INTERSECTION COHOMOLOGY OF THE MODULI SPACE OF HIGGS BUNDLES ON A GENUS 2 CURVE

Let C be a smooth projective curve of genus $2$ . Following a method by O’Grady, we construct a semismall desingularisation $\tilde {\mathcal {M}}_{Dol}^G$ of the moduli space $\mathcal {M}_{Dol}^G$ of semistable G-Higgs bundles of degree 0 for $G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$ . By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of $\tilde {\mathcal {M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal {M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal {M}_{Dol}^G$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

Stability of BPS states and weak coupling limits

We study the stability and spectrum of BPS states in $$ \mathcal{N} $$ N = 2 supergravity. We find evidence, and prove for a large class of cases, that BPS stability exhibits a certain filtration which is partially independent of the value of the gauge couplings. Specifically, for any perturbative value of any gauge coupling g ≪ 1, a BPS state can only decay to some constituents if those constituents do not become infinitely heavier than it in the vanishing coupling limit g → 0. This stability filtration can be mathematically formulated in terms of the monodromy weight filtration of the limiting mixed Hodge structure associated to the vanishing coupling limit. We study various implications of the result for the Swampland program which aims to understand such weak-coupling limits, specifically regarding the nature and presence of an infinite tower of light charged BPS states.

MIXED HODGE STRUCTURES WITH MODULUS

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

Log Picard Algebroids and Meromorphic Line Bundles

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve the prequantization problem, showing that under the appropriate integrality condition, a log Picard algebroid is the Lie algebroid of symmetries of what is called a meromorphic line bundle, a generalization of the usual notion of line bundle in which the fibers degenerate along the divisor. We give a geometric description of such bundles and establish a classification theorem for them, showing that they correspond to a subgroup of the holomorphic line bundles on the divisor complement. Importantly, these holomorphic line bundles need not be algebraic. Finally, we provide concrete methods for explicitly constructing examples of meromorphic line bundles, such as for a smooth cubic divisor in the projective plane.

Hodge correlators

Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

Variations of Mixed Hodge Structure

This chapter discusses the definition of admissible variations of mixed Hodge structure (VMHS), the results of M. Kashiwara in A study of variation of mixed Hodge structure (1986), and applications to the proof of algebraicity of the locus of certain Hodge cycles. It begins by recalling the relations between local systems and linear differential equations as well as the Thom–Whitney results on the topological properties of morphisms of algebraic varieties. The definition of a VMHS on a smooth variety is given, and the singularities of local systems are discussed. The chapter then studies the properties of degenerating geometric VMHS. Next it gives the definition and properties of admissible VMHS and reviews important local results of Kashiwara. Finally, the chapter recalls the definition of normal functions and explains recent results on the algebraicity of the zero set of normal functions.

Hodge Theory (MN-49)

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.