F. J. Calderon-Moreno, L. Narvaez-Macarro
We find explicit free resolutions for the $\scr D$-modules ${\scr D} f^s$ and ${\scr D}[s] f^s/{\scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper
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Normal split divisors in rational homogeneous spaces
研究有理齐性空间中具有分裂正常序列的除子性质，揭示其与投影空间或二次曲面超平面截面的关系。
Let $X$ be a nonsingular complex projective variety and $D$ a locally quasi-homogeneous free divisor in $X$. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on $X$ with respect to $D$, and the Chern-Schwartz-MacPherson class of the complement of $D$ in $X$. Our result confirms a conjectural formula for these classes, at least after push-forward to projective space; it proves the full form of the conjecture for locally quasi-homogeneous free divisors in $\mathbb P^n$. The result generalizes several previously known results. For example…
We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.
Let $X$ be an $n$-dimensional complex Fano manifolds $(n\geq 3)$. Assume that $X$ contains a divisor $A$, which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle $\mathscr{N}^*_{A/X}$ is ample over $A$. Building on the works of Tsukioka, Watanabe and Casagrande-Druel, we give a complete classification of such pairs $(X,A)$.
One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors which do not come from hyperplane arrangements nor as explicit discriminants from singularity theory.
Divisors on coherent schemes and homogeneous spaces
本文研究相干基上齐性空间上除子的正性与可延展性，发展了局部相干概形上的估值除子理论，建立了有效估值除子与秩为一的反射sheaf的精确对应关系，推导出非Noetherian的Ramanujam-Samuel定理。
Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.
We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-α}$ associated to the effective $\mathbb{Q}$-divisor $D=α\cdot Z$, where $0<α\le1$ and $Z=(f=0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic functio…
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。
We prove two theorems on the locally finite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of the numerical linear systems in order to simplify the argument on the Zariski decompositions.
We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric …
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.
Isotropy subgroups of homogeneous locally nilpotent derivations
本文研究了齐次局部幂零导出在仿射尖点簇和某些三元多项式超曲面上的同构群，并给出了判断这些导出为最大导出的条件。
The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Th…
The Chern class of the sheaf of logarithmic derivations along a simple normal crossing divisor equals the Chern-Schwartz-MacPherson class of the complement of the divisor. We extend this equality to more general divisors, which are locally analytically isomorphic to free hyperplane arrangements.
We describe two situations where adding the adjoint divisor to a divisor D with smooth normalization yields a free divisor. Both also involve stability or versality. In the first, D is the image of a corank one stable germ of a map from complex n-space to complex (n+1)-space, and is not free. In the second, D is the discriminant of a versal deformation of a weighted homogeneous function with isolated critical point (subject to certain numerical conditions on the weights). Here D itself is already free. We also prove an elementary result, inspired by these first two, from which we obtain a plet…
Twisted logarithmic complexes of positively weighted homogeneous divisors
本文研究正加权齐次除子的扭曲对数复形的拟同构条件，证明LCT与Bernstein-Sato多项式根及一阶算子生成零化子等价，并给出超平面排列情形的简单证明。
For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric (Heegner) divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.
Let $X$ be an affine algebraic variety endowed with an action of complexity one of an algebraic torus $\mathbb{T}$. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions $\mathbb{K}[X]$ can be described in terms of proper polyhedral divisors corresponding to $\mathbb{T}$-variety $X$. We prove that homogeneous locally nilpotent derivations commute if an only if some combinatorial criterion holds. These results are used to describe actions of unipotent groups of dimension two on affine $\mathbb{T}$-varieties.
An infinite family of irreducible homogeneous free divisors in $K[x, y, z]$ is constructed. Indeed, we identify sets of monomials $X$ such that the general polynomial supported on $X$ is a free divisor.
We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coincide. As a consequence, we find that a quiver can only give rise to a linear free divisor if it has no (oriented or unoriented) cycles. We also deduce that the linear free divisors which appear in Sato and…
We establish normal form theorems for a large class of singular flat connections on complex manifolds, including connections with logarithmic poles along weighted homogeneous Saito free divisors. As a result, we show that the moduli spaces of such connections admit the structure of algebraic quotient stacks. In order to prove these results, we introduce homogeneous Lie groupoids and study their representation theory. In this direction, we prove two main results: a Jordan-Chevalley decomposition theorem, and a linearization theorem. We give explicit normal forms for several examples of free div…
We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational equivalences and as a corollary obtain a formula for rational equivalence between intersections of two locally principal Cartier divisors. Such canonical rational equivalence applies quite naturally to the setting of algebraic stacks. We present two applications: (i) a simplification of the development of Fulton-MacPherson-style intersection theory on Deligne…
In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the $D[s]$-module $D[s] h^s$ admits a Spencer logarithmic resolution satisfies the symmetry property $b(-s-2) = \pm b(s)$. This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection $E$ and of its dual $E^*$ with respect to a free divisor of linear Jacobian type are related by the equality $b_{E}(s)=\…
We show that in locally-ringed connected topoi the primes dividing the period and index of a Brauer class coincide. The result applies in particular to Brauer classes on connected schemes, algebraic stacks, topological spaces and to the projective representation theory of profinite groups.
Advances on a conjecture about free divisors
本文通过研究强欧拉同次性和弱科兹尔自由性，证明了关于自由除子的猜想在新条件下成立，并推翻了线性自由除子的猜想。
Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition that X has a K-point implies that X has an R-point.
On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors
本文研究复杂超曲面的强欧拉齐次性和Saito全纯性，证明了自由除子满足LCT的猜想在新条件下成立，并推翻了关于线性自由除子的对称b-函数根的猜想。
In this paper we develop the "local part" of our local/global approach to globally valued fields (GVFs). The "global part", which relies on these results, is developed in a subsequent paper.We study virtual divisors on projective varieties defined over a valued field $K$, as well as sub-valuations on polynomial rings over $K$ (analogous to homogeneous polynomial ideals). We prove a Nullstellensatz-style duality between projective varieties equipped with virtual divisors (analogous to projective varieties over a plain field) and certain sub-valuations on polynomial rings over $K$ (analogous to …
Locally Homogeneous Holomorphic Geometric Structures on Projective Varieties
本文研究带齐次代数簇模型的光滑射影簇上的全纯局部齐次结构，并刻画所有能展开到模型的子簇。
A plane curve on a the projective space over a field of characteristic zero is free if its associated sheaf T of tangent vector fields tangent is a free module. Relatively few free curves are known. Here we prove that a divisor consisting of a union of curves of a pencil of plane projective curves with the same degree and with a smooth base locus is a free divisor if and only this union contains all the singular members of the pencil and its Jacobian ideal is locally a complete intersection.
We describe homogeneous locally nilpotent derivations of the algebra of regular functions for a class of affine trinomial hypersurfaces. This class comprises all non-factorial trinomial hypersurfaces.
Homogeneous locally nilpotent derivations on trinomial varieties
研究三元多项式簇的正则函数代数最细分级下的同质局部nilpotent导出的显式描述。
Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear equivalence. When $\dim(X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive $\mathcal{O}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.
Let A be a commutative associative integrally closed k-algebra without zero divisors effectively graded by a lattice. We obtain a criterion of local nilpotency of the sum of two homogeneous locally nilpotent derivations (LNDs) of fiber type on A in terms of their degrees. The same problem is solved for commutators of two homogeneous LNDs.
For g a split semisimple Lie algebra over a field of characteristic 0, we link locally trivial principal homogeneous spaces over Spec(R) to the question of conjugacy of maximal abelian diagonalizable subalgebras of g(R).
In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $σ$-ample divisor, where $σ$ is an automorphism of a projective scheme X. Many open questions regarding $σ$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $σ$-ample. As a consequence, we show right and left $σ$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension.…
Reason for withdrawal: There is a serious mistake in the calculation of the divisor of the rational section used in the proof of Prop. 2.2.1., and with the correct value the argument does not work.
We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.
We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give a sufficient criterion for the property, a simple proof of F.J. Calderon-Moreno's theorem that free divisors have the property, a proof that divisors in dimension 3 with only isolated quasi-homogeneous singularities have the property, an example of a non-free divisor with non-isolated singularity having the property, an example of a divisor not having the property, and an algorithm to compute the V-filtration…
$\mathbb T$-homogeneous locally nilpotent derivations of trinomial algebras
本文研究了三项多项式代数中$\mathbb T$-齐次的局部 nilpotent 导出，这些导出在自然的复杂度为一的 torus 作用下保持齐次。
On Euler-homogeneity for free divisors
本文研究满足对数比较定理的自由除子需具备强 Euler 齐性的猜想，通过将奇异导子分解为可交换的半单与拓扑幂零导子之和并证明其保持对数性质，为已知的低维情形提供了替代性证明。
We introduce the concept of a prehomogeneous determinant as a nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the $b$-function are symmetric about -1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under which this symmetry still holds. Combined with Kashiwara's theorem on the roots of b-functions, our symmetry result shows that -1 is the only integer root of the b-function. This gives a positive answer to a probl…
In this paper we consider an effective divisor on the complex projective line and associate with it the module D consisting of all the derivations $θ$ such that $θ(I_i)\subset I_i^{m_i}$ for every $i$, where $I_i$ is the ideal of $p_i$. The module D is graded and free of rank 2; the degrees of its homogeneous basis, called the exponents, form an important invariant of the divisor. We prove that under certain conditions on $(m_i)$ the exponents do not depend on $\{p_i\}$. Our main result asserts that if these conditions do not hold for $(m_i)$ then there exists a general position of $n$ points …
We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space $S=Γ\backslash G/K$ is compact. More precisely, given a sequence of homogeneous probability measures on $S$, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including $S$ itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when $G={\rm SL}_3(\mathbb{R})$ and $Γ={\rm SL}_3(\mathbb{Z})$.
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