Real algebraic varieties and their intersection cohomology

Technical Summary: Real Algebraic Varieties and Their Intersection Cohomology

Problem Statement
The paper addresses a fundamental question posed by Goresky and MacPherson regarding the existence of intersection cohomology groups for real algebraic varieties. In the complex setting, for a singular variety $Y$, the intersection cohomology groups $IH^*(Y(\mathbb{C}), k)$ provide a Poincaré self-dual theory that factors the cap product with the fundamental class. They arise intrinsically as the unique Verdier self-dual perverse sheaf extending the constant sheaf on the smooth locus, and extrinsically as a direct summand of the pushforward of the constant sheaf under any resolution of singularities (via the Decomposition Theorem).

For real varieties $Y$, the cap product with the fundamental class in $H^*(Y(\mathbb{R}), \mathbb{F}_2)$ does not generally induce an isomorphism, and no canonical family of "intersection cohomology" groups existed that satisfied analogous duality and factorization properties. Previous constructions by McCrory-Parusiński and van Hamel offered solutions but differed significantly in their geometric or equivariant approaches, sometimes lacking finiteness or perfect pairing properties. This paper seeks to construct a real analogue that behaves like intersection cohomology, specifically utilizing $\mathbb{F}_2$ coefficients.

Methodology
The author adopts an extrinsic approach, constructing the desired object as a specific direct summand within the derived pushforward of the constant sheaf along a resolution of singularities. The methodology relies on the formalism of constructible sheaves on real algebraic varieties with $\mathbb{F}_2$ coefficients.

Key technical steps include:

1. Krull-Schmidt Decomposition: The paper utilizes the fact that the derived category of constructible sheaves $D^b_c(Y(\mathbb{R}), \mathbb{F}_2)$ is Krull-Schmidt. This ensures that the pushforward $Rf_* \mathbb{1}_{X(\mathbb{R})}$ for any resolution $f: X \to Y$ decomposes uniquely into a finite direct sum of indecomposable objects.
2. Splitting of Birational Maps: A crucial lemma establishes that for a proper birational map $g: W \to X$ between smooth real varieties, the canonical map $\mathbb{1}_{X(\mathbb{R})} \to Rg_* \mathbb{1}_{W(\mathbb{R})}$ is a split monomorphism. This splitting is induced by the compatibility of fundamental classes under pushforward.
3. Independence of Resolution: By showing that any two resolutions are dominated by a third, and utilizing the splitting property, the author proves that the "dense" summand of the pushforward (the part extending the constant sheaf on the smooth locus) is independent of the choice of resolution.
4. Motivic Variant: A secondary construction, $E_{num}(Y_R)$, is defined using the image of the real cycle class map from Chow groups to Borel-Moore homology. This variant is designed to be Zariski-local, a property not immediately obvious for the primary construction.

Key Contributions and Results

• Construction of the Real Geometric Extension ($E(Y_R)$):
  The paper defines a complex of sheaves $E(Y_R)$ on $Y(\mathbb{R})$, called the real geometric extension. It is characterized as the unique direct summand of $Rf_* \mathbb{1}_{X(\mathbb{R})}$ (for any resolution $f$) that extends the constant sheaf on the smooth locus $Y^{sm}(\mathbb{R})$ and has no proper direct summand with this property.

• Poincaré Duality and Self-Duality:
  The complex $E(Y_R)$ is shown to be Verdier self-dual up to a shift by the dimension $d$ of the variety ($DE(Y_R) \cong E(Y_R)[d]$). Consequently, the hypercohomology groups $E^i(Y(\mathbb{R}), \mathbb{F}_2)$ satisfy a Poincaré duality isomorphism:
  $$E^i(Y(\mathbb{R}), \mathbb{F}_2) \cong E^{d-i}(Y(\mathbb{R}), \mathbb{F}_2)^\vee$$
  This duality is induced by a choice of isomorphism in the derived category, which is non-canonical even over $\mathbb{F}_2$.

• Factorization and Lower Bounds:
  The construction provides canonical comparison maps $\gamma^i: H^i(Y(\mathbb{R}), \mathbb{F}_2) \to E^i(Y(\mathbb{R}), \mathbb{F}_2)$ that factor the cap product with the fundamental class. Furthermore, the stalks of $E(Y_R)$ provide lower bounds for the Betti numbers of the fibers in any resolution of singularities:
  $$\dim H^i(i_y^* E(Y_R)) \leq \dim H^i(F_y(\mathbb{R}), \mathbb{F}_2)$$

• Explicit Descriptions in Special Cases:

  • Small Resolutions: If $Y$ admits a small resolution $f: X \to Y$, then $E(Y_R) \cong Rf_* \mathbb{1}_{X(\mathbb{R})}$.
  • Isolated Singularities: For varieties with isolated singularities satisfying specific topological conditions on the exceptional divisor (e.g., trivial normal bundle), $E(Y_R)$ is identified with the pushforward of the constant sheaf from the blowup.
  • Schubert Varieties: On real flag varieties, the category of real geometric extensions is shown to be equivalent to the category of even parity sheaves on the corresponding complex flag variety. This equivalence doubles the degrees: $E^i(X(w)(\mathbb{R})) \cong H^{2i}(E(X(w)(\mathbb{C})))$.

• Motivic Variant ($E_{num}$):
  The paper introduces $E_{num}(Y_R)$, a variant defined via the real cycle class map. This object is Zariski-local (its restriction to an open set is the extension on that open set) and shares the duality and finiteness properties of $E(Y_R)$. The paper notes that while $E_{num}$ has better formal properties, it is less explicit to compute, and it is currently unknown if $E(Y_R)$ and $E_{num}(Y_R)$ differ in any case.

Significance and Claims
The paper claims to provide a solution to the Goresky-MacPherson question by constructing a family of $\mathbb{F}_2$ vector spaces for real varieties that behave like intersection cohomology. The significance lies in:

1. Extrinsic Definition: Unlike the intrinsic definition of complex intersection cohomology (via perversity bounds), this construction is purely extrinsic, defined via resolutions. This is necessitated by the lack of a perverse t-structure in the real setting and the fact that real geometric extensions do not satisfy perversity bounds.
2. Connection to Parity Sheaves: The equivalence between real geometric extensions on Schubert varieties and even parity sheaves on complex varieties suggests a deep structural link between real algebraic geometry and the representation theory of complex groups.
3. Complement to Existing Work: The author positions this work as a natural generalization of geometric extensions and parity sheaves to the real setting, complementing the geometric approach of McCrory-Parusiński and the equivariant approach of van Hamel.

The paper remains modest regarding the uniqueness of the duality isomorphism (noting it is a choice) and the potential difference between the primary construction and the motivic variant. It explicitly states that the construction depends on the variety structure, not just the underlying topology, as demonstrated by examples of non-isomorphic varieties with homeomorphic real point sets.