Even-dimensional intersection theorem and conditional odd case for Lagrangian spheres in quadrics

Definitions and mathematical statement

Question 1.18 conjectures that a smooth complex quadric contains no two disjoint Lagrangian spheres and calls the even-dimensional case trivial. Its displayed coordinates $z_0,\ldots,z_{n+1}$ conflict with the printed ambient ${\mathbb{C}}{\mathbb{P}}^n$. For the aid's dimensionally consistent interpretation use \[ Q^n=\{[z_0:\cdots:z_{n+1}]\in{\mathbb{C}}{\mathbb{P}}^{n+1}: z_0^2+\cdots+z_{n+1}^2=0\}, \] of complex dimension $n$, equipped with the standard restricted Fubini--Study symplectic form. Here ${\mathbb{C}}{\mathbb{P}}^j$ denotes complex projective space. The ambient correction and symplectic convention are explicit working assumptions, not a verified correction of the canonical typography. A Lagrangian sphere means a smoothly embedded $S^n\subset Q^n$ on which the symplectic form restricts to zero. The intended question in dimensions $n\ge2$ is \[ \forall L_0,L_1\subset Q^n\text{ Lagrangian spheres},\qquad L_0\cap L_1\ne\varnothing. \] The source does not state this lower bound; in the literal $n=1$ case the conic is a symplectic two-sphere and two disjoint embedded latitude circles are Lagrangian, so that version is false.

For every even $n\ge2$, the aid reports a stronger integral intersection proposition. Let $h\in H^2(Q^n;{\mathbb{Z}})$ be the restricted hyperplane class. Then $H_n(Q^n;{\mathbb{Z}})$ has rank two and \[ K=\ker\left(H_n(Q^n;{\mathbb{Z}})\longrightarrow{\mathbb{Z}}, \quad a\longmapsto\langle h^{n/2},a\rangle\right) \] is a primitive rank-one subgroup. If $\delta$ is a primitive generator of $K$, every oriented Lagrangian sphere has \[ [L]=\delta\ \text{or}\ -\delta, \qquad |[L]\cdot[L]|=|\delta\cdot\delta|=\chi(S^n)=2. \] Consequently any two such spheres satisfy \[ |[L_0]\cdot[L_1]|=2, \] so they intersect; if their intersection is transverse, it contains at least two points. The absolute-value statement avoids choices of orientation and intersection-sign convention. The Lagrangian condition is used to place the class in $K$, not merely to assert that a sphere is topologically embedded.

For odd $n$, the aid discusses Entov--Polterovich and Biran--Cornea arguments but explicitly disclaims verification of the characteristic-zero/integral orientation extension on which those arguments depend. Thus it records that route only conditionally; no unconditional odd-dimensional theorem is asserted in this draft. The needed extension and its exact hypotheses are not reproduced. The final sentence introducing a question asked by Leonid Polterovich belongs to the next item and does not attribute Question 1.18 to him.

Short English statement

The source asks whether two Lagrangian spheres in a quadric must meet. The aid gives an integral even-dimensional result with algebraic intersection of absolute value two, while keeping its odd-dimensional route conditional. The draft flags the inconsistent projective ambient dimension and the false literal one-dimensional version.

Sources

• Local UnsolvedMath registry, permanent ID 20002076, exact source code catalogue entry; canonical aim-geometry-notes.json, notes[413], tag question; AIM Workshop Problem Lists, Geometry, Holomorphic curves in contact geometry, Question 1.18. Created and updated 14 August 2026.
• Source URL https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The supplied verification repeats the inconsistent coordinate/ambient display. The PDF was not accessed. The trailing Leonid Polterovich introduction is next-item context.
• Entov--Polterovich and Biran--Cornea are named in the aid for conditional odd-dimensional arguments, without full publication entries or exact orientation-extension statements. No separate authoritative reference is supplied for the even-dimensional proposition.
• Attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved. Candidate: even-dimensional primitive-class and intersection theorem. The orientation-extension disclaimer and $n=1$ caveat are retained as supplied.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.