Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

Galois theory of linear differential equations

Introduction 221 § 1. Relative Monodromy Filtration 227 §2. Mixed Hodge Modules on Complex Spaces (2. a) Vanishing Cycle Functors and Specializations (Divisor Case) 236 (2.b) Extensions over Locally Principal Divisors 243 (2.c) Direct Images 251 (2.d) Mixed Hodge Modules 256 (2.e) Beilinson's Functor and its Applications 261 (2.f) Extensions over Closed Subspaces 265 (2.g) Kodaira Vanishing 273 §3. Mixed Hodge Modules of Normal Crossing Type (3.a) Constant Case 277 (3. b) Pure Case 288 (3.c) Mixed Case 304 §4. Algebraic Case 314

Mixed hodge modules

We investigate degeneracies of BPS states of D-branes on compact Calabi-Yau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial D-brane systems, and to clarify the subtle relation of Donaldson-Thomas invariants to BPS indices of stable D6-D2-D0 states, realized in supergravity as “hole halos.” We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are “split” D6-anti-D6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor $ g_{\text{top}}^{{ - 2}}{e^{{ - K}}} $
 and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of “swing states” — D6 states which exist at large radius but do not form stable D6-anti-D6 bound states. We point out a likely breakdown of the OSV conjecture at small g
 top (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kahler moduli, a charge Λ Γ supporting single centered black holes of entropy ~ Λ2
 S(Γ) also admits two-centered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.

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Split states, entropy enigmas, holes and halos

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

It has been conjectured that in theories consistent with quantum gravity infinite distances in field space coincide with an infinite tower of states becoming massless exponentially fast in the proper field distance. The complex-structure moduli space of Calabi-Yau manifolds is a good testing ground for this conjecture since it is known to encode quantum gravity physics. We study infinite distances in this setting and present new evidence for the above conjecture. Points in moduli space which are at infinite proper distance along any path are characterised by an infinite order monodromy matrix. We utilise the nilpotent orbit theorem to show that for a large class of such points the monodromy matrix generates an infinite orbit within the spectrum of BPS states. We identify an infinite tower of states with this orbit. Further, the theorem gives the local metric on the moduli space which can be used to show that the mass of the states decreases exponentially fast upon approaching the point. We also propose a reason for why infinite distances are related to infinite towers of states. Specifically, we present evidence that the infinite distance itself is an emergent quantum phenomenon induced by integrating out at one-loop the states that become massless. Concretely, we show that the behaviour of the field space metric upon approaching infinite distance can be recovered from integrating out the BPS states. Similarly, at infinite distance the gauge couplings of closed-string Abelian gauge symmetries vanish in a way which can be matched onto integrating out the infinite tower of charged BPS states. This presents evidence towards the idea that also the gauge theory weak-coupling limit can be thought of as emergent.

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Infinite distances in field space and massless towers of states

Degeneration of hodge-structures

Let S be a complex algebraic variety and {Xs}s∈S a family of non singular projective varieties parametrized by S: the Xs are the fibers of f : X → S, with X projective and smooth over S. Fix s ∈ S, an integer p, and a class h ∈ H(Xs,Z) of Hodge type (p, p). Let U be an open, simply connected neighborhood of s. The H(Xt,Z), t ∈ S, form a local system on S, necessarily trivial on U , so that for t ∈ U they can all be identified with H(Xs,Z). The Hodge filtration Ft of H (Xt,C), t ∈ U , can be viewed as a variable filtration on the fixed complex vector space H(Xs,C). It varies holomorphically with t. It follows that the locus T ⊂ U where h remains of type (p, p), i.e., in F, is a complex analytic subspace of U .

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On the locus of Hodge classes

Introduction. Inthis paper we make a detailed analysis of the -hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We describe all rational solutions and show in Theorem 1.10 that they are, in fact, Laurent polynomials. We also show that for any exponent there are at most two linearly independent Laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically Cohen-Macaulay. We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial (0.5) associated with . We also determine in Theorem 3.7 the holonomic rank r(α)for all α ∈ Z 2 and show that d ≤ r(α)≤ d +1, where d is the degree of the curve. Moreover, the value d +1 is attained only for those exponents α for which there are two linearly independent rational solutions, and, therefore, r(α)= d for all α if and only if the curve is arithmetically Cohen-Macaulay. Inorder to place these results intheir appropriate con text, we recall the defin ition

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The A-hypergeometric system associated with a monomial curve

© Société mathématique de France, 1989, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Degenerating variations of hodge structure

The Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in various contexts: they impose restrictions on the cohomology algebra of a smooth compact Kahler manifold; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the f-vectors of convex polytopes. While the statements of these theorems depend on the choice of a Kahler class, or its analog, there is usually a cone of possible choices. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note, we present a unified approach to proving the mixed HLT and HRR, generalizing the known results, and proving it in new cases, such as the intersection cohomology of nonrational polytopes.

/pdf/mixed-lefschetz-theorems-and-hodge-riemann-bilinear-28lwp03anw.pdf

Eduardo Cattani

Papers

Degeneration of hodge-structures

On the locus of Hodge classes

The A-hypergeometric system associated with a monomial curve

Degenerating variations of hodge structure

Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations