# Support varieties for finite tensor categories: Complexity, realization, and connectedness

## TL;DR

This paper develops support variety theory for finite tensor categories, linking support variety dimensions to projective resolution growth, realizing subvarieties, and proving connectedness of indecomposable objects' support varieties.

## Contribution

It introduces a new framework connecting support varieties with growth rates, realization of subvarieties, and connectedness properties in finite tensor categories.

## Key findings

- Support variety dimension equals the growth rate of minimal projective resolutions.
- Every conical subvariety can be realized as a support variety.
- Support variety of an indecomposable object is connected.

## Abstract

We advance support variety theory for finite tensor categories. First we show that the dimension of the support variety of an object equals the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension. Then we show that every conical subvariety of the support variety of the unit object may be realized as the support variety of an object. Finally, we show that the support variety of an indecomposable object is connected.