D. Guzzetti
We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.
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We define monodromy maps for tropical Dolbeault cohomology of algebraic varieties over non-Archimedean fields. We propose a conjecture of Hodge isomorphisms via monodromy maps, and provide some evidence.
We describe the relationship between intersection cohomology with twisted coefficients and the perverse sheaves which play the role of the eigenspaces for the Milnor monodromy of an affine hypersurface.
In this paper we study the monodromy action on the first Betti and de Rham non-abelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski dense monodromy orbit. In particular we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.
We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen to be equivalent to $X$ being a homology manifold. Given an analytic function $f$ on an intersection cohomology manifold, we describe a simple relation between $V(f)$ being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of $f$. We then describe how the Sebastiani-Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Fin…
We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If ther…
We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.
Motivic monodromy and p-adic cohomology theories
本文构建统一框架研究局部域上簇的上同调理论的 monodromy 与权重滤过，并据此简化定义 Hyodo-Kato 上同调及构造 Clemens-Schmid 链复形。
In this paper we study the map associating to a linear differential operator with rational coefficients its monodromy data. The operator has one regular and one irregular singularity of Poincare' rank 1. We compute the Poisson structure of the corresponding Monodromy Preserving Deformation Equations on the space of the monodromy data.
Given a complex analytic function with a one-dimensional critical locus at the origin, we examine the monodromy action on the integral cohomology of the Milnor fiber. We relate this monodromy to that of a generic hyperplane slice through the origin, and to that of a generic hyperplane slice near the origin. We thereby obtain number-theoretic restrictions on the monodromy and on the cohomology of the original Milnor fiber.
For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical Čech spectral sequence $(C)_T$ abutting to the Hyodo-Kato (log crystalline) cohomology $H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent cohomology of its various component intersections $Y^i$. We compare the filtration on $H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $(C)_T$ with the monodromy operator $N$ on $H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $N$ through residue maps and study relations with singular cohomology. If $Y$ lifts to a proper str…
In this note we consider the Milnor fiber $F$ associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.
Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlevé equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations.
Let $f$ and $g$ be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of $fg$ to that of $f$ and $g$ separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu.
We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first characteristic variety of the Milnor fiber in degree 1, and to prove that the monodromy action is non-trivial in degree 1. For an arbitrary line arrangement, we prove that the triviality of the monodromy in de…
We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete discrete valuation ring $K$ of mixed characteristic. For this we introduce log rigid cohomology and generalize the so called Hyodo-Kato isomorphism to versions for non-proper $Y$, for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of $Y$. We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel'd's …
We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space Γ_X. We give a geometric interpretation of the cohomology of Γ_X in terms of the action of a monodromy operato…
We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.
For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient variety and also by the cycle classes of the irreducible components of the subvariety. Using Deligne's semisimplicity theorem together with Steenbrink's theory for semistable degenerations, we give a simpler proof of the first author's theorem (with a better bound of the degree of hypersurfaces) that this monodromy representation is irreducible.
Let S be a K3 surface and Aut D(S) the group of auto-equivalences of the derived category of S. We construct a natural representation of Aut D(S) on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on S. The main result of this paper is the precise relation of this action with the monodromy of the Hilbert schemes S^[n] of points on the surface. A formula is provided for the monodromy representation, in terms of the Chern character of the universal sheaf. Isometries of the second cohomology of S^[n] are lifted, via this formula, to monodromy operators of the …
Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equi…
Meromorphic Projective Structures, Opers and Monodromy
本文将 Hejhal 关于复射影结构单值映射为局部微分同胚的定理推广到带极点情形，证明相应单值映射（含 Stokes 数据）为局部双全纯映射。
Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of $X$. This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, étale and $p$-adic settings). In the specific case in which $k=\mathbb C$ and $X$ is the central fiber of a semistable degeneration over the complex disc, our …
Polystability of Stokes representations and differential Galois groups
研究Stokes表示的多项稳定性，通过微分Galois群（推广 tame情况下的monodromy群 Zariski闭包）进行表征，扩展Richardson的结果，并建立内在方法。
We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $Γ$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection groups of rank two, we establish some general results which relate the isotypic componenents of the monodromy on the cohomology, to the Hodge structure and to the cohomology degree. Using eigenspace theory for reflection groups, we prove some sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler c…
Formality for rigid-analytic spaces satisfying the weight-monodromy conjecture
研究\(\mathbf{Q}_p\)有限扩张上光滑恰当刚性解析空间，证明满足权重 - 单值性猜想时其平展和德拉姆上同调代数是形式的，并给出非形式的此类曲面例子。
Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields
本文研究p-进域上超凯勒簇的p-进与ℓ-进单值算子，提出并验证了Nagai猜想在算术情形的类比，并引入基于Sen理论的新方法分析p-进上同调。
Reflection Vectors and Quantum Cohomology of Blowups
针对半单量子上同调簇的一点爆破，本文给出其量子上同调单值群中若干反射生成元仅依赖例外除子几何的显式公式。
Let $X$ be a projective manifold, and $D$ be a normal crossing divisor of $X$. By Jost-Zuo's theorem that if we have a reductive representation $ρ$ of the fundamental group $π_{1}(X^{*})$ with unipotent local monodromy, where $X^*=X-D$, then there exists a tame pluriharmonic metric $h$ on the flat bundle $\mathcal V$ associated to the local system $\mathbb V$ obtain from $ρ$ over $X^*$. Therefore, we get a harmonic bundle $(E, θ, h)$, where $θ$ is the Higgs field, i.e. a holomorphic section of $End(E)\otimesΩ^{1,0}_{X^*}$ satisfying $θ^2=0$. In this paper, we study the harmonic bundle $(E,θ,h)…
Mod p Monodromy of Cyclic Covers of the Projective Line
本文证明了对于模p系数的项目线循环覆盖的monodromy的大monodromy定理，扩展了Achter和Pries关于二阶和三阶循环覆盖的结果，采用积分上同调的证明方法。
We prove the weight-monodromy conjecture for varieties which are p-adically uniformized by a product of the Drinfeld upper half spaces. It is an easy consequence of Dat's work on the cohomology complex of the Drinfeld upper half space.
We calculate the monodromy action on the mod 2 cohomology for SL(2,C) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2,R) Higgs bundles.
We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception.
Assume that the local universal deformation of a Calabi-Yau 3-manifold X has an automorphism which does not act by 1 or -1 on the third cohomology. We show that the $F^2$ bundle in the Variation of Hodge structures of each maximal family containing $X$ is constant in this case. Thus X cannot be a fiber of a maximal family with maximally unipotent monodromy, if such an automorphism exists. Moreover we classify the possible actions of such an automorphism on the third cohomology, construct examples and show that the period domain is a complex ball containing a dense set of complex multiplication…
We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.
Given a family of local systems on a punctured Riemann sphere, with moving singularities, its first parabolic cohomology is a local system on the base space. We study this situation from different points of view. For instance, we derive universal formulas for the monodromy of the resulting local system. We use a particular example of our construction to prove that the simple groups $\PSL_2(p^2)$ admit regular realizations over the field $\QQ(t)$ for primes $p\not\equiv 1,4,16\mod{21}$. Finally, we compute the monodromy of the Euler-Picard equation, reproving a classical result of Picard.
I describe the monodromy of smooth hypersurfaces $X$ of high degree in a fixed smooth variety $Y$ containing a fixed subvariety $W$ of $Y$. The cohomology of $X$ in middle degree spanned by the pull-back of the cohomology of $Y$ and by the classes of the irreducible components of $W$ is monodromy invariant. I show that the monodromy representation on the orthogonal of those classes is irreducible. The proof is essentially topological. Difficulties arise from the fact that $W$ may have arbitrary singularities.
We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of smooth cubic surfaces. The proof is based on a toric degeneration of cubic surfaces, the Givental's mirror theorem for toric manifolds, and the Picard-Lefschetz theory.
Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the compl…
Slopes and weights of $\ell$-adic cohomology of rigid spaces
研究p进局部域上刚性空间的ℓ进上同调，证明其弗罗贝尼乌斯特征值是代数整数并给出p进赋值界，推导权重界证明相关猜想，还给出特定单值纯反常层例子回答问题。
In this paper we consider a conjecture formulated by the second author in occasion of the 1998 ICM in Berlin (arXiv:math/9807034v2). This conjecture states the equivalence, for a Fano variety $X$, of the semisimplicity condition for the quantum cohomology $QH^\bullet(X)$ with the existence condition of full exceptional collections in the derived category of coherent sheaves $\mathcal D^b(X)$. Furthermore, in its quantitative formulation, the conjecture also prescribes an explicit relationship between the monodromy data of $QH^\bullet(X)$ and characteristic classes of both $X$ and objects of th…
Virasoro constraints for K3 surfaces and monodromy operators
本研究利用Markman单值化算子证明K3曲面的Virasoro约束，将问题约化到秩1情形，同时证明秩0的新Virasoro约束，并引入负次数新型Virasoro算子，与已有算子共同构成中心荷为24的Virasoro代数表示。
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narv\' ez Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one p…
Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with res…
Semistable Lefschetz Pencils
研究离散赋值环上半稳定概形的 Lefschetz 铅笔几何与上同调，证明在低维假设 monodromy-weight 猜想下可完整理解其相对上同调，从而将该猜想归结为射影直线上 Kashiwara 猜想的算术变体。
Given an arrangement of hyperplanes in $¶^n$, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of $q$-forms with logarithmic poles along our arrangement. We give a basis for the ideal $\cal J$ of relations for the Orlik-Solomon's algebra. Under certain genericity conditions it was shown by H.~Esnault, V.~Schechtman and E.~Viehweg that the cohomology of a local system is given by the Aomoto complex. We generalize this result to a deformation of local systems obtained via a deformation of our arrangement. We calculate the Gauß-Manin connection for thi…
We show how some of our recent results clarify the relationship between the Lê numbers and the cohomology of the Milnor fiber of a non-isolated hypersurface singularity. The Lê numbers are actually the ranks of the free Abelian groups -- the Lê modules -- appearing in a complex whose cohomology is that of the Milnor fiber. Moreover, the Milnor monodromy acts on the Lê module complex, and we describe the traces of these monodromy actions in terms of the topology of the critical locus.
Let X be a smooth complex projective variety of dimension d. We show that its primitive cohomology in degree d is generated by certain "tube classes," constructed from the monodromy of the family of smooth hyperplane sections on X. The proof makes use of a result about the group cohomology of certain representations that may be of independent interest.
We study the vanishing cycle complex $φ_fA_X$ for a holomorphic function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the monodromy eigenvalue decomposition may hold after a localization of $A$). Assuming the perversity of the shifted constant sheaf $A_X[d_X]$, we show that the lowest possibly-non-zero vanishing cohomology at $0\in X$ can be calculated by the restriction of $φ_fA_X$ to an appropriate nearby curve in the singular locus $Y$ of $f$, which is given by intersecting $Y$ with …
We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.
Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy
本文研究线性约束 Kloosterman 指数和族的同调，通过 Newton 多面体分析确定权滤过与顶权秩，并借助中值卷积和四阶矩计算证明几何单群为特殊线性群。
We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of the Grassmannian. The proof is based on the relation between the quantum cohomology of the Grassmannian and that of the projective space.
We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential equations was defined by Katz and includes the classical Airy equation. In addition, it includes differential equations which are not rigid. Our approach is based on the topological computation of Stokes matrices of the enhanced Fourier-Sato transform of a perverse sheaf due to D'Agnolo, Hien, Morando and Sabbah.
We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A3-Frobenius manifold, as well as at the…
The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of…
Arithmetic Properties Of $\ell$-adic Étale Cohomology and Nearby Cycles of Rigid-Analytic Spaces
本文研究p进非阿基米德局部域上刚性解析簇的ℓ进平展上同调，建立Frobenius特征值界、强局部单值化定理、邻近环混合性，并用perfectoid分析证明Deligne权-单值化猜想的局部版本，同时发展Deligne拓扑斯上的可构造ℓ进复形理论。
Zariski density of monodromy groups via Picard-Lefschetz type formula
本文证明沿一般位置超平面排列分支的射影空间循环覆盖万有族的单值群在相应线性群中Zariski稠密，作为Carlson-Toledo结果的退化类比，证明依赖于Picard-Lefschetz型公式。
We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milno…
For a proper semistable curve $X$ over a DVR of mixed characteristics we reprove the "invariant cycles theorem" with trivial coefficients (see Chiarellotto, 1999) i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber of $X$ coincides with the first rigid cohomology group of its special fiber, without the hypothesis that the residue field of $\cal V$ is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of $X$ with coefficients convergent $F$-isocr…
Dubrovin conjecture and the second structure connection
该研究围绕杜布罗文猜想，借助量子上同调的第二结构联络及扭曲反射向量概念，重新表述猜想，并基于拉普拉斯变换单值数据给出量子联络单值数据的优美描述。
In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.
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