[Paper Review] Spin(7) instantons and the Hodge Conjecture for certain abelian four-folds: a modest proposal

Ramadas T. Ramakrishnan|ArXiv.org|Sep 23, 2008

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper proposes using Spin(7) instantons on abelian four-folds to address the Hodge Conjecture by constructing smooth vector bundles with Chern characters equal to Hodge classes, then showing that an instanton connection endows them with holomorphic structures. The key result is a strategy to prove such bundles are algebraic via deformation of known holomorphic structures using complex multiplication and stability arguments.

ABSTRACT

The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which can in fact be constructed in terms of of smooth line bundles - whose Chern characters are given Hodge classes. An instanton connection on these vector bundles would endow them with a holomorphic structure and thus prove that these classes are algebraic. I use complex multiplication to exhibit Cayley cycles representing the given Hodge classes. I find alternate complex structures with respect to which the given bundles are holomorphic, and close with a suggestion (due to G. Tian) as to how this may possibly be put to use.

Motivation & Objective

• To test a gauge-theoretic approach to the Hodge Conjecture using Spin(7) instantons on specific abelian four-folds.
• To construct smooth vector bundles with Chern characters equal to given Hodge (2,2)-classes on Mumford's abelian four-folds.
• To demonstrate that an instanton connection on such bundles would induce a holomorphic structure, thus proving the Hodge class is algebraic.
• To use complex multiplication to exhibit Cayley cycles representing the Hodge classes.
• To explore deformation methods (suggested by G. Tian) to construct instantons from known holomorphic bundles.

Proposed method

• Construct smooth vector bundles from line bundles on abelian four-folds, ensuring their Chern characters match rational Hodge (2,2)-classes.
• Utilize complex multiplication to define alternate complex structures on the four-fold, making the bundles holomorphic with respect to these structures.
• Apply the continuity method to deform known holomorphic bundles into instanton connections, leveraging the existence of a starting holomorphic structure (from the conjugate complex structure).
• Use Butler’s Theorem to prove stability of kernel bundles arising from evaluation maps of global sections of line bundles.
• Construct exact sequences involving dual bundles and tensor products to build a quotient bundle E with polystable structure.
• Ensure the slope of the summands in the final exact sequence is equal via Riemann-Roch and choice of integer parameters k and k₁.

Experimental results

Research questions

• RQ1 Can Spin(7) instantons be used to prove that certain Hodge (2,2)-classes on abelian four-folds are algebraic?
• RQ2 Under what conditions can a smooth vector bundle with a given Chern character be endowed with a holomorphic structure via an instanton connection?
• RQ3 How can complex multiplication on abelian four-folds be used to construct Cayley cycles dual to Hodge classes?
• RQ4 Is it possible to deform a known holomorphic bundle structure into a Spin(7) instanton using the continuity method?
• RQ5 What conditions ensure the quotient bundle constructed from global sections is polystable with respect to a Kähler polarization?

Key findings

• Explicit smooth vector bundles with Chern characters equal to Hodge classes are constructed from line bundles on Mumford's abelian four-folds.
• The bundle E, constructed as a quotient of a direct sum of line bundles, admits a holomorphic structure when equipped with a Hermite-Einstein metric.
• Polystability of E is established via Butler’s Theorem applied to the kernel of evaluation maps of global sections.
• The slope equality of the summands in the final exact sequence is ensured by choosing k such that (kk₁³ − (3/2)k₁⁴)⟨ω⁴⟩ = c²Δ.
• The existence of such a bundle with a holomorphic structure implies the corresponding Hodge class is algebraic, assuming the instanton connection exists.
• The strategy relies on a starting holomorphic structure (from the conjugate complex structure) and deformation via the continuity method to obtain a Spin(7) instanton.