  From the perspective of Sen entropy function, we study the geometric and
algebraic properties of a class of (extremal) black holes in $ D \geq 4 $
spacetimes. For a given moduli space manifold, we describe the thermodynamic
geometry away from attractor fixed point configurations with and without higher
derivative corrections. From the notion of embedding theory, the present
investigation offers a set of generalized complex structures and associated
properties of differentiable manifolds. We have shown that the convexity of
arbitrary entropy function can be realized in an extended subfield of the
eigenvalues of the Hessian $\mathcal B$ of Sen entropy function. Thus, the
spectra of $\mathcal B$ are analyzed by defining Krull of the corresponding
semisymplectic algebras. From the framework of commutative algebra, we find
that the convex hull of the eigenvalues defines a generalized spectrum of
$\mathcal B$. The corresponding complexification is established for finitely
many eigenvalues of $ \mathcal B $. For the minimally extended subfield, we
show that the spectrum of $ \mathcal B $ reduces to the thermodynamic type
spectra, at the attractor fixed point(s). In the limit of $ AdS_2 \times
S^{D-2} $ near horizon geometry, the attractor flow analysis offers the
stability of arbitrary extremal black hole. From the perspective of string
compactifications, our investigation implies a set of deformed S-duality
transformations, which contain both the duality invariant charges and monodromy
invariant parameters. The role of the algebraic geometry is discussed towards
the viewpoints of attractor stability conditions, rational conformal field
theory, elliptic curves, deformed quantization(s), moduli manifolds and
Calabi-Yau.
