanned by divisors and trivial H^1. We prove that if the relative motive of X/S is finite-dimensional the Chow group CH^2(X) with coefficients in Q is generated by a multisection and vertical cycles, i.e. one-dimensional cycles lying in closed fibers of the map X --> S. If S is the projective line P^1 then CH^2(X) is a direct sum of n+1 copies of Q where n<=b_2 and b_2 is the second Betti number of the generic fiber. Vertical generators in CH^2(X) can be concretel…
We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.
On the topology of fibers of complex polynomial maps
该研究综述复多项式映射纤维上同调的已知结果，建立分岔值处消失上同调新结果，推导贝蒂数上界，将相关定理推广至含任意奇点的多项式映射。
We study the topology of toric maps. We show that if $f\colon X\to Y$ is a proper toric morphism, with $X$ simplicial, then the cohomology of every fiber of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an explicit formula for the Betti numbers of the fibers in terms of a relative version of the $f$-vector, extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theorem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of $Y$, which is…
Arakelov inequalities for fibered surfaces in positive characteristic
本文在正特征下证明半稳定纤维化曲面的Arakelov不等式，并构造达到Nguyen下界和Hodge-Witt最优性的亏格2例子，进而导出典范类与Szpiro型不等式。
We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…
Towards refined curve counting on the Enriques surface II: Motivic refinements
本文研究 Enriques Calabi-Yau 三维簇纤维曲线类中的 motivic PT 不变量，借助 Toda 穿墙公式提出紧化 Jacobian 纤维化反常 Hodge 数的 Betti 数猜想公式。
Surfaces elliptiques réelles et inégalité de Ragsdale-Viro
本文研究无多重纤维的实正则椭圆曲面，证明可通过形变使Betti数$h_1$等于Hodge数$h^{1,1}$，且每个同调类可由实代数曲线表示，方法为模构造的实版本。
Milnor fibrations of arrangements with trivial algebraic monodromy
本文研究具有平凡代数单值性的超平面配置 Milnor 纤维化，重点分析上同调跳跃轨迹与基本群下中心序列商，并构造出 Betti 数相同但基本群不同构的一对配置。
For a K3 surface S, we study motivic invariants of stable pairs moduli spaces associated to 3-fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of Kawai-Yoshioka, imply a full determination of the theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The work may be viewed as the third in a sequence of formulas starting with Yau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical data suggest the motivic invariants are linked to the Mathieu M_…
We compute the combinatorial Aomoto-Betti numbers $β_p(\mathcal{A})$ of a complex reflection arrangement. When $\mathcal{A}$ has rank at least $3$, we find that $β_p(\mathcal{A})\le 2$, for all primes $p$. Moreover, $β_p(\mathcal{A})=0$ if $p>3$, and $β_2(\mathcal{A})\ne 0$ if and only if $\mathcal{A}$ is the Hesse arrangement. We deduce that the multiplicity $e_d(\mathcal{A})$ of an order $d$ eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto-Betti number, when $d$ is prime. We give a uniform combinatorial characterizatio…
In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an ext…
In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the Betti numbers as well as on the number of different stable homotopy types of certain fibers of semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials, i…
Fibrations on the 6-sphere and Clemens threefolds
该研究针对满足一阶、二阶Betti数为零且欧拉非零的3维复流形Z，证明其不存在到2维复空间的满全纯映射，仅可能纤维化到曲线，还为S⁶的假想复结构提供了新限制。
Varieties with free tangent sheaves
提出T-平凡代数簇概念，研究特征p下其与阿贝尔簇的关联，证明可分支配的T-平凡簇仅在p≤3存在，还得到有限平展覆盖后的纤维化结果及提升特征零情形的覆盖结论。
Cyclic coverings of rational normal surfaces which are quotients of a product of curves
本文研究作为曲线乘积之商的有理正规曲面的循环覆盖，推广 Esnault-Viehweg 方法证明单值群在第一 Betti 群及 Hodge 结构上的作用可分解为 P^1 上特定循环覆盖的直和，并应用于 Lê-Yomdin 曲面奇点和等平凡纤维化曲面。
Vafa-Witten invariants from modular anomaly
本文利用模反常的普适公式，提取了Hirzebruch和del Pezzo曲面上U(N) Vafa-Witten理论的全纯生成函数，推导出所有N的瞬子模空间Betti数闭式公式，并揭示了fiber-base对偶及新不变量。
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