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arxiv: 2606.02921 · v1 · pith:SQNV4EHTnew · submitted 2026-06-01 · 🧮 math.AG · math.AC

Syzygies of Isotropic Kalman Varieties

Suhas Vadan Gondi , Sarah Kumar , Abhik Pal This is my paper

Pith reviewed 2026-06-28 12:13 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords isotropic Kalman varietiessymplectic morphismstype Cdefining equationssingularitiessyzygiesorthogonal Kalman varietiesLagrangian case
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The pith

Isotropic Kalman varieties in symplectic spaces satisfy defining equations, geometric invariants, and singularity structures analogous to type A cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines isotropic Kalman varieties as the sets of symplectic endomorphisms of a symplectic vector space V that admit an invariant coisotropic subspace of fixed dimension inside W perpendicular, where W is an isotropic subspace satisfying W perpendicular equals W direct sum L. It establishes that these varieties behave like their type A counterparts by determining their defining equations, computing geometric invariants, and analyzing their singularities, with a focus on the Lagrangian case. Analogues are also described for endomorphisms in the orthogonal cases of types B and D. A conjecture posits the existence of a long exact sequence relating the structure sheaves of these varieties.

Core claim

In the symplectic vector space setting, with an isotropic subspace W satisfying W^perp = W ⊕ L, the isotropic Kalman variety is the set of symplectic morphisms having an invariant coisotropic subspace of prescribed dimension inside W^perp. Analogues of the type A results hold: the defining equations can be determined, geometric invariants computed, and singularities analyzed. This is established particularly for the Lagrangian case, with extensions to orthogonal cases in types B and D.

What carries the argument

The isotropic Kalman variety, the locus of symplectic endomorphisms of V that preserve an invariant coisotropic subspace of prescribed dimension inside W^perp.

If this is right

  • The defining equations of the variety can be determined explicitly from the symplectic condition.
  • Geometric invariants such as dimension and degree can be computed in direct analogy with the type A case.
  • The singularities of these varieties admit analysis and classification parallel to known type A results.
  • Kalman variety analogues exist for endomorphisms of orthogonal vector spaces in types B and D.
  • The structure sheaves of the varieties are conjectured to fit into a long exact sequence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The pattern indicates that Kalman varieties may admit similar explicit descriptions for other groups preserving bilinear forms.
  • Computing the syzygies in concrete cases could produce an explicit minimal free resolution of the ideal.
  • Verification of the conjectured exact sequence in small dimensions would confirm or refute the relation among structure sheaves.
  • These constructions connect naturally to questions about invariant subspaces under classical group actions.

Load-bearing premise

The construction requires an isotropic subspace W such that W perpendicular equals W direct sum L inside the symplectic space, with the morphisms required to be symplectic and preserve a coisotropic subspace of prescribed dimension.

What would settle it

An explicit calculation in a low-dimensional symplectic space such as dimension 4 or 6 that shows the ideal generators or geometric invariants of the isotropic Kalman variety differ from those predicted by the type A analogy.

Figures

Figures reproduced from arXiv: 2606.02921 by Abhik Pal, Sarah Kumar, Suhas Vadan Gondi.

Figure 1
Figure 1. Figure 1: Block decomposition of an arbitrary symplectic endomorphism φ with respect to the decomposition V = R ⊕ W ⊕ (R⊥/R) ⊕ W∗ ⊕ R∗ . The four quadrants defined by the solid lines correspond to A, B, C, and D from (2.3.2). The colored blocks are used in proof of Proposition 4.1.1 and the decomposition (4.2.1) is simply a reorganization of the blocks here. 4.2. Defining equations. Recall that the defining equation… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
read the original abstract

Let $L$ be a subspace of a complex vector space $V$ and fix $s \leq \dim{L}$. The (type A) Kalman variety consists of all endomorphisms of $V$ that have an $s$-dimensional invariant subspace in $L$. We introduce a generalization where $V$ and $L$ are symplectic vector spaces. We fix an isotropic subspace $W \subseteq V$ satisfying $W^\perp = W \oplus L$. The isotropic (type C) Kalman variety consists of symplectic morphisms of $V$ that have an invariant coisotropic subspace of a prescribed dimension inside $W^\perp$. We are mainly interested in studying the Lagrangian case. In type C, we prove analogues of results known for type A Kalman varieties; in particular, we determine the defining equations, compute geometric invariants, and analyze their singularities. We conjecture the existence of a long exact sequence relating the structure sheaves. Based on the results in the symplectic case, we describe Kalman variety analogues with respect to endomorphisms of odd orthogonal (type B) and even orthogonal (type D) vector spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper defines isotropic (type C) Kalman varieties in the symplectic setting: given a symplectic vector space V and isotropic W with W^perp = W ⊕ L, the variety consists of symplectic endomorphisms of V possessing an invariant coisotropic subspace of prescribed dimension inside W^perp. It focuses on the Lagrangian case and proves analogues of type-A results by determining the defining equations, computing geometric invariants, and analyzing singularities. It conjectures a long exact sequence relating the structure sheaves and extends the construction to describe Kalman-variety analogues for odd orthogonal (type B) and even orthogonal (type D) vector spaces.

Significance. If the claimed proofs hold, the work supplies explicit defining equations and singularity analysis for a natural symplectic generalization of Kalman varieties, together with geometric invariants. These results parallel known type-A statements and furnish a uniform framework that also covers orthogonal cases, which may be useful for further study of syzygies and degeneracy loci in algebraic geometry.

minor comments (2)
  1. The abstract states that the Lagrangian case is the main focus, yet the precise dimension or rank conditions that distinguish the Lagrangian subcase from the general coisotropic case are not indicated in the provided description; a short clarifying sentence would help readers.
  2. The conjecture on the long exact sequence is stated without any supporting computation or partial result; if space permits, a brief remark on why the conjecture is plausible (e.g., a low-dimensional check) would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on isotropic Kalman varieties. The recommendation of minor revision is appreciated; however, no specific major comments were provided in the report, so we have no points to address point-by-point at this time. We are prepared to make any minor editorial adjustments requested by the editor.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the isotropic Kalman variety explicitly via the condition on the isotropic subspace W (W^perp = W ⊕ L) inside a symplectic space and states that it proves analogues of type-A results (defining equations, invariants, singularities) as a direct generalization. No equations, parameters, or self-citations are shown reducing any claimed result to its own inputs by construction; the derivation chain is self-contained against the standard definitions of symplectic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard axioms of symplectic vector spaces and the existence of the isotropic splitting W^perp = W ⊕ L; no free parameters or new entities with independent evidence are introduced beyond the definition of the variety itself.

axioms (2)
  • domain assumption V is a symplectic vector space and W is isotropic with W^perp = W ⊕ L
    Invoked in the definition of the isotropic Kalman variety in the abstract.
  • domain assumption Morphisms are required to be symplectic
    Standard assumption for type C setting stated in the abstract.
invented entities (1)
  • Isotropic (type C) Kalman variety no independent evidence
    purpose: Object whose defining equations, invariants, and singularities are studied
    Newly defined in the paper; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.1-grok · 5725 in / 1321 out tokens · 24314 ms · 2026-06-28T12:13:56.385096+00:00 · methodology

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