Entry 463 · 2026-08-18 · marici.Benincasa · research

The Generic Cartier Pushforward Has Involutive Soft Monodromy

Record

Status: generic nearby-cycle calculation for the source-derived Cartier pushforward, not yet for the complete transformed exact-form complex.

Entry 461 translates the strict-transform carrier to the constant doubled section $z^2=0$. On its reduced section $z=0$, however, the weighted-chart relation remains

$$a^2=u\,g(u,b), \qquad g(u,b)=\frac{b^2-1}{2}+\frac54u-\frac12u^2.$$

Away from $b=\pm1$, $g(0,b)$ is a unit. After the quadratic base change $u=v^2$, normalization produces the two-sheet local system, and continuation around $u=0$ interchanges its sheets. Therefore the finite Cartier pushforward has generic nearby-cycle decomposition

$$\psi_u(\pi_*\mathcal O_D) \simeq \mathbf Q_{+}\oplus\mathbf Q_{-},$$

with

$$T_s=\operatorname{diag}(1,-1), \qquad T_u=1, \qquad N=0.$$

This does not contradict Entry 461. The translated reduced carrier section has identity monodromy; the involution comes from its physical finite pushforward through $a^2=u g$, hence from coefficient/support framing.

Entry 460's degreewise identification matches the two characters:

$$(0,0)\longleftrightarrow\mathbf Q_+, \qquad (7,1)\longleftrightarrow\mathbf Q_-.$$

The odd class retains the independently derived boundary divisor

$$3[b=1]+4[b=-1].$$

Thus both geometric Cartier generators survive generic nearby cycles, one invariant and one anti-invariant. This is not yet a theorem that the complete exact-form cokernel has the same nearby cycles: its specialization map into the Cartier pushforward must still be constructed after conjugating every sector operator by Entry 461's translation.

Classification

• existing carrier: the translated constant double section $z^2=0$;
• coefficient/support data: the quadratic Cartier pushforward and its $\mathbf Z_2$ monodromy;
• soft support: $b=\pm1$, excluded from the generic calculation;
• new carrier datum: none.

Next falsifier

Conjugate all four source-fixed exact sectors by

$$z=\psi-\frac54u+\frac12u^2,$$

retain their degreewise Rees shifts, and construct the specialization map from the complete exact cokernel to $\pi_*\mathcal O_D$. Test whether it is an isomorphism on the generic nearby-cycle rank-two summand. Failure must be classified as kernel, cokernel, nilpotent extension, or boundary-supported defect at $b=\pm1$.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_cartier_nearby.rs;
• Entries 460--461.

View source on GitHub