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Hard Lefschetz fails for a non-dominant degree-two element: the rank-one case

Statement refuted

In the rank-one case $n = 2$ the element $ρ^{'} : = x_{1} + x_{2}$ is a degree-two element of $S$ with $ρ^{'} (α_{s}^{\lor}) = 0$ , so it is not strictly dominant. Its action on the reduced rank-one Soergel module is zero: $ρ^{'} = e_{1} = x_{1} + x_{2}$ is $s$ -invariant and of positive degree, and $x_{1} + x_{2} = 1 \cdot e_{1}$ acts on the right basis element $\overline{c_{i d}}$ as the right action by an element of $S_{+}$ , hence $ρ^{'} \cdot \overline{c_{i d}} = 0$ ; a fortiori it cannot map $(\overline{B_{s}})^{- 1} = R \overline{c_{i d}}$ isomorphically to $(\overline{B_{s}})^{1}$ . Therefore for $ρ^{'}$ the hard Lefschetz property fails (the map on the $- 1$ degree is the zero map), even though $ρ^{'}$ is a symmetric degree-two endomorphism of the reduced module; the same failure occurs for every degree-two element with $ρ^{'} (α_{s}^{\lor}) = 0$ . Consequently the hard Lefschetz and Hodge-Riemann conclusions do not hold for arbitrary degree-two elements when strict dominance is omitted. This does not assert that every Lefschetz element is strictly dominant.

Facts & Assumptions

Given: The rank-one setting $n = 2$ , $S = R [x_{1}, x_{2}]$ with $\deg x_{i} = 2$ , the simple reflection $s = s_{1}$ with coordinate root $α = α_{s} = x_{1} - x_{2}$ , the generator $B_{s} = S \otimes_{S^{s}} S (1)$ with right basis $c_{i d} = 1 \otimes 1$ of degree $- 1$ and $c_{s} = \frac{1}{2} (α \otimes 1 + 1 \otimes α)$ of degree $+ 1$ , its reduced module $\overline{B_{s}} = B_{s} / B_{s} S_{+}$ with the induced basis $\overline{c_{i d}}, \overline{c_{s}}$ , and the degree-two element $ρ^{'} = x_{1} + x_{2}$ (The Soergel bimodule $B_{i}$ of a simple reflection, The real type-A Soergel category by scalar extension, The reduced Soergel module and the dominant Lefschetz operator).

[F1]

The left and right actions on $B_{s}$ satisfy $r \cdot c_{i d} = c_{i d} \cdot s r + \partial_{s} (r) c_{s}$ and $r \cdot c_{s} = c_{s} \cdot r$ , and $c_{s} c_{s} = c_{s} α$ , $c_{i d}$ is the unit of the commutative ring $B_{s} (- 1)$ (The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction, The Soergel bimodule $B_{i}$ of a simple reflection).

[F2]

The reduction functor kills right multiplication by $S_{+}$ : $\overline{M} = M / M S_{+}$ , and $m r + M S_{+} = M S_{+}$ for $r \in S_{+}$ (The reduced Soergel module and the dominant Lefschetz operator).

[F3]

Hard Lefschetz for a degree-two symmetric operator $L$ on a graded space with lowest degree $- d$ requires $L : ()^{- 1} \to ()^{1}$ to be an isomorphism in the case $d = 1$ ; definiteness and symmetry conventions as in Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR and Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms.

Counterexample

technique · direct computation in the rank-one reduced module

1.1givenF1

Non-dominance. The reflection $s$ exchanges $x_{1}$ and $x_{2}$ , so $ρ^{'} = x_{1} + x_{2}$ is $s$ -invariant and $ρ^{'} (α^{\lor}) = (x_{1} + x_{2}, x_{1} - x_{2}) = 1 - 1 = 0$ in the standard form of the coordinate normalization; in particular $ρ^{'}$ is not strictly dominant.

2.1step 1.1givenF1

The left action of $ρ^{'}$ equals its right action. Since $ρ^{'} \in S^{s}$ , the balanced tensor relation gives $ρ^{'} \otimes 1 = 1 \otimes ρ^{'}$ in $B_{s}$ ; hence for every $b \in B_{s}$ the left multiplication $ρ^{'} \cdot b$ equals the right multiplication $b \cdot ρ^{'}$ : indeed both actions are $S$ -linear and the identity holds on the two basis elements $c_{i d}$ (where $ρ^{'} \cdot c_{i d} = ρ^{'} \otimes 1 = 1 \otimes ρ^{'} = c_{i d} \cdot ρ^{'}$ ) and $c_{s}$ (where both equal $c_{s} ρ^{'}$ by [F1]).

3.1step 2.1F1F2

The operator vanishes on the reduced module. By step 2.1, $ρ^{'} \cdot c_{i d} = c_{i d} \cdot ρ^{'} = 1 \otimes ρ^{'}$ lies in $B_{s} S_{+}$ , since it is the right product of $c_{i d} = 1 \otimes 1$ with $ρ^{'} \in S_{+}$ ; by [F2] its class in $\overline{B_{s}}$ is zero, so $ρ^{'} \cdot \overline{c_{i d}} = 0$ . Similarly $ρ^{'} \cdot c_{s} = c_{s} \cdot ρ^{'} = c_{s} ρ^{'}$ lies in $B_{s} S_{+}$ and reduces to zero. Since $\overline{c_{i d}}$ and $\overline{c_{s}}$ span $\overline{B_{s}}$ , the induced operator $L = ρ^{'} \cdot (-)$ is the zero endomorphism of $\overline{B_{s}}$ .

4.1step 3.1F3

Failure of hard Lefschetz. The reduced module has $\dim (\overline{B_{s}})^{- 1} = \dim (\overline{B_{s}})^{1} = 1$ with bases $\overline{c_{i d}}$ and $\overline{c_{s}}$ . The degree-one Lefschetz map $L : (\overline{B_{s}})^{- 1} \to (\overline{B_{s}})^{1}$ is zero by step 3.1, hence not an isomorphism, and hard Lefschetz fails for $L$ in the sense of [F3]. The bottom Lefschetz form $⟨ -, L (-) ⟩$ is likewise identically zero, so the Hodge-Riemann relations with any nonzero sign also fail.

5.1step 4.1∎

The general degree-two case. Let $ρ^{''} = a x_{1} + b x_{2}$ satisfy $ρ^{''} (α^{\lor}) = a - b = 0$ ; then $a = b$ and $ρ^{''} = a ρ^{'}$ is a scalar multiple of $ρ^{'}$ . If $a = 0$ the operator is trivially zero; if $a \neq 0$ the same computation as in steps 2.1-4.1 applies because invariance and positivity of degree are unaffected by the scalar. Hence every degree-two element with vanishing evaluation on $α_{s}^{\lor}$ yields the zero operator on the reduced rank-one module and fails hard Lefschetz. This proves the claimed failure and shows that allowing arbitrary degree-two elements invalidates the conclusion; no choice principle is used.

Remark

The counterexample isolates the hypothesis at the very bottom of the induction: the strict dominance of $ρ$ is used on this page exactly through the positivity of the scaling factors $(w ρ) (α_{s}^{\lor})$ for reduced prefixes, and at the simple reflection itself the vanishing of the coroot evaluation makes the operator vanish on the whole reduced module. Since $ρ^{'}$ is still symmetric of degree two, the failure is not an artifact of a badly normalized form.

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Sources

Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136) (standard reference, not scraped)
Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525) (standard reference, not scraped)