Anti-Symplectic Isomorphisms

• Anti-symplectic isomorphisms are mappings that reverse the symplectic form (φ*ω = -ω) on manifolds, K3 surfaces, and moduli spaces.
• They interact intricately with cohomological structures and lattice theory, impacting derived categories and moduli space birational geometry.
• The fixed loci of these involutions form Lagrangian submanifolds, providing key insights into Bloch's conjecture and the topology of algebraic cycles.

Anti-symplectic isomorphisms refer to automorphisms, involutions, or derived autoequivalences acting on symplectic manifolds, algebraic surfaces, or their moduli spaces, that reverse the symplectic form or structure. Precisely, an (anti-)symplectic isomorphism is a morphism $\varphi: X \rightarrow X$ such that $\varphi^*\omega = -\omega$, where $(X,\omega)$ is a symplectic manifold or variety. In algebraic and derived contexts, this notion generalizes to actions on cohomology, Chow groups, and moduli spaces, leading to profound consequences for the geometry, topology, and arithmetic of these spaces.

1. Formal Definition and Framework

An anti-symplectic isomorphism is an automorphism (often of finite order) $\sigma$ or a derived autoequivalence $\Phi$ such that:

• For a symplectic manifold $(X, \omega)$,

$$\sigma^* \omega = -\omega, \qquad \sigma^2 = \operatorname{id}$$

• In the context of a K3 surface $X$,

$\Phi^{\tilde{H}}|_{T(X)} = -\operatorname{id}$

where $\Phi^{\tilde{H}}$ is the induced action on the Mukai lattice $\widetilde{H}(X,\mathbb{Z})$ and $T(X)$ is the transcendental lattice.

In involutive cases, the fixed locus $F = \operatorname{Fix}(\sigma)$ is a smooth Lagrangian submanifold of $X$ (Beauville, 2010), i.e., $\dim F = \frac{1}{2} \dim X$ and $\omega|_F = 0$. In Hilbert schemes and moduli spaces of sheaves on K3 surfaces, anti-symplectic involutions constructed via spherical twists and dualities produce birational involutions that act as reflections on the Mukai lattice (Faenzi et al., 19 Sep 2024).

2. Cohomological and Derived-Categorical Aspects

Anti-symplectic isomorphisms interact with cohomological and categorical structures:

• On the Mukai lattice of a K3 surface, such an autoequivalence reverses the transcendental part:

$$T(X) = \{ v \in H^2(X,\mathbb{Z}) : (v,w)=0 \ \forall w \in NS(X) \}$$

$\Phi^{\tilde{H}}|_{T(X)} = -\operatorname{id}$

• Derived autoequivalences constructed using spherical twists $T_S$ and duality functors $R(-,L)$:

$\Phi_{S,L}^p(E) = R(T_S(E), L[p])$

When $S$ is self-dual up to twist (e.g., $S \cong R(S,L[0])$), $\Phi$ becomes involutive and anti-symplectic on moduli spaces (Faenzi et al., 19 Sep 2024).

• Such involutions yield birational transformations of moduli spaces $\mathcal{M}(v)$, reflecting Mukai vectors and potentially preserving stability conditions (subject to congruence or vanishing constraints on $v$) (Faenzi et al., 19 Sep 2024).

3. Fixed Loci and Lagrangian Submanifolds

The fixed locus $F$ of an anti-symplectic involution on a holomorphic symplectic manifold is necessarily Lagrangian (Beauville, 2010). The geometry of $F$ is tightly controlled:

• $F$ is a smooth, half-dimensional submanifold satisfying $N_{F/X}^* \cong T_F$.
• The $\hat{A}$-genus of $F$ is $1$ for $\dim X$ divisible by $4$ (Beauville, 2010).
• Chern numbers and signatures of $F$ can be explicitly computed via Lefschetz fixed point formulas:
  • For $X$ deformation of $K3^{[2]}$, invariants such as $K_F^2$, $\chi(\mathcal{O}_F)$, and $e(F)$ are determined via trace formulas:

  $$K_F^2 = t^2 - 1;\quad \chi(\mathcal{O}_F) = \frac{t^2+7}{8};\quad e(F) = \frac{t^2+23}{2}$$

  where $t$ is the trace of $\sigma^*$ on $H^{1,1}(X)$ (Beauville, 2010).

In the Hilbert scheme setting $(S^{[n]})^{\tau}$ for a surface $S$ with anti-symplectic involution $\tau$, the fixed locus is likewise Lagrangian and smooth, and its mixed Hodge structure is determined by those of $S^{\tau}$ and $S/\tau$ (Baird, 2023).

4. Cartan–Dieudonné Decomposition and Reflective Involutions

For autoequivalences (possibly twisted) on K3 surfaces, the Cartan–Dieudonné theorem for even lattices plays a crucial role (Li et al., 2023). Every orientation-preserving isometry of the Mukai lattice (acting as $\pm$ identity on $T(X)$) admits a decomposition into a product of reflective involutions:

• For primitive $v$ with $v^2 = \pm 2$ or $0$, the reflection is given by $s_v(x) = x - 2(x,v)/(v,v) v$.
• Symplectic and anti-symplectic autoequivalences in $\operatorname{Aut}(D^b(X))$ decompose into compositions of these involutions, and their action on $CH_0(X)_{\text{hom}}$ is $\pm \operatorname{id}$.
• This gives an integral version of Cartan–Dieudonné adapted to the Mukai lattice topology, confirming Bloch's conjecture for (anti-)symplectic autoequivalences when the Picard number is at least 3 (Li et al., 2023).

5. Chow Groups, Bloch's Conjecture, and Constant Cycle Property

Bloch's conjecture, in the context of (anti-)symplectic automorphisms, states that the action on the group of 0-cycles modulo homological equivalence $CH_0(X)_{\text{hom}}$ mirrors the action on transcendental cohomology:

• For (anti-)symplectic autoequivalences $\Phi$, $\Phi_* = \pm \operatorname{id}$ on $CH_0(X)_{\text{hom}}$ (Li et al., 2023).
• For hyperkähler varieties $Y$ of $K3^{[n]}$-type, (anti-)symplectic birational automorphisms act as $\pm 1$ on the graded pieces of the Beauville–Voisin filtration:

$$\varphi_* = (\pm 1)^i \operatorname{id} \text{ on } CH_0(Y)_i$$

especially for finite order automorphisms $\neq 2,4$.

A crucial application is the constant cycle property: under mild conditions (e.g., $n\leq 2$ or invariant sublattice of rank $\geq 2$), the fixed locus $F$ of an anti-symplectic involution is a constant cycle subvariety. That is, all points of $F$ are rationally equivalent in the ambient variety, which has deep consequences for the structure of algebraic cycles and moduli (Li et al., 2023).

6. Examples and Applications

Notable examples include:

• Beauville involutions, Markman–O’Grady reflections, and Beri–Manivel constructions (Faenzi et al., 19 Sep 2024), realized via combinations of spherical twists, dualizing functors, and line bundle shifts.
• Fixed loci in Hilbert schemes $(S^{[n]})^{\tau}$ classified by genus, component count, and mixed Hodge structure (Baird, 2023). For K3 surfaces, Nikulin’s invariants $(r, a, \delta)$ specify the topology and Hodge data of fixed curves and their impact on the ambient geometry.
• Rational symplectic 4-manifolds with anti-symplectic involution classified up to equivariant deformation by the involution's diffeomorphism type, with explicit computation in the cases of $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ (Kharlamov et al., 2020).

7. Mathematical Significance and Research Directions

Anti-symplectic isomorphisms represent a categorical and geometric symmetry that impacts the topology, period mapping, and algebraic cycles of symplectic and hyperkähler manifolds. Their paper unifies reflection phenomena in lattice theory, derived category operations, and moduli space birational geometry. The confirmation of Bloch's conjecture in wide generality, explicit constructions of involutions on moduli spaces, and the constant cycle property for fixed loci highlight the interplay between symplectic topology, algebraic geometry, and representation theory. Current advances open further avenues, including the classification of involutions via lattice-theoretic invariants, implications for mirror symmetry, and refined period map analysis for moduli spaces and their automorphism groups.