According to Kant, geometry is possible because of our intuition of space. But, this intuition is presumably 3-dimensional, as we experience the world 3-dimensionally. So, how would higher-dimensional (4 dimensions, 10 dimensions, 1000 dimensions?) geometry fit into this narrative? Humans are capable of this, but I don't see how this would fit into his view of geometry being possible because of our intuition of space.
Kant's claim was that we have intuitive knowledge of things like "between any two points there is a straight line", and "two lines that cross, intersect in a point". He didn't foresee modern geometry and wasn't making any claims about it.
Modern geometry is arithmetic-based, meaning it's done with numbers and symbols instead of a compass and straight edge. This allows one to explore far beyond our intuitions of space, but truth is based on formal consistency rather than perceptible spatial relationships. No intuition of space is required, but consequently, the results don't need to conform to anything real in space either.
It is important to understand what one means by the term 'geometry' specifically. The Kantian implication clearly refers to a 'geometric' intuition, a spatial understanding of 'basic' concepts that are a priori, or in certain ways intrinsic, things that you are aware of naturally, and don't require an experience or stimulation from the so called 'real world'. There are two considerations when Kant makes such claims:
The level of abstraction he is operating at, for instance he could talk about 'science' and not refer to the natural sciences(In this case he's clearly operating on a highly generalised plane), which brings me to the second point
How Kant defines words; In the Critique of Pure Reason, Kant defines intuition as "that through which knowledge refers to its objects immediately" Now, pesky word choices, what does immediately mean in a temporal sense? Well, it is trying to communicate some sort of a spontaneous understanding. This clearly differs from the modern use of the word, which I've noticed is quite a common error made when one tends to interpret Kant.
Coming back to your question, what we mean when the refer to the term 'geometry' is usually the modern Mathematical treatment of the word, so, as pointed above, not what Kant had in mind.
Secondly, from a pure mathematical standpoint, we make generalisations in order to operate in higher dimensions. That is, our intuition is still restricted to 3-space, no human being is capable of imagining 3 vectors perpendicular to one another and magically adding a 4th that is also normal to all three of them. However that is the concept of a 'dimension' roughly. We saw how we moved from 1 space to 2 space to 3 space and therefore arrived at a rough idea of how higher evolutions might proceed. In the human brain we are still operating by means of 3-space analogies, or mathematical intuitions(here not in the Kantian sense, in the comprehension/internalisation of a concept sense) that is readily available to us as a consequence of our 'actual' reality.