Moduli space holography and the finiteness of flux vacua

A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2, ℂ)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincaré metric with Sl(2, ℝ) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture.

On the Hodge conjecture for quasi-smooth intersections in toric varieties

We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

Notes on the Hodge conjecture for Fermat varieties

We review a combinatoric approach to the Hodge conjecture for Fermat varieties and announce new cases where the conjecture is true. We show the Hodge conjecture for Fermat fourfolds $ {X}_m^4 $ of degree m ≤ 100 coprime to 6, and also prove the conjecture for $ {X}_{21}^n $ and $ {X}_{27}^n $ , for all n.

Real rectifiable currents, holomorphic chains and algebraic cycles

We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.

Finite descent obstruction for Hilbert modular varieties

Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb {Z}_{S}$ -points on integral models of Hilbert modular varieties, extending a result of D. Helm and F. Voloch about modular curves. Let L be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre’s conjecture for mod $\ell $ representations of the absolute Galois group of L, we prove that the same holds also for the $\mathcal {O}_{L,S}$ -points.