F. J. Castro-Jimenez, J. M. Ucha
The relationship between ${\cal D}$-modules and free divisors has been studied in a general setting by L. Narváez and F.J. Calderón. Using the ideas of these works we prove in this article a duality formula between two ${\cal D}$-modules associated to a class of free divisors on ${\bf C}^n$ and we give some applications
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We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these D-modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.
We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized K3s.
We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor due to Hashimoto.
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 study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.
We describe new relations among conformal block divisors in $\operatorname{Pic}(\bar{\operatorname{M}}_{0,n})$. These relations appear from various rank-level dualities of conformal blocks on $\mathbb{P}^1$ with $n$ marked points. We also give a coordinate free description of rank-level duality maps on $\overline{\mathcal{M}}_{g,n}$.
We prove the injectivity and vanishing theorem for R-Hodge modules and R-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in \cite{Wu15}. In particular, the injectivity generalizes the fundamental injectivity of Esnault-Viehweg for normal crossing Q-divisors, while the vanishing generalizes Kawamata-Viehweg vanishing for Q-divisors. As a main application, we also deduce a Fujita-type freeness result for R-Hodge modules in the normal crossing case.
The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak $\Q$--Fano, $\Q$--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studi…
The moduli stack M_X(E_8) of principal E_8-bundles over a smooth projective curve X carries a natural divisor Delta. We study the pull-back of the divisor Delta to the moduli stack M_X(P), where P is a semi-simple and simply connected group such that its Lie algebra Lie(P) is a maximal conformal subalgebra of Lie(E_8). We show that the divisor Delta induces "Strange Duality"-type isomorphisms between the Verlinde spaces at level one of the following pairs of groups (SL(5), SL(5)), (Spin(8), Spin(8)), (SL(3), E_6) and (SL(2), E_7).
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
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural intersection-theoretic volume functional for 1-cycles over compact Kähler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional.
We study the wall-crossing phenomena of BPS D4-D2-D0 states on the conifold and orbifold C^2/Z_2, from the viewpoint of the quiver quantum mechanics on the D-branes. The Kahler moduli dependence of the BPS index is translated into the FI parameter dependence of the Witten index. The wall-crossing phenomena are related to the Seiberg dualities of the quiver quantum mechanics. All the differences from the D6-D2-D0 case arise from the additional superpotential and "anti-quark" induced by the D4-brane. When the D-branes are on the conifold, the flop transition changes the duality cascade. When the…
We prove a duality theorem for the $p$-adic etale motivic cohomology of a variety $U$ which is the complement of a divisor on a smooth projective variety over $\F_p$. This extends the duality theorems of Milne and Jannsen-Saito-Zhao. The duality introduces a filtration on $H^1_{\etl}(U, {\Q}/{\Z})$. We identify this filtration to the classically known Matsuda filtration when the reduced part of the divisor is smooth. We prove a reciprocity theorem for the idele class groups with modulus introduced by Kerz-Zhao and Rulling-Saito. As an application, we derive the failure of Nisnevich descent for…
We use filtered log-$\mathscr{D}$-modules to represent the (dual) localization of Saito's Mixed Hodge Modules along a smooth hypersurface, and show that they also behave well under the direct image functor and the dual functor in the derived category of filtered log-$\mathscr{D}$-modules. The results of this paper can be used to generalize the result of M. Popa and C. Schnell about Kodaira dimension and zeros of holomorphic one-forms into the log setting.
We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saito's criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation.
We prove that the pull back of the canonical theta divisor for E_8-bundles at level one induces a strange duality between Verlinde spaces for G_2 and F_4 at level one on smooth curves of genus g. We also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in the Picard group of \bar-M_{g,n}.
Graded modules associated with permissible $C^{\infty}$-divisors on tropical manifolds
本文利用 Strominger--Yau--Zaslow 猜想和微局部层论，为紧致热带流形上的 $C^{\infty}$-除子定义分次模，并证明 Riemann--Roch 公式，区别于已有热带定理。
Hermite Reciprocity and Self-Duality of Generalized Eagon-Northcott Complexes
本文通过选取具有良好性质的Koszul模作为输入，证明了广义Eagon-Northcott复形自对偶性的一个新情形，并在特殊化为Weyman模时引入了一种新的Hermite互反性版本。
The restriction, on the spectral variables, of the Baker-Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of differential operators in $k$ variables. The remaining g-k derivations define evolution equations for generators of the BA-module. As a corollary new examples of commutative ring of partial differential operators with matrix coefficients and their non-trivial evolution equ…
The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincare duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic Groups and Class Fields, combined with an invariance result for the topology on the dual of the space of repartitions. Our Abel ma…
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 …
Let G be a finite group acting faithfully on an irreducible non-singular projective curve defined over an algebraically closed field F. Does every G-invariant divisor class contain a G-invariant divisor? The answer depends only on G and not on the curve. We answer the same question for degree 0 divisor (classes). We investigate the question for cycles on varieties.
We obtain several new characterizations of splayedness for divisors: a Leibniz property for ideals of singularity subschemes, the vanishing of a `splayedness' module, and the requirements that certain natural morphisms of modules and sheaves of logarithmic derivations and logarithmic differentials be isomorphisms. We also consider the effect of splayedness on the Chern classes of sheaves of differential forms with logarithmic poles along splayed divisors, as well as on the Chern-Schwartz-MacPherson classes of the complements of these divisors. A postulated relation between these different noti…
Duality theory of tautological systems
该研究探讨重言系统的全纯对偶，通过推广GKZ理论中的Euler–Koszul技术引入Chevalley–Eilenberg型复形，证明其等变性与全纯性，为线性自由除子和齐性空间的相关研究提供新工具。
Let $D$ be a divisor in ${\bf C}^n$. We present methods to compare the ${\mathcal D}$-module of the meromorphic functions ${\mathcal O}[* D]$ to some natural approximations. We show how the analytic case can be treated with computations in the Weyl algebra.
We establish a relationship between the graded quotients of a filtered holonomic D-module, their sheaf-theoretic duals, and the characteristic variety, in case the filtered D-module underlies a polarized Hodge module on a smooth algebraic variety. The proof is based on Saito's result that the associated graded module is Cohen-Macaulay, and on local duality on the cotangent bundle.
Using the duality of positive cones, we show that applying the polar transform from convex analysis to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants have nice properties similar to those for divisors. In particular, this enables us to give a characterization of the divisorial components of the non-Kahler locus of a big class.
We introduce tautological system defined by prehomogenous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.
We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both …
We prove a structure theorem for the differential operator in the 0-term of the ${\cal V}$-filtration with respect to a free divisor. Using this theorem, we give a formula for the logarithmic de Rham complex in terms of ${\cal V}_0$-modules. We also give a sufficient condition for perversity of the logarithmic de Rham complex.
In this paper we survey the role of D-module theory in the comparison between logarithmic and meromorphic de Rham complexes of integrable logarithmic connections with respect to free divisors, and we present some new linearity conditions on the Jacobian ideal which arise in this setting.
Duality for t- modules: The Difficult Cases
本文研究特定约化形式的二维三角t-模，通过符号计算揭示满足ALD条件的t-模的约化模式，证明其与双对偶同构，并将Cartier-Nishi定理和Weil-Barsotti公式推广到更广的t-模类。
The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\ms{D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\ms{D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will lead us to an analog of "Weil II" and a product formula for $p$-adic epsilon factors.
For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the structure sheaf. We also prove that the local cohomology sheaves can be calculated by using the higher direct images of the twists of the associated sheaf complex on the projective cotangent bundle. Thes…
Duality for KGL-modules in motivic homotopy theory
本文在稳定动机同伦论框架下，针对任意特征零拟优概形，证明了KH-理论模的对偶性，明确其对偶化对象为G-理论，丰富了动机同伦论的对偶理论成果。
Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_λ$ and $L_λ$, which satisfy the identities as contraction and Lie derivative do for smooth differential forms. Out of this data one defines cohomology of the invariants and equivariant cohomology of $M$. We establish Koszul duality between each other.
We give an interpretation of quantum Serre of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E^\vee over X, and (3) the quantum D-module of a submanifold Z\subset X cut out by a regular section of E. When E is the anticanonical line bundle K_X^{-1}, we identify these twisted quantum D-modules with second structure connections with differ…
Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.
Cartier duality via Mittag-Leffler modules
本文通过Mittag-Leffler模构建交换有限群方案的Cartier对偶，并在特定条件下构造了导出范畴之间的傅里叶-穆卡伊变换。
Cyclic covers via mixed Hodge modules
本文针对复代数簇上由线丛整体截面定义的有效卡蒂埃除子，用混合霍奇模理论描述其N重循环覆盖的构造方法。
Duality and polyhedrality of cones for Mori dream spaces
该研究针对Mori dream空间，证明余维数$k$丰沛除子锥的有理多面性，借助Mori腔分解研究其与$k$-移动曲线锥的对偶性，给出弱对偶性质的新证明并展示满足强对偶性的示例族。
On nefness of the lowest piece of Hodge modules
本文研究了复Hodge结构外的Hodge模最低部分商线丛的度数下界，探讨了非单位元单子情况下nef性失效的问题，并通过代数方法证明了相关结论。
We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modify…
Consider a complex symplectic manifold $X$ and the algebroid $W_X$ of quantization-deformation. For two regular holonomic modules $L_i$ ($i=0,1$) supported by smooth Lagrangian manifolds, we prove that the complex $Rhom_{W_X}(L_1,L_0)$ is constructible and perverse and dual to the complex $Rhom_{W_X}(L_0,L_1)$.
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.
A refinement of the strange duality conjecture on K3 surfaces
针对K3曲面上奇异对偶猜想，本文构造三类反例说明原猜想不成立，并提出改进猜想，要求相关theta线丛为大且数值有效。
Correspondences between convex geometry and complex geometry
本文建立凸几何与复几何（全纯线丛理论）的类比，研究正乘积与混合体积的关系，定义除子类的布拉施克加法并证明新不等式，同时将亚历山德罗夫体构造和闵可夫斯基存在定理分别对应为除子扎里斯基分解和锥对偶性的凸几何版本。
In this paper we study the comparison between the logarithmic and the meromorphic de Rham complexes along a divisor in a complex manifold. We focus on the case of free divisors, starting with the case of locally quasihomogeneous divisors, and we explain how D-module theory can be used for this comparison.
We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.
We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing …
In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of the de Rham functor, sending logarithmic D-modules to certain graded sheaves on the so-called Kato-Nakayama space. For holonomic modules we show that the associated sheaves have finitely generated stalks and that the de Rham functor intertwines duality for D-modules with a version of Poincaré-Verdier duality on the Kato-Nakayama space. Finally, we explain how…
Let (X;OX) be a locally noetherian scheme with a dualizing complex D. We prove that DOX - : K(PinfX)----> K(InjX) is an equivalence of triangulated categories where K(InjX) is the homotopy category of injective quasi-coherent OX- modules and K(PinfX) is the homotopy category of pure injective flat quasi-coherent OX-modules. Where X is affine, we show that this equivalence is the infinite completion of the Grothendieck duality theorem. Furthermore, we prove that D OX - induces an equivalence between the pure derived category of flats and the pure derived category of absolutely pure quasi-cohere…
We give a Hodge-theoretic interpretation of the multiplier ideal of an effective divisor on a smooth complex variety. More precisely, we show that the associated graded coherent sheaf with respect to the jumping-number filtration can be recovered from the smallest piece of M. Saito's Hodge filtration of the D-module of vanishing cycles.
Finiteness and duality of cohomology of $(φ,Γ)$-modules and the 6-functor formalism of locally analytic representations
本文通过分析堆和6- functor形式主义，研究了局部解析表示的上同调性质，并推广了Kedlaya-Pottharst-Xiao的结果。
Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $\mathcal{P}$ a separated smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $X$ a smooth closed subscheme of $P$, $T$ a divisor in $P$ such that $T_X = T \cap X$ is a divisor in $X$ and $\smash{\D}^†_{\mathcal{P}}(\hdag T)$ the weak completion of the sheaf of differential operators on $\mathcal{P}$ with overconvergent singularities along $T$. We construct a fully faithful functor denoted by $ \sp_{X \hookrightarrow \mathcal{P},T,+}$ from the category of isocrystal on $X \setminus T_X$ overconvergent alo…
Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.
Frobenius--Cartier duality via lax equalizers
本文证明稳定∞-范畴松弛等化子的对偶定理，应用于特征p的Noetherian F-有限概形上Frobenius与Cartier模，恢复Baudin导出Hom对偶，并实现凝聚模对偶中标准t-结构与perverse t-结构的互换。
We show that (quantum) D-modules of Givental type on affine flag varieties are transformed to D-modules of Beilinson-Drinfeld type on $n$-reduced Sato Grassmannians via affine Miura transformation.
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