This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. It can be seen that $P(H)$ is a complex Hilbert manifold (modelled on $H$) and has homotopy type $K(\Bbb{Z},2)$.
Suppose $A$ is a continuous symmetric $d$-linear functional on $H$. Define the homogeneous polynomial of degree $d$ w.r.t. $A$ as $p_A : H \rightarrow \Bbb{C} : v \mapsto A(\underbrace{v,v,...,v}_{d \text{ times}})$. Each $p_A$ has a well-defined zero locus in $P(H)$ due to the homogenity. Call $p_A$ non-singular if its Fréchet derivative (which, at a given point, is a continuous linear functional on $H$ and can be canonically identified with an element in $H$) is non-zero everywhere aside from the origin. In this case, the zero locus of $p_A$ is also a complex Hilbert manifold and can be safely called a variety.
Question: Is the transverse intersection of finitely many (can be $1$) varieties always non-generic, e.g. they have finitely generated total homotopy groups (like the varieties of degree $1$ which have the same homotopy type $K(\Bbb{Z},2)$), infinite automorphism groups, etc.? In particular, since the unit sphere in $H$ is diffeomorphic to $H$ itself which admits a complex structure, is it also realizable as a transverse intersection of finitely many varieties? (probably has a negative answer if my attempt below is valid)
Attempt: Assuming a modified infinite-dimensional version of Lefschetz hyperplane theorem (LHT) yields a rather surprising answer: Choose a countable complete orthonormal basis of $H$ (denoting it $\{ e_i \}_{i=0}^{\infty}$). Let $x_i$ be the inner product with $e_i$, then $[x_i]:=[x_0 : x_1 : \cdots]$ is a homogeneous coordinate system on $P(H)$. Consider the degree-$d$ Veronese map $$V_d: P(H) \rightarrow P(H): [x_i] \mapsto \left[ \sqrt{\frac{d!}{l_{1}! \cdots l_{d}!}} x_{k_1}^{l_1} \cdots x_{k_d}^{l_d} \right]$$ where $0 \le l_1 \le \cdots \le l_d \le d$ is an integer partition of $d$ and the coordinates of the image are arranged so that $l_1 k_1 + \cdots + l_d k_d$ is increasing. The map is well-defined since $\lVert V_d(x) \rVert = \lVert x \rVert^d$ is finite. Because $V_d$ is a holomorphic injection, $S_d:=\mathrm{Im} (V_d)$ is a complex (closed?) submanifold of $P(H)$ isomorphic to $P(H)$ itself, being both infinite-dimensional and co-infinite-dimensional. A large number of degree-$d$ varieties (to be precise, the varieties correspond to completely continuous polynomials) are mapped to closed hyperplane sections of $S_d$, and formally apply LHT to them leads to the result that all codimension-$1$ varieties are homotopically indistinguishable regardless of the degree. Moreover, since the varieties are Hilbert manifolds, a homotopy equivalence will imply a diffeomorphism (see this Wiki page). This sounds ridiculous, but things related to infinity are sometimes ridiculous, so I'm willing to know whether my attempt can be actually transformed into a strict proof.
Motivation: Low-degree (compared to the dimension of the ambient space) complex hypersurfaces and their intersections have simpler topology and richer symmetries in contrast to the high-degree generic ones. I'm wondering if every finite number is "low enough" compared to the infinity of the dimension of a Hilbert space.
Related question: Since the continuity of $p_A$ influences the formal application of LHT, what will happen if the projectivization at the start is performed on $H$ with weak topology? It does not seem to change the homotopy type of $P(H)$, but the varieties are no longer Hilbert manifolds.
