The question: is there a significant geometric form of these logics?
Here, "these logics" refers to Boolean algebras, Heyting algebras and modal algebra. The various representation theorems for these algebras as set-algebras related to certain topological spaces seem to provide a positive answer to this question. These representation theorems are non-trivial, since they are equivalent to the Boolean prime ideal theorem, which cannot be derived from the axioms of Zermelo–Fraenkel set theory without the axiom of choice.
It is well known that every Boolean algebra can be represented as the set-algebra of clopen sets of its associated Stone space.
There are multiple closely related representation theorems for Heyting algebras. First note that a Heyting algebra is bounded and distributive as a lattice, and every bounded distributive lattice can be represented as the set-algebra of clopen upper sets of its associated Priestley space. In case of a Heyting algebra, this associated Priestley space is an Esakai space, which is a Priestley space for which the downward closure of each clopen set is clopen. There are also representation theorems for bounded distributive lattices and Heyting algebras using pairwise Stone spaces instead of Priestley spaces, and representation theorems using spectral spaces.
The wikipedia article on modal algebras says
Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame.
I haven't tried to understand this in detail, but some wikipedia authors apparently believe that this is a geometric representation:
The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust completeness of the latter.