# On type II degenerations of hyperkähler manifolds

D. Huybrechts & M. Mauri  Mathematisches Institut and Hausdorff Center for
Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany & Max
Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
<EMAIL_ADDRESS>&<EMAIL_ADDRESS>
###### Abstract.

We give a simple argument to prove Nagai’s conjecture for type II
degenerations of compact hyperkähler manifolds and cohomology classes of
middle degree. Under an additional assumption, the techniques yield the
conjecture in arbitrary degree. This would complete the proof of Nagai’s
conjecture in general, as it was proved already for type I degenerations by
Kollár, Laza, Saccà, and Voisin [10] and independently by Soldatenkov [18],
while it is immediate for type III degenerations. Our arguments are close in
spirit to a recent paper by Harder [8] proving similar results for the
restrictive class of _good_ degenerations.

††footnotetext: The first author is supported by the ERC Synergy Grant HyperK.
The second author is supported by the Max Planck Institute for
Mathematics.††margin:

##  1\. Introduction

Any one-dimensional degeneration ${\cal X}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ of
compact Kähler manifolds induces monodromy operators $T_{k}$ acting on the
cohomology groups $H^{k}(X,\mathbb{C})$ of a smooth fibre $X\coloneqq{\cal
X}_{t}$, ${t\neq 0}$. After a base change, one can assume that $T_{k}$ is
unipotent, i.e. the operator $T_{k}-{\rm id}$ is nilpotent. Alternatively,
this can be expressed by saying that the logarithmic monodromy operator
$N_{k}\coloneqq\log T_{k}$ acting on $H^{k}(X,\mathbb{C})$ is nilpotent. Thus,
the _index of nilpotence_ defined as

${\rm nilp}(N_{k})\coloneqq\max\\{i\mid N^{i}\neq 0\\}$

is finite and it is known that ${\rm nilp}(N_{k})\leq k$, see [7, Ch. IV].
Clearly, if ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ is a
smooth family, then $N_{k}=0$ for all $k$. In general, there is no direct link
between the ${\rm nilp}(N_{k})$ for different $k$. However, following the
general philosophy that the geometry of a compact hyperkähler manifold $X$ is
largely determined by its Hodge structure of weight two $H^{2}(X,\mathbb{Z})$,
it was conjectured by Nagai [14] that $N_{2}$ determines all even $N_{2k}$.

######  Conjecture 1.1 (Nagai).

The even logarithmic monodromy operators $N_{2k}$, $k\leq n$, of a
degeneration of compact hyperkähler manifolds ${\cal X}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
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16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ with
unipotent monodromy satisfy

${\rm nilp}(N_{2k})=k\cdot{\rm nilp}(N_{2}).$

Using Verbitsky’s result [1, 20] that cup-product yields inclusions
$S^{k}H^{2}(X,\mathbb{C})\subset H^{2k}(X,\mathbb{C})$ for $k\leq n$, Nagai
already observed that

${\rm nilp}(N_{2k})\geq k\cdot{\rm nilp}(N_{2}).$

Thus, for type III degenerations, i.e. ${\rm nilp}(N_{2})=2$, the conjecture
follows from the two inequalities $2k\geq{\rm nilp}(N_{2k})\geq k\cdot{\rm
nilp}(N_{2})$, see also [10, Sec. 6] or [6, Sec. 5.1].

For projective degenerations of type I, i.e. $N_{2}=0$, the conjecture was
first established by Kollár, Laza, Saccà, and Voisin [10]. An independent
proof, also valid in the non-projective case, was given by Soldatenkov [18,
Cor. 3.6]. For type II degenerations, i.e. ${\rm nilp}(N_{2})=1$, it was
proved in [10, Thm. 6.19] that $k\leq{\rm nilp}(N_{2k})\leq 2k-2$ for all
$2\leq k\leq n-1$, which follows from the observation that the level of the
Hodge structure of $H^{2k}(X,\mathbb{Q})/S^{k}H^{2}(X,\mathbb{Q})$ does not
exceed $2k-2$.

Thus, in order to establish Nagai’s conjecture in full, only the case of type
II degenerations remains open. In fact, for all known examples of compact
hyperkähler manifolds the conjecture was established by Green, Kim, Laza, and
Robles [6], relying on a complete understanding of the cohomology ring as a
representation of the LLV algebra. Special cases of degenerations of Hilbert
schemes and generalized Kummer varieties have been treated in [14].

The purpose of this note is to give an elementary proof of the following.

######  Theorem 1.2.

Let ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ be a
type II degeneration of compact hyperkähler manifolds of dimension $2n$. Then
$N^{n+1}=0$ on $H^{2n}(X,\mathbb{C})$, i.e.

${\rm nilp}(N_{2n})=n.$

Furthermore, ${\rm nilp}(N_{2k})\leq n-1$ for all $k<n$ and, assuming
condition _( 2.7)_, in fact ${\rm nilp}(N_{2k})=k$ for all $k\leq n$.

Assuming that the degeneration is good, the result for ${\rm nilp}(N_{2n})$
and ${\rm nilp}(N_{2n-2})$ was proved before using different but related
arguments by Harder [8]. In this sense, our arguments show that the
restrictive assumption in [8] that the degeneration is good can be dropped.
Good degenerations had earlier been studied by Nagai [14] who observed already
that ${\rm nilp}(N_{2k})\not\in\\{1,\ldots,k-1\\}$. In particular, for good
type II degenerations one has ${\rm nilp}(N_{2n-2})=n-1$ which in fact holds
for general type II degenerations.

In Section 3 we establish a link between the perverse and the monodromy
filtration. It leads to the following reformulation of Nagai’s conjecture.

######  Corollary 3.4.

The Nagai conjecture (for type II degenerations) holds if and only if the
Hodge structure on the graded pieces ${\rm Gr}^{P}_{i}H^{2k}(X,\mathbb{Q})$ of
the perverse filtration has level at most $2k-2|i-k|$.

The techniques that will be used to prove Theorem 1.2 also yield information
about the logarithmic monodromy action in odd degree. In Section 4 we shall
prove the following.

######  Theorem 4.1.

Let ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ be a
type II degeneration of compact hyperkähler manifolds of dimension $2n$. Then
the odd logarithmic monodromy operators $N_{2k-1}$, $k\leq n$, satisfy

${\rm nilp}(N_{2k-1})\leq\min\\{2k-3,n-1\\}.$

Furthermore, assuming condition _( 2.7)_, in fact ${\rm nilp}(N_{2k-1})\leq
k-1$ for all $k\leq n$.

There is also a lower bound for ${\rm nilp}(N_{2k-1})$ depending on the level
of the Hodge structure, see Corollary 4.2.

##  2\. Even degree

In the following, ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ will
always be a degeneration of compact hyperkähler manifolds with unipotent
monodromy and $X={\cal X}_{t}$, $t\neq 0$, denotes a fixed smooth fibre.

### 2.1.

We begin by recalling some well-known properties of the logarithmic monodromy
operator $N=\bigoplus N_{k}$.

######  Lemma 2.1.

For classes $\alpha\in H^{k}(X,\mathbb{C})$ and $\alpha^{\prime}\in
H^{\ell}(X,\mathbb{C})$ one has

(2.1)
$N_{k+\ell}(\alpha\wedge\alpha^{\prime})=N_{k}(\alpha)\wedge\alpha^{\prime}+\alpha\wedge
N_{\ell}(\alpha^{\prime}).$

Furthermore, for a degeneration of compact hyperkähler manifolds, the operator
$N_{2}$ is compatible with the Beauville–Bogomolov form:

(2.2) $q(N_{2}(\alpha),\alpha^{\prime})+q(\alpha,N_{2}(\alpha^{\prime}))=0.$

###### Proof.

The monodromy operator $T$ on $H^{\ast}(X,\mathbb{C})$ is an algebra
isomorphism and, hence,

(2.3) $T^{m}(\alpha\wedge\alpha^{\prime})=T^{m}(\alpha)\wedge
T^{m}(\alpha^{\prime})$

for all $m\geq 0$. Now use $T^{m}=\exp(mN)$ and expand (2.3) in $m$. Comparing
linear terms, yields the first assertion. The proof of the second assertion is
similar, relying on the fact that the monodromy operator $T_{2}$ is
orthogonal, i.e.
$q(T_{2}(\alpha),T_{2}(\alpha^{\prime}))=q(\alpha,\alpha^{\prime})$. ∎

Let us now consider a degeneration ${\cal X}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
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16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ of
compact hyperkähler manifolds with $T_{2}-{\rm id}$ of order two or,
equivalently, such that $N_{2}^{2}=0$ but $N_{2}\neq 0$, i.e. a degeneration
of type II. For the convenience of the reader we recall the following
observation due to Schreieder and Soldatenkov [16, Prop. 4.1].

######  Lemma 2.2.

For a type II degeneration of compact hyperkähler manifolds the image of the
logarithmic monodromy operator $N_{2}\colon
H^{2}(X,\mathbb{C})\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces
H^{2}(X,\mathbb{C})$ is an isotropic plane.

Furthermore, for any basis $w_{1},w_{2}\in{\rm Im}(N_{2})$ one has

(2.4) $N_{2}(\alpha)=q(w_{2},\alpha)\,w_{1}-q(w_{1},\alpha)\,w_{2}$

up to scaling $N_{2}$ by a factor.

###### Proof.

Due to (2.2), one has
$q(N_{2}(\alpha),N_{2}(\alpha^{\prime}))=-q(\alpha,N_{2}^{2}(\alpha^{\prime}))=0$,
which shows that the image is indeed isotropic. A computation with Hodge
numbers, see [7, Ch. VI], yields $\dim{\rm Im}(N_{2})=2$.

Thus, one can write $N_{2}=\lambda_{1}(~{})\,w_{1}+\lambda_{2}(~{})\,w_{2}$
for certain linear forms $\lambda_{1},\lambda_{2}$ on $H^{2}(X,\mathbb{C})$.
Then, since by (2.2) one has $q(N_{2}(\alpha),\alpha)=0$ for all $\alpha\in
H^{2}(X,\mathbb{C})$, one finds
$\lambda_{1}(\alpha)\,q(w_{1},\alpha)+\lambda_{2}(\alpha)\,q(w_{2},\alpha)=0$,
which proves
$[\lambda_{1}(~{}):\lambda_{2}(~{})]=[q(w_{2},~{}):-q(w_{1},~{})]$. ∎

### 2.2.

Let us now fix a non-trivial isotropic class $\beta\in H^{1,1}(X)$ and denote
by $L_{\beta}$ the operator $\alpha\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
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1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\alpha\wedge\beta$
which is of bidegree $(1,1)$. Furthermore, we let $\Lambda^{c}_{\sigma}$ be
the operator of bidegree $(-2,0)$ defined as the contraction by the
holomorphic symplectic form $\sigma$, see also Remark 2.3. Then the operator

$M\coloneqq[L_{\beta},\Lambda^{c}_{\sigma}]$

is of bidegree $(-1,1)$ and enjoys similar properties as the monodromy
operator $N$. For example,

(2.5)
$M_{k+\ell}(\alpha\wedge\alpha^{\prime})=M_{k}(\alpha)\wedge\alpha^{\prime}+\alpha\wedge
M_{\ell}(\alpha^{\prime})$

for classes $\alpha\in H^{k}(X,\mathbb{C})$ and $\alpha^{\prime}\in
H^{\ell}(X,\mathbb{C})$. Furthermore, $M$ is of type II, i.e. $M_{2}^{2}=0$,
and the image of $M_{2}$ is an isotropic plane, namely ${\rm
Im}(M_{2})=\langle\beta,\bar{\sigma}\rangle$. More precisely, up to a scaling
factor

$M(\sigma)=-q(\bar{\sigma},\sigma)\,\beta\text{ and
}M(\alpha)=q(\beta,\alpha)\,\bar{\sigma}$

for any $\alpha\in H^{1,1}(X)$. So, similarly to Lemma 2.2, we can write, up
to a scaling factor,

(2.6) $\displaystyle
M_{2}=q(\beta,~{})\,\bar{\sigma}-q(\bar{\sigma},~{})\,\beta.$

The quickest way to prove these facts is via the (complex)
Looijenga–Lunts–Verbitsky Lie algebra ${\mathfrak{g}}(X)$, see [12, 20] or the
summaries [6, Thm. 2.7] and [19, Sec. 2]. It is generated by all $L_{x}$ and
$\Lambda_{x}$, where $x\in H^{2}(X,\mathbb{C})$ can be any class satisfying
the Lefschetz property. As $L_{\beta}$ and $\Lambda_{\sigma}^{c}$ are linear
combinations of certain $L_{x}$ and $\Lambda_{x}$, cf. Remark 2.3 below, the
known commutator relations in ${\mathfrak{g}}(X)$ also hold for
$[L_{\beta},\Lambda^{c}_{\sigma}]$ and the property that ${\mathfrak{g}}(X)$
acts by derivation on the algebra $H^{\ast}(X,\mathbb{C})$ extends to the
operator $M$.

######  Remark 2.3.

(i) If one writes $\sigma=\omega_{J}+\sqrt{-1}\omega_{K}$, where
$\omega_{J},\omega_{K}$ are the Kähler classes of complex structures $J$ and
$K$ defined by the choice of a hyperkähler metric, then
$\Lambda^{c}_{\sigma}=\Lambda_{\omega_{J}}-\sqrt{-1}\Lambda_{\omega_{K}}$.
Here, $\Lambda_{\omega_{J}}$ and $\Lambda_{\omega_{K}}$ are the usual dual
Lefschetz operators associated to the Lefschetz classes $\omega_{J}$ and
$\omega_{K}$. In other words, if $\Lambda_{\omega}$ depending on a Lefschetz
class $\omega$ is extended linearly on the whole $H^{2}(X,\mathbb{C})$, then
$\Lambda_{\sigma}^{c}=\Lambda_{\bar{\sigma}}$. For an explanation of the sign
we refer to [5, p. 118], see also [17, Sec. 3.1].

(ii) There exists an isomorphism
${\mathfrak{g}}(X)\simeq\mathfrak{so}(\tilde{H}(X,\mathbb{C}))$, see [12, 20].
Here, $\tilde{H}(X,\mathbb{C})$ denotes the Mukai extension of
$H^{2}(X,\mathbb{C})$, i.e. its direct sum with a hyperbolic plane.
Furthermore,
${\mathfrak{g}}(X)\simeq{\mathfrak{g}}_{-2}\oplus\left({\mathfrak{g}}^{\prime}\oplus\mathbb{C}\,h\right)\oplus{\mathfrak{g}}_{2}$
where ${\mathfrak{g}}_{\pm 2}\simeq H^{2}(X,\mathbb{C})$,
${\mathfrak{g}}^{\prime}\simeq\mathfrak{so}(H^{2}(X,\mathbb{C}))$, and $h$ is
the standard counting operator, i.e. $h(v)=(d-2n)v$ for $v\in
H^{d}(X,\mathbb{C})$, cf. [12, 6, 19, 20].

### 2.3.

Nagai’s conjecture for the operator $M_{2n}$ can be verified by a computation.
This is the content of the next result. For $k=n$ the assertion will give
Nagai’s conjecture for the middle cohomology, see Section 2.6. For $k\leq n/2$
one in fact has the stronger result $M^{2k+1}=0$ on $H^{2k}(X,\mathbb{C})$,
simply because $M$ is of bidegree $(-1,1)$.

######  Proposition 2.4.

For all $k\leq n$ we have $M^{n+1}=0$ on $H^{2k}(X,\mathbb{C})$.

###### Proof.

Since $M$ is of bidegree $(-1,1)$, we have $M^{\ell+1}=0$ on $H^{p,q}(X)$ for
$p\leq\ell$. Thus, to prove the assertion, it suffices to show
$M^{n+1}(\alpha)=0$ for $\alpha\in H^{p,q}(X)$ with $p+q=2k$ and $p>n$. Using
the isomorphism $\sigma^{p-n}\colon\Omega_{X}^{2n-p}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
3.19443pt\raise 3.8674pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
2.58334pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
2.58334pt}}}}}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Omega_{X}^{p}$
for $n<p\leq 2n$, we can write $\alpha=\sigma^{p-n}\wedge\gamma$ for some
$\gamma\in H^{2n-p,q}$. Then, by (2.5)

$\displaystyle M^{n+1}(\alpha)=M^{n+1}(\sigma^{p-n}\wedge\gamma)$
$\displaystyle=$ $\displaystyle\sum_{i+j=n+1}{n+1\choose
i}\,M^{i}(\sigma^{p-n})\wedge M^{j}(\gamma)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i+j=n+1\\\ j\leq
2n-p\end{subarray}}{n+1\choose i}\,M^{i}(\sigma^{p-n})\wedge M^{j}(\gamma),$

where we use that for bidegree reasons $H^{2n-p,q}(X)$ is annihilated by
$M^{j}$ for $j>2n-p$. On the other hand, since $M^{2}(\sigma)=0$, we have
$M^{i}(\sigma^{p-n})=0$ for $i>p-n$. This yields the assertion. ∎

### 2.4.

The arguments in the previous proof can be used to deduce $M^{k+1}=0$ on
$H^{2k}(X,\mathbb{C})$ assuming the following holds for all $p+q\leq 2n-2$:

(2.7) $\\{\gamma\in H^{p,q}(X)\mid\gamma\wedge\beta=0,\
\gamma\wedge\bar{\sigma}=0\\}=0.$

In fact, it suffices to show triviality for those $\gamma\in
H^{2k}(X,\mathbb{C})$, $2k<2n$, that are furthermore contained in the image of
$M^{k+1}$ or satisfy $\Lambda_{\bar{\sigma}}(\gamma)\wedge\beta=0$.

######  Corollary 2.5.

Assuming _( 2.7)_, one has $M^{k+1}=0$ on $H^{2k}(X,\mathbb{C})$ for all
$2k\leq 2n$.

###### Proof.

It suffices to prove that $M^{k+1}=0$ on $H^{2k}(X,\mathbb{C})$, $k<n$, is
implied by $M^{k+2}=0$ on $H^{2k+2}(X,\mathbb{C})$, which reduces the
assertion to the middle degree $2n$ covered by Proposition 2.4.

Let $\alpha\in H^{p,q}(X)$ with $p+q=2k<2n$ (and $q<k$) and assume that $\ell$
is maximal with $M^{\ell}(\alpha)\neq 0$. Suppose $\ell\geq k+1$. Since
$\ell+1\geq k+2$, $\alpha\wedge w\in H^{2k+2}(X,\mathbb{C})$, and $M^{2}(w)=0$
for all classes $w\in H^{2}(X,\mathbb{C})$, we have

$0=M^{\ell+1}(\alpha\wedge w)=(l+1)M^{\ell}(\alpha)\wedge M(w).$

As the image of $M$ acting on $H^{2}(X,\mathbb{C})$ is spanned by
$\bar{\sigma}$ and $\beta$, assumption (2.7) yields the contradiction
$M^{\ell}(\alpha)=0$. ∎

######  Remark 2.6.

The condition (2.7) is trivially satisfied for $q<n$, for $L_{\bar{\sigma}}$
is injective on $H^{p,q}(X)$ for $q<n$. Also, if $\beta$ were a Kähler class
or simply a class with $q(\beta)\neq 0$, then $L_{\beta}$ would be injective
for $2k<2n$. At this point, we do not know how realistic (2.7) is. Note that,
unlike condition (5.2) in [6, Thm. 5.2], which is shown to be equivalent to
Nagai’s conjecture, condition (2.7) may fail without contradicting Nagai’s
conjecture.

######  Remark 2.7.

(i) Note that the argument in the above proof reducing $M^{k+1}=0$ on
$H^{2k}(X,\mathbb{C})$ to $M^{k+2}=0$ on $H^{2k+2}(X,\mathbb{C})$ works for
$2k<n$, for in this case $L_{\bar{\sigma}}$ is injective on
$H^{2k}(X,\mathbb{C})$. In other words, without assuming (2.7), to confirm
Nagai’s conjecture (for the operator $M$) it suffices to show $M^{k+1}=0$ on
$H^{2k}(X,\mathbb{C})$ for $n\leq 2k<2n$.

(ii) Also, one can combine the arguments in the proofs of Proposition 2.4 and
Corollary 2.5 to show that $M^{n}=0$ on $H^{2k}(X,\mathbb{C})$ for $2k<2n$,
because in this case one only needs (2.7) for $(0,2k)$-classes which holds
true. Indeed, $M^{n}(\alpha)=0$ for $\alpha\in H^{p,q}(X)$ with $p<n$. For
$p\geq n$, write again $\alpha=\sigma^{p-n}\wedge\gamma$ and compute
$M^{n}(\alpha)=\sum_{i+j=n}{n\choose i}\,M^{i}(\sigma^{p-n})\wedge
M^{j}(\gamma)={n\choose p-n}\,M^{p-n}(\sigma^{p-n})\wedge
M^{2n-p}(\gamma)=c\cdot\beta^{p-n}\wedge M^{2n-p}(\gamma)$ for some constant
$c$ with $M^{2n-p}(\gamma)\in H^{0,2(k-p+n)}(X)$ and so
$M^{2n-p}(\gamma)=c^{\prime}\cdot\bar{\sigma}^{k-p+n}$.

Hence, for $w\in H^{2}(X,\mathbb{C})$ with $M(w)=\bar{\sigma}$ we obtain
$0=M^{n+1}(w\wedge\alpha)=c\cdot
c^{\prime}\cdot(n+1)\cdot\beta^{p-n}\wedge\bar{\sigma}^{k-p+n+1}$. Since
$\bar{\sigma}^{2n-p}$ induces an isomorphism $H^{p-n,p-n}(X)\simeq
H^{p-n,3n-p}(X)$ and since $k-p+n+1\leq 2n-p$, multiplying classes in
$H^{p-n,p-n}(X)$ with $\bar{\sigma}^{k-p+n+1}$ is injective and, in
particular, $\beta^{p-n}\wedge\bar{\sigma}^{k-p+n+1}\neq 0$. This proves
$c\cdot c^{\prime}=0$ and, therefore, $M^{n}(\alpha)=0$.

### 2.5.

Let $X$ be a compact hyperkähler manifold and $H^{*}(X,\mathbb{C})$ its
complex cohomology algebra. In addition, $H^{2}(X,\mathbb{C})$ is endowed with
the complex linear extension of the Beauville–Bogomolov pairing $q$.

Assume $V\subset H^{2}(X,\mathbb{C})$ is an isotropic plane with a basis
$v_{1},v_{2}\in V$. Since $v^{n+1}=0$ in $H^{2n+2}(X,\mathbb{C})$ for any
isotropic class $v$, the choice of $V=\langle v_{1},v_{2}\rangle$ endows
$H^{\ast}(X,\mathbb{C})$ with the structure of a graded algebra over the Artin
algebra $\mathbb{C}[x_{1},x_{2}]/(x_{1},x_{2})^{n+1}$.

The next proposition is a two-parameter version of the observation [9, Prop.
2.2].

######  Proposition 2.8.

Any element in the special orthogonal group ${\rm SO}(H^{2}(X,\mathbb{C}))$
lifts to an automorphism of the cohomology ring $H^{\ast}(X,\mathbb{C})$
provided that $b_{2}\geq 4$.

###### Proof.

We use the shorthand $H^{\ast}\coloneqq H^{\ast}(X,\mathbb{C})$ and denote by
${\rm Aut}(H^{\ast})$ the complex algebraic group of automorphisms of the
graded $\mathbb{C}$-algebra $H^{\ast}$. Then, we have to show that the image
$G$ of the restriction map ${\rm Aut}(H^{\ast})\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces{\rm
GL}(H^{2})$ contains ${\rm SO}(H^{2},q)$. For this we use that monodromy
defines a discrete subgroup in ${\rm Aut}(H^{\ast})$ whose image in ${\rm
GL}(H^{2})$ contains a finite index subgroup of the integral special
orthogonal group ${\rm SO}(H^{2}(X,\mathbb{Z}))$. Since by [2] the latter is
Zariski dense in ${\rm SO}(H^{2},q)$ when $q$ is indefinite, which holds by
our assumption $b_{2}\geq 4$, we indeed have ${\rm SO}(H^{2},q)\subset G$. ∎

######  Corollary 2.9.

Assume $b_{2}\geq 5$, and let $V=\langle v_{1},v_{2}\rangle$ and
$V^{\prime}=\langle v_{1}^{\prime},v_{2}^{\prime}\rangle$ be two isotropic
planes in $H^{2}(X,\mathbb{C})$. Then the induced graded
$\mathbb{C}[x_{1},x_{2}]/(x_{1},x_{2})^{n+1}$-algebra structures on
$H^{\ast}(X,\mathbb{C})$ are isomorphic.

###### Proof.

If $b_{2}\geq 5$, any two isotropic planes $V=\langle v_{1},v_{2}\rangle$ and
$V^{\prime}=\langle v_{1}^{\prime},v_{2}^{\prime}\rangle$ are contained in the
same orbit of the action of the complex special orthogonal group ${\rm
SO}(H^{2},q)$, i.e. there exists $g\in{\rm SO}(H^{2}(X,\mathbb{C}),q)$ with
$g(V)=V^{\prime}$. Since $g$ lifts to an automorphism of the graded
$\mathbb{C}$-algebra $H^{\ast}(X,\mathbb{C})$, the induced
$\mathbb{C}[x_{1},x_{2}]/(x_{1},x_{2})^{n+1}$-algebra structures are
isomorphic. ∎

######  Corollary 2.10.

Let ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ be a
type II degeneration of compact hyperkähler manifolds with $b_{2}\geq 5$, and
let $\beta\in H^{1,1}(X)$ be a non-trivial isotropic class on a smooth fibre
$X\coloneqq{\cal X}_{t}$, $t\neq 0$. Then the monodromy operator $N$ and the
operator $M=[L_{\beta},\Lambda^{c}_{\sigma}]$, for $0\neq\sigma\in
H^{2,0}(X)$, are conjugate under a graded automorphism of the algebra
$H^{*}(X,\mathbb{C})$.

###### Proof.

According to (2.4) and (2.6) we can write
$N_{2}=q(w_{2},~{})\,w_{1}-q(w_{1},~{})\,w_{2}$ for a certain basis
$w_{1},w_{2}$ of the isotropic plane ${\rm Im}(N_{2})\subset
H^{2}(X,\mathbb{C})$ and, up to a scaling factor,
$M_{2}=q(\beta,~{})\,\bar{\sigma}-q(\bar{\sigma},~{})\,\beta$. Now choose a
graded algebra automorphism of $H^{*}(X,\mathbb{C})$ that maps the basis
$w_{1},w_{2}$ of the image of $N_{2}$ to the basis $\beta,\bar{\sigma}$ of the
image of $M_{2}$. Let $N^{\prime}$ denote the conjugate of the logarithmic
monodromy operator $N$ under this algebra isomorphism.

Then $N^{\prime}_{2}=M_{2}$ up to a scaling factor and both operators
$N^{\prime}$ and $M$ are contained in the total Lie algebra
${\mathfrak{g}}(X)$. For $N^{\prime}$ this follows from the corresponding
assertion for $N$, which is a consequence of [18, Prop. 3.5], see also [6,
Sec. 4], and for $M=[L_{\beta},\Lambda^{c}_{\sigma}]$ it holds by definition.
Also, $N^{\prime}$ and $M$ are both of degree zero and hence contained in the
degree zero part of ${\mathfrak{g}}(X)$. Now use that the degree zero part of
${\mathfrak{g}}(X)$ is isomorphic to
$\mathfrak{so}(H^{2}(X,\mathbb{C}))\oplus\mathbb{C}h$, where $h$ is the degree
operator. Hence, any two degree zero elements of ${\mathfrak{g}}(X)$ with
identical action on $H^{2}(X,\mathbb{C})$ coincide. Thus, the fact that
$N^{\prime}_{2}=M_{2}$ immediately implies $N^{\prime}=M$. ∎

######  Remark 2.11 ($b_{2}=4$).

Note that the existence of an isotropic plane implies $b_{2}\geq 4$. If
$b_{2}=4$, weaker versions of Corollaries 2.9 and 2.10 hold. In this case, the
set of isotropic planes is the Fano variety of lines on a quadric in
$\mathbb{P}^{3}$, which is the disjoint union of two lines. In particular,
${\rm SO}(H^{2},q)$ does not act transitively on it. However, the special
orthogonal group exchanges any two lines in the same component or,
equivalently, whenever their corresponding isotropic planes are transverse.
Therefore, Corollary 2.9 holds for $b_{2}=4$ too provided that $V$ and
$V^{\prime}$ are transverse.

Furthermore, if $b_{2}=4$, the proof of Corollary 2.10 does not work, because
the isotropic planes $\mathrm{Im}(N_{2})$ and
$\langle\beta,\bar{\sigma}\rangle$ may not be transverse. Nonetheless, we can
always replace $\beta$ by a non-trivial isotropic class in $H^{1,1}(X)$ with
this property. For instance, if $\beta$ does not work, exchange it with the
class $\eta$ to be constructed in the proof of Theorem 3.2. We conclude that
even for $b_{2}=4$, the monodromy operator $N$ in Corollary 2.10 is conjugate
to $M=[L_{\beta},\Lambda^{c}_{\sigma}]$ for some non-trivial isotropic class
$\beta\in H^{1,1}(X)$. In particular, we do not need any restriction on Betti
number $b_{2}$ for Theorems 1.2 and 3.2.

###  2.6.  Proof of Theorem 1.2

According to Corollary 2.10 (and Remark 2.11), the monodromy operator $N$ and
the operator $M$ correspond to each other under a graded algebra automorphism
of $H^{\ast}(X,\mathbb{C})$. Consider their actions on $H^{2k}(X,\mathbb{C})$
for $k\leq n$. Since $M^{n+1}=0$ on $H^{2k}(X,\mathbb{C})$ by Proposition 2.4,
we can conclude that $N^{n+1}=0$ holds as well. The stronger statement
$N^{n}=0$ on $H^{2k}(X,\mathbb{C})$ for $2k<2n$ follows from Remark 2.7, (ii).

Furthermore, assuming (2.7), we deduce from Corollary 2.5 that $M^{k+1}=0$ on
$H^{2k}(X,\mathbb{C})$, which again by Corollary 2.10 implies the assertion
$N^{k+1}=0$. ∎

##  3\. Nagai’s conjecture and the perverse filtration

Let $X$ be a compact hyperkähler manifold of complex dimension $2n$. In this
section we show that, quite remarkably, Nagai’s conjecture prescribes the
shape of the perverse filtration associated to a Lagrangian fibration $f\colon
X\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces B$.

### 3.1.

We start by recalling the filtration associated to a nilpotent operator, see
[4, §1.6].

Given a nilpotent endomorphism $N$ of a finite dimensional vector space $V$ of
index of nilpotence $k$, i.e. $N^{k}\neq 0$ and $N^{k+1}=0$, the _weight
filtration of $N$ centred at $k$_ is the unique increasing filtration
$W^{N}_{0}V\subset W^{N}_{1}V\subset\cdots\subset W^{N}_{2k-1}V\subset
W^{N}_{2k}V=V$, such that

$NW^{N}_{i}\subseteq W^{N}_{i-2}\text{ and }N^{i}\colon{\rm
Gr}^{N}_{k+i}V\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
3.19443pt\raise 3.8674pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
2.58334pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
2.58334pt}}}}}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces{\rm
Gr}^{N}_{k-i}V.$

Here, we use $N$ to denote also the induced maps on the graded pieces ${\rm
Gr}^{N}_{i}V\coloneqq W^{N}_{i}V/W^{N}_{i-1}V$.

######  Examples 3.1.

Let us consider some geometric examples.

(i) For $0\neq\bar{\sigma}\in H^{0,2}(X)$, the weight filtration of the
nilpotent operator $L_{\bar{\sigma}}$ on $H^{*}(X,\mathbb{C})$ centred at $n$
is the conjugate Hodge filtration, i.e.
$W^{L_{\bar{\sigma}}}_{i}H^{*}(X,\mathbb{C})=\sum_{q\geq 2n-i}H^{p,q}(X)$, see
[5, Prop. 2.6].

(ii) Assume $f\colon X\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces B$
is a Lagrangian fibration. Up to a scalar factor, there exists a unique class
$\beta\in H^{2}(X,\mathbb{Q})$ which is the pullback of an ample class in
$H^{2}(B,\mathbb{Q})$, see [13]. The weight filtration of $L_{\beta}$ on
$H^{*}(X,\mathbb{Q})$ centred at $n$ is the perverse filtration

$P_{i}H^{d}(X,\mathbb{Q})\coloneqq\sum_{j\geq 0}\beta^{j}\cdot{\rm
Ker}\left(\beta^{n-(d-2j)+i+1}\colon
H^{d-2j}(X,\mathbb{Q})\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces
H^{2n-d+2j+2i+2}(X,\mathbb{Q})\right),$

i.e. $W^{L_{\beta}}_{i}H^{*}(X,\mathbb{Q})\cap
H^{d}(X,\mathbb{Q})=P_{d+i-2n}H^{d}(X,\mathbb{Q})$, see [3, Thm. 2.1.5]. In
particular, the graded pieces ${\rm Gr}^{P}_{i}H^{d}(X,\mathbb{Q})$ are pure
Hodge structures of weight $d$.

(iii) Let ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ be a
type II degeneration of compact hyperkähler manifolds with unipotent monodromy
and $X={\cal X}_{t}$, $t\neq 0$, a fixed smooth fibre. Up to a shift, the
weight filtration of the logarithmic monodromy operator $N$ on
$H^{*}(X,\mathbb{C})$ centred at $n$ coincides with the monodromy filtration
$W_{\rm mon}$ defined in [15, Sec. 11.2.5]. On the other hand, by Corollary
2.10, the weight filtration of $N$ is conjugate to the weight filtration of
$M=[L_{\beta},\Lambda^{c}_{\sigma}]$. Together this becomes

${\rm Gr}^{W_{\rm mon}}_{k+j}H^{k}(X,\mathbb{C})\simeq{\rm
Gr}^{N}_{n+j}H^{k}(X,\mathbb{C})\simeq{\rm Gr}^{M}_{n+j}H^{k}(X,\mathbb{C}).$

Observe that the dimension of ${\rm Gr}^{N}_{n+j}H^{k}(X,\mathbb{C})$ is
independent of the choice of ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ by
Corollary 2.10, see [10, 18] for the analogous statements in type I and III.

### 3.2.

The goal of this section is to provide an equivalent formulation of Nagai’s
conjecture, see [6, Prop. 1.15] for another one using the representation
theory of the LLV algebra. We begin by establishing a link between the
monodromy filtration and the perverse filtration.

As before, $X$ denotes a compact hyperkähler manifold of complex dimension
$2n$.

######  Theorem 3.2.

Let $N$ be the logarithmic monodromy operator on $H^{*}(X,\mathbb{C})$ of a
type II degenerations and assume that there exists a Lagrangian fibration
$f\colon X\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces B$.
For the two induced filtrations there exists an isomorphism

(3.1) ${\rm Gr}^{N}_{n+j}H^{\ell}(X,\mathbb{C})\simeq\bigoplus_{p+q=\ell}{\rm
Gr}^{P}_{j+q}H^{p,q}(X).$

###### Proof.

Let $(g,I,J,K)$ be a hyperkähler structure on the complex manifold $X$, i.e. a
Riemannian metric $g$ which is Kähler with respect to three complex structures
$I$ (the one defining $X$), $J$, and $K$ satisfying the quaternion relations
$I^{2}=J^{2}=K^{2}=IJK=-1$. 0 Assume
$q(\omega_{I})=q(\omega_{J})=q(\omega_{K})=2$ and let $\beta=f^{*}\alpha$ be
the pullback of an ample class $\alpha\in H^{2}(B,\mathbb{Q})$ normalized such
that $q(\omega_{I},\beta)=1$. The class $\eta\coloneqq\omega_{I}-\beta$ is
isotropic, i.e. $q(\eta)=0$. Note also that
$q(\beta,\eta)=q(\beta,\omega_{I})=q(\bar{\sigma},\sigma)=1$. We shall denote
by $U\subset H^{2}(X,\mathbb{C})$ the orthogonal complement of
$\langle\beta,\omega_{I},\omega_{J},\omega_{K}\rangle$.

By Proposition 2.8, Corollary 2.9 and Remark 2.11, there exists an
automorphism $\varphi\in\operatorname{Aut}(H^{*}(X,\mathbb{C}))$ of the graded
algebra $H^{*}(X,\mathbb{C})$ which is an involution on $H^{2}(X,\mathbb{C})$
with $\varphi(\sigma)=\beta$, $\varphi(\eta)=\bar{\sigma}$, and
$\varphi|_{U}={\rm id}$. Let $y\coloneqq\varphi(\omega_{J})$,
$z\coloneqq\varphi(\omega_{K})\in H^{2}(X,\mathbb{C})$. Then the following are
all $\mathfrak{sl}_{2}$-triples:

$E_{\sigma}\coloneqq
L_{\sigma}=\frac{1}{2}L_{\omega_{J}}+\frac{\sqrt{-1}}{2}L_{\omega_{K}},~{}F_{\sigma}\coloneqq\Lambda_{\sigma}^{c}=\Lambda_{\bar{\sigma}}=\frac{1}{2}\Lambda_{\omega_{J}}-\frac{\sqrt{-1}}{2}\Lambda_{\omega_{K}},~{}H_{\sigma}=(p-n)\,{\rm
id}\text{ on }H^{p,q}(X),$ $E_{\bar{\sigma}}\coloneqq
L_{\bar{\sigma}}=\frac{1}{2}L_{\omega_{J}}-\frac{\sqrt{-1}}{2}L_{\omega_{K}},~{}F_{\bar{\sigma}}\coloneqq\Lambda^{c}_{\bar{\sigma}}=\Lambda_{\sigma}=\frac{1}{2}\Lambda_{\omega_{J}}+\frac{\sqrt{-1}}{2}\Lambda_{\omega_{K}},~{}H_{\bar{\sigma}}=(q-n)\,{\rm
id}\text{ on }H^{p,q}(X),$ $E_{\beta}\coloneqq\varphi
E_{{\sigma}}\varphi^{-1}=L_{\beta},~{}F_{\beta}\coloneqq\varphi
F_{{\sigma}}\varphi^{-1}=\Lambda_{\eta}=\frac{1}{2}\Lambda_{y}-\frac{\sqrt{-1}}{2}\Lambda_{z},~{}H_{\beta}=[E_{\beta},F_{\beta}]=\varphi
H_{\sigma}\varphi^{-1},$

and

$E_{\eta}\coloneqq\varphi
E_{\bar{\sigma}}\varphi^{-1}=L_{\eta},~{}F_{\eta}\coloneqq\varphi
F_{\bar{\sigma}}\varphi^{-1}=\Lambda_{\beta},~{}H_{\eta}=\varphi
H_{\bar{\sigma}}\varphi^{-1}.$

Since $H_{\beta}$ is a morphism of Hodge structures, we obtain the
decomposition

(3.2) $H^{p,q}(X)=\bigoplus_{i}V^{p,q,i}$

such that $H_{\beta}=(p+q-i-n)\,{\rm id}$ on $V^{p,q,i}$. In particular, the
representation theory of $\mathfrak{sl}_{2}$-triples gives
$V^{p,q,i}\simeq{\rm Gr}^{P}_{i}H^{p,q}(X)$. Next one observes that

(3.3) $E_{M}\coloneqq
2M=2[E_{\beta},F_{{\sigma}}]=2[L_{\beta},\Lambda_{\sigma}^{c}],~{}F_{M}\coloneqq
2[F_{\bar{\sigma}},E_{\eta}]=2[\Lambda_{\bar{\sigma}}^{c},L_{\eta}],\\\
~{}H_{M}\coloneqq[E_{M},F_{M}]=H_{\beta}-H_{\sigma}$

is an $\mathfrak{sl}_{2}$-triple. Indeed, by [11, Lem. 3.9] or [6, Thm. 2.7],
see also Section 2.2,

$E_{M}|_{H^{2}(X,\mathbb{C})}=q(\beta,\,)\,\bar{\sigma}-q(\bar{\sigma},\,)\,\beta\text{
and }F_{M}|_{H^{2}(X,\mathbb{C})}=q(\sigma,\,)\,\eta-q(\eta,\,)\,\sigma.$

From here it is straightforward to check that $(E_{M},F_{M},H_{M})$ is an
$\mathfrak{sl}_{2}$-triple in the Lie algebra
$\mathfrak{so}(H^{2}(X,\mathbb{C}))$ and, hence, also in the LLV algebra
${\mathfrak{g}}(X)\simeq\mathfrak{so}(\tilde{H}(X,\mathbb{C}))$.

Finally by (3.3), the eigenspaces of $H_{M}$ are
$\bigoplus_{p+q=\ell}V^{p,q,j+q}\simeq{\rm
Gr}^{M}_{n+j}H^{\ell}(X,\mathbb{C})$, and hence

${\rm Gr}^{N}_{n+j}H^{\ell}(X,\mathbb{C})\simeq{\rm
Gr}^{M}_{n+j}H^{\ell}(X,\mathbb{C})\simeq\bigoplus_{p+q=\ell}{\rm
Gr}^{P}_{j+q}H^{p,q}(X),$

which concludes the proof of the theorem. ∎

######  Remark 3.3.

Despite being evocative from a geometric viewpoint, it is not necessary to
assume the existence of a Lagrangian fibration in Theorem 3.2. Indeed, in the
above proofs we can replace $\beta$ with any non-zero isotropic class of type
$(1,1)$ by [9, Prop. 2.2].

The theorem immediately yields an equivalent reformulation of Nagai’s
conjecture in terms of the perverse filtration.

######  Corollary 3.4.

The Nagai conjecture (for type II degenerations) holds if and only if the
Hodge structure ${\rm Gr}^{P}_{i}H^{2k}(X,\mathbb{Q})$ has level at most
$2k-2|i-k|$.

###### Proof.

The Nagai conjecture for $N$ says that ${\rm nilp}(N_{2k})=k$, i.e. ${\rm
Gr}^{N}_{n+j}H^{2k}(X,\mathbb{C})=0$ for $|j|>k$, see Section 3.1. By Theorem
3.2 this condition is equivalent to ${\rm Gr}^{P}_{i}H^{p,q}(X)=0$ for
$|i-q|>k$, i.e. ${\rm Gr}^{P}_{i}H^{2k}(X,\mathbb{Q})$ has level at most
$2k-2|i-k|$. ∎

The $(q,i)$-entry in the following picture is the direct summand $V^{2-q,q,i}$
of $H^{2}(X,\mathbb{C})$.

${2}$${\eta}$${1}$${\sigma}$${U}$${\bar{\sigma}}$${0}$${\beta}$
${0}$${1}$${2}$$q$$i$${\rm Gr}^{M}_{n+1}$${\rm Gr}^{M}_{n}$${\rm
Gr}^{M}_{n-1}$

The sums along the columns give the Hodge decomposition of
$H^{2}(X,\mathbb{Q})$; the sums along the rows split the perverse filtration
on $H^{2}(X,\mathbb{Q})$; the sums along the northeast-southwest diagonals
split the weight filtration of $M$. Complex conjugation and the conjugate via
$\varphi$ of complex conjugation account for the symmetries of this diamond,
respectively the reflection about the vertical and horizontal middle axes. The
Nagai conjecture predicts the existence of these diamonds in any cohomological
degree.

The arguments above also yield the following observation concerning the
cohomology of odd degree of a compact hyperkähler manifold, see also [5, Lem.
1.2] or [21, Cor. 8.1] where the hypothesis $b_{2}\geq 4$ is not needed.

######  Corollary 3.5.

If $b_{2}\geq 4$, then all odd Betti numbers $b_{2k-1}$ are divisible by $4$.

###### Proof.

If $b_{2}\geq 4$, there exists a non-zero isotropic class $\beta$ of type
$(1,1)$. Consider then the involution $\varphi$ in the proof of Theorem 3.2
and the decomposition (3.2), both well-defined by Remark 3.3. The map
$\varphi$ exchanges the eigenvectors of $H_{\bar{\sigma}}$ and $H_{\beta}$,
i.e. $\varphi(V^{p,q,i})=V^{i,p+q-i,p}$. By complex conjugation, we obtain

(3.4) $V^{q,p,i}\simeq V^{p,q,i}\simeq V^{i,p+q-i,p}\simeq V^{p+q-i,i,p}.$

We conclude that

$H^{2k-1}(X,\mathbb{C})=\bigoplus_{p+q=2k-1}V^{p,q,i}\simeq\bigoplus_{p+q=2k-1,p<k,i<k}(V^{p,q,i})^{\oplus
4},$

which proves the assertion. ∎

##  4\. Odd degree

We shall now discuss the action of the logarithmic monodromy on the odd degree
cohomology. Again, ${\cal X}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\Delta$ is a
type II degeneration of compact hyperkähler manifolds of dimension $2n$.

######  Theorem 4.1.

The odd logarithmic monodromy operators $N_{2k-1}$, $k\leq n$, satisfy

${\rm nilp}(N_{2k-1})\leq\min\\{2k-3,n-1\\}.$

Furthermore, assuming condition _( 2.7)_, in fact ${\rm nilp}(N_{2k-1})\leq
k-1$ for all $k\leq n$.

For any degeneration types one has ${\rm nilp}(N_{2k-1})\leq 2k-3$, as the
level of the Hodge structure $H^{2k-1}(X,\mathbb{C})$ is at most $2k-3$. For
type III degenerations, i.e. ${\rm nilp}(N_{2})=2$, Soldatenkov proves [18,
Prop. 3.15] that the upper bound $2k-3$ is attained if
$H^{3}(X,\mathbb{C})\neq 0$. In fact, one can show that for type III
degenerations ${\rm nilp}(N_{d})$ is always the level of the Hodge structure
$H^{d}(X,\mathbb{C})$, see [9, Prop. 3.12]. For type I degenerations all
logarithmic monodromy operators are trivial, cf. [10].

###  4.1. Proof of Theorem 4.1

As we argue in Section 2.6, it suffices to show the analogous statements for
$M_{2k-1}$. Since $M$ is of bidegree $(-1,1)$ and the Hodge structure of
$H^{2k-1}(X,\mathbb{Q})$ has level at most $2k-3$, we have ${\rm
nilp}(M_{2k-1})\leq 2k-3$.

To show that ${\rm nilp}(M_{2k-1})\leq n-1$, we imitate the arguments in the
proof of Proposition 2.4 to show $M^{n}=0$ on $H^{2k-1}(X,\mathbb{C})$. Assume
$\alpha\in H^{p,q}(X)$ with $p+q=2k-1$. If $p<n$, then clearly
$M^{n}(\alpha)=0$ and for $p=n$, we have $M^{n}(\alpha)\in H^{0,2k-1}(X)=0$.
For $p>n$, we write again $\alpha=\sigma^{p-n}\wedge\gamma$ for some
$\gamma\in H^{2n-p,q}(X)$. Then use $M^{i}(\sigma^{p-n})=0$ for $i>p-n$ and
$M^{j}(\gamma)=0$ for $j\geq 2n-p$ to conclude $M^{n}(\alpha)=0$.∎

Note that the induction in the proof of Corollary 2.5 under the assumption
(2.7) will work here as well. So, assuming (2.7), one can actually show ${\rm
nilp}(N_{2k-1})\leq k-1$ for all $k\leq n$.

### 4.2.

Let us now turn to lower bounds, which can not be obtained directly from ${\rm
nilp}(N_{2})$.

######  Corollary 4.2.

Assume that the level of the Hodge structure $H^{2k-1}(X,\mathbb{Q})$, $k\leq
n$, is $2\ell-1>0$. Then

$\ell\leq{\rm nilp}(N_{2k-1}).$

If $H^{3}(X,\mathbb{Q})\neq 0$, then $\ell=k-1$ and, therefore,

$k-1\leq{\rm nilp}(N_{2k-1}).$

In particular, ${\rm nilp}(N_{3})=1$ and ${\rm nilp}(N_{2n-1})=n-1$.

###### Proof.

We prove first that $\ell\leq{\rm nilp}(M_{2k-1})$. By assumption,
$H^{k-\ell,k+\ell-1}(X)\neq 0$, and so there exists an integer $i$ such that
$V^{k-\ell,k+\ell-1,i}\neq 0$. Then, by (3.4) and (3.1), ${\rm
Gr}^{M}_{n+j}H^{2k-1}(X,\mathbb{C})\neq 0$ if $j=|k-\ell-i|$ and
$|k+\ell-1-i|$. Therefore, we obtain

$\ell=\min_{i}\max\\{|k-\ell-i|,|k+\ell-1-i|\\}\leq{\rm nilp}(M_{2k-1}).$

If $H^{3}(X,\mathbb{Q})\neq 0$, the level of the Hodge structure
$H^{2k-1}(X,\mathbb{Q})$, for $k\leq n$, is exactly $2(k-1)-1$. Indeed, since
$\bar{\sigma}^{n-2}$ induces an isomorphism $H^{1,2}(X)\simeq H^{1,2n-2}(X)$,
the map $\bar{\sigma}^{k-2}\colon H^{1,2}(X)\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces
H^{1,2k-2}(X)$ is injective, and so $H^{1,2k-2}(X)\neq 0$.∎

## Acknowledgements

We wish to thank Oliver Debarre for the reference [2], Andrey Soldatenkov for
comments on a first version of this note, and an anonymous referee for helpful
remarks.

## References

  * [1] F. Bogomolov, _On the cohomology ring of a simple hyper-Kähler manifold (on the results of Verbitsky)_ , Geom. Funct. Anal. 6 (1996), no. 4, 612–618.
  * [2] A. Borel, _Density and maximality of arithmetic subgroups_ , J. Reine Angew. Math. 224 (1966) 78–89.
  * [3] M. de Cataldo and L. Migliorini, _The Hodge theory of algebraic maps_ , Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 693–750.
  * [4] P. Deligne, _La conjecture de Weil. II_ , Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252.
  * [5] A. Fujiki, _On the de Rham cohomology group of a compact Kähler symplectic manifold_ , in Algebraic geometry, Sendai, 1985, Vol. 10 of _Adv. Stud. Pure Math._ , 105–165, North-Holland, Amsterdam (1987).
  * [6] M. Green, Y.-J. Kim, R. Laza, and C. Robles, _The LLV decomposition of hyper-Kähler cohomology_ (2019).
  * [7] P. Griffiths, Topics in algebraic and analytic geometry, Princeton University Press, Princeton, NJ (2015).
  * [8] A. Harder, _Mixed Hodge structures in log symplectic geometry_ (2020).
  * [9] D. Huybrechts and M. Mauri, _Lagrangian fibrations_ (2021).
  * [10] J. Kollár, R. Laza, G. Saccà, and C. Voisin, _Remarks on degenerations of hyper-Kähler manifolds_ , Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2837–2882.
  * [11] N. Kurnosov, A. Soldatenkov, and M. Verbitsky, _Kuga-Satake construction and cohomology of hyperkähler manifolds_ , Adv. Math. 351 (2019) 275–295.
  * [12] E. Looijenga and V. A. Lunts, _A Lie algebra attached to a projective variety_ , Invent. Math. 129 (1997), no. 2, 361–412.
  * [13] D. Matsushita, _On fibre space structures of a projective irreducible symplectic manifold_ , Topology 38 (1999), no. 1, 79–83.
  * [14] Y. Nagai, _On monodromies of a degeneration of irreducible symplectic Kähler manifolds_ , Math. Z. 258 (2008), no. 2, 407–426.
  * [15] C. Peters and J. Steenbrink, Mixed Hodge structures, Vol. 52 of _Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics_ , Springer-Verlag, Berlin (2008).
  * [16] S. Schreieder and A. Soldatenkov, _The Kuga-Satake construction under degeneration_ , J. Inst. Math. Jussieu 19 (2020), no. 6, 2165–2182.
  * [17] J. Shen and Q. Yin, _Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds_ (2018).
  * [18] A. Soldatenkov, _Limit mixed Hodge structures of hyperkähler manifolds_ , Mosc. Math. J. 20 (2020), no. 2, 423–422.
  * [19] ———, _On the Hodge structure of compact hyperkähler manifolds_ , Math. Res. Lett. 28 (2021), no. 2, 623–635.
  * [20] M. Verbitsky, _Cohomology of compact hyperkähler manifolds and its applications_ , Geometric & Functional Analysis GAFA 6 (1996), no. 4, 601–611.
  * [21] H. Wakakuwa, _On Riemannian manifolds with homogeneous holonomy group ${\rm Sp}(n)$_, Tohoku Math. J. (2) 10 (1958) 274–303.


#  ManyQuadrupeds: Learning a Single Locomotion Policy for Diverse Quadruped
Robots

Milad Shafiee1, Guillaume Bellegarda1, and Auke Ijspeert1 1 This research is
supported by the Swiss National Science Foundation (SNSF) as part of project
No.197237. The authors are with the BioRobotics Laboratory, Ecole
Polytechnique Federale de Lausanne (EPFL). (e-mail:
<EMAIL_ADDRESS>
###### Abstract

Learning a locomotion policy for quadruped robots has traditionally been
constrained to a specific robot morphology, mass, and size. The learning
process must usually be repeated for every new robot, where hyperparameters
and reward function weights must be re-tuned to maximize performance for each
new system. Alternatively, attempting to train a single policy to accommodate
different robot sizes, while maintaining the same degrees of freedom (DoF) and
morphology, requires either complex learning frameworks, or mass, inertia, and
dimension randomization, which leads to prolonged training periods. In our
study, we show that drawing inspiration from animal motor control allows us to
effectively train a single locomotion policy capable of controlling a diverse
range of quadruped robots. The robot differences encompass: a variable number
of DoFs, (i.e. $12$ or $16$ joints), three distinct morphologies, a broad mass
range spanning from $2$ $\mathrm{k}\mathrm{g}$ to $200$
$\mathrm{k}\mathrm{g}$, and nominal standing heights ranging from $16$
$\mathrm{c}\mathrm{m}$ to $100$ $\mathrm{c}\mathrm{m}$. Our policy modulates a
representation of the Central Pattern Generator (CPG) in the spinal cord,
effectively coordinating both frequencies and amplitudes of the CPG to produce
rhythmic output (Rhythm Generation), which is then mapped to a Pattern
Formation (PF) layer. Across different robots, the only varying component is
the PF layer, which adjusts the scaling parameters for the stride height and
length. Subsequently, we evaluate the sim-to-real transfer by testing the
single policy on both the Unitree Go1 and A1 robots. Remarkably, we observe
robust performance, even when adding a 15 $\mathrm{k}\mathrm{g}$ load,
equivalent to $125\%$ of the A1 robot’s nominal mass.

##  I Introduction and Related Work

The oldest group of vertebrates which have appendages (i.e. fins or legs) are
the elasmobranchs (sharks and rays). These vertebrates have followed a
separate evolutionary path from mammals for over 420 million years. However,
neural circuits controlling elasmobranch fins and mammalian limbs have
exhibited remarkable similarities at the molecular, cellular, and behavioral
levels [1]. This suggests that the neural substrate responsible for limb
control was already conserved over 420 million years ago, and that the motor
control scheme of tetrapods is preserved across various vertebrate species,
each with their own unique size, inertia, and morphology [2]. In contrast, in
robotics, it is common practice to design and train a new control policy for
each new specific robot. In this paper, we demonstrate how employing a
biology-inspired motor-control scheme can streamline the training process,
enabling the development of a single control policy applicable to quadruped
robots with diverse sizes, inertias, morphologies, and degrees of freedom
(DoF).

Figure 1:  Simulation and experiment snapshots of training and deploying a
single policy to control 16 different robots. Videos:
https://miladshafiee.github.io/ManyQuadrupeds/

###  I-A Central Pattern Generators

Quadruped animal motor control can be described as an intricate interplay
between the Central Pattern Generator (CPG), sensory feedback, and supraspinal
drive signals from the brain. In robotics, abstract models of CPGs are
commonly used for locomotion pattern generation [3, 4, 5, 6], as well as to
investigate biological hypotheses [7, 8]. Besides the intrinsic oscillatory
behavior of CPGs, several other properties such as robustness and
implementation simplicity make CPGs desirable for locomotion control [9]. For
legged robots, the CPG is usually designed for feedforward rhythm generation,
and dynamic balancing is achieved with optimization [3], or hand-tuned
feedback [10, 4].

###  I-B Learning Locomotion

Deep Reinforcement Learning (DRL) has emerged as a powerful approach for
training robust legged locomotion control policies in simulation, and
deploying these sim-to-real on hardware [11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24]. Most of these works view the trained artificial neural
network (ANN) as a “brain” which has full authority to directly control the
joint movements. These methods involve training custom policies from the
ground up for each new quadruped’s unique morphology. Recent research aims to
eliminate the need for policy retraining by investigating the potential of
graph learning [25, 26, 27, 28]. In these approaches, the agent’s morphology
is typically considered as a graph, with the graph’s structure mirroring the
agent’s physical body. Most of these methods utilize agent-agnostic
reinforcement learning combined with transfer learning techniques. While these
methods based on graph learning exhibit promise in simulating toy examples,
they have yet to be validated in real-world robotic systems [25, 26, 27].
Additionally, the gaits generated in these simulations often lack a natural
sense of locomotion, which could make the transition from simulation to the
real world challenging. Furthermore, such approaches demand significant
computational resources and can require days of training on a standard GPU
[27].

Feng et al. [29] introduced a unified policy training approach for learning
locomotion through animal motion imitation for a diverse set of quadrupeds,
encompassing a range of sizes and masses, yet sharing identical DoFs. This
method demonstrated successful simulation-to-reality transfer capabilities;
however, it requires a two-week training period on a 16-core CPU, and is
limited to robots with the same number of DoFs per leg.

###  I-C Contribution

In this paper, we employ a biology-inspired learning framework to learn a
single locomotion policy able to control a diverse range of quadrupeds,
encompassing significant variations in size, mass, morphology, and DoF. Our
proposed framework seamlessly integrates Central Pattern Generators and Deep
Reinforcement Learning [30, 31, 32], where we use a simple Multi-Layer
Perceptron (MLP) to represent higher control centers. This control center
orchestrates the precise modulation of the CPG within the spinal cord and
adeptly maps the rhythm generation network onto a pattern formation layer. We
list below the advantages of the proposed architecture:

Generality and Versatility: The pattern formation layer, which shapes the feet
trajectories in task-space, allows us to train a single locomotion policy for
different robots with only a few parameter adjustments specific to each robot.
Therefore, the action space does not rely on the number of joints or
morphology, and we do not need to include joint information in the observation
either. This leads to a constant size for both the action and observation
spaces for all four-limb robots, which facilitates training robots with
varying morphologies and DoFs with a simple MLP architecture. Mapping feet
trajectories to joint positions is accomplished through inverse kinematics
(IK). Solving the IK problem for legged robots, which fall under the category
of serial manipulators, is a well-established research area. Numerous
numerical and analytical methods are readily available for efficiently solving
the IK problem for various types of serial limbs, each with different DoFs. By
entrusting the task of handling morphology to IK, in contrast to the current
state-of-the-art in learning quadrupedal locomotion for various robots [29],
our method accommodates diverse morphologies and varying DoF. Furthermore, we
tested the generalization of the framework by excluding three robots with
extreme mass and size properties across three different morphologies from the
training process. During these tests, we observed a stable gait for the
previously unseen robots, even though the policy had not been specifically
trained for them.

Computational Efficiency: Additionally, our approach doesn’t necessitate
extensive dynamics randomization or motion imitation [29], and it relies on a
simple MLP architecture for the Artificial Neural Network. This architectural
simplicity, in addition to training robot policies in parallel on a single GPU
in Isaac Gym [33], results in high computational efficiency, enabling us to
train a single policy for 16 diverse robots in less than two hours, giving
orders of magnitude speed-up over previous works.

Robustness: Furthermore, we achieve stable trotting in sim-to-real quadruped
experiments, even with an additional load of 15.0 $\mathrm{k}\mathrm{g}$,
equivalent to $125\%$ of the A1 robot’s nominal mass. To the best of our
knowledge, this accomplishment represents the pinnacle of robustness against
additional loads ever achieved on the A1 robot. In comparison, learning based
methods managed a load of up to 12 kg [18], and a model-based approach
achieved an 11 kg load capacity [34].

##  II Central Pattern Generators

Figure 2:  Quadrupedal locomotion in animals is governed by interactions
between the spinal CPG, sensory feedback, and supraspinal brain signals. In
our approach, we employ DRL to train a neural network policy that mimics
supraspinal drive behavior, enabling it to modulate CPG dynamics. To simulate
the CPG within the spinal cord, we utilize nonlinear amplitude-controlled
phase oscillators to represent the Rhythm Generator (RG) layer. The RG’s
outputs are then transformed into foot positions and, using inverse
kinematics, converted into motor commands via a Pattern Formation (PF) layer.

The vertebrate locomotor system is structured such that spinal CPGs are
responsible for generating primary rhythmic patterns, while higher-level
centers, such as the motor cortex, cerebellum, and basal ganglia, adjust these
patterns in response to environmental conditions [1]. Rybak et al. [35]
propose that biological CPGs exhibit a dual-level functional organization,
with a half-center rhythm generator (RG) determining locomotion frequency, and
pattern formation (PF) circuits shaping the precise form of muscle activation
signals [35]. This two-tier functional model has also found application in
robotics, particularly for quadruped locomotion research [30, 36].

###  II-A Rhythm Generator (RG) Layer

We utilize non-linear phase oscillators to model the RG layer of the CPG
circuits in the spinal cord. These oscillators have been effectively applied
in the control of quadrupedal locomotion [3, 7, 30] with the following
dynamics:

$\displaystyle\ddot{r}_{i}$
$\displaystyle=\alpha\left(\frac{\alpha}{4}\left(\mu_{i}-r_{i}\right)-\dot{r}_{i}\right)$
(1) $\displaystyle\dot{\theta}_{i}$ $\displaystyle=\omega_{i}\vspace{-1.18em}$
(2)

where $\theta_{i}$ is the phase of the oscillator, $r_{i}$ is the amplitude of
the oscillator, $\omega_{i}$ and $\mu_{i}$ are the intrinsic frequency and
amplitude, $\alpha$ is a positive convergence factor. Notably, we do not
consider explicit phase coupling between different oscillators. Consequently,
the phase relationships between the legs must be learned and managed by the
control policy by effectively modulating the intrinsic frequency for each
limb. The control policy also learns to manipulate stride length by modifying
the intrinsic amplitude.

###  II-B Pattern Formation (PF) Layer

To establish a mapping from the output of the RG layer to joint commands, we
first determine desired foot positions, and then we employ inverse kinematics
to compute the corresponding desired joint positions. We formulate the desired
foot position coordinates as follows:

$\displaystyle x_{i,\text{foot}}$ $\displaystyle=\ \
x_{off,i}-L_{step}(r_{i})\cos(\theta_{i})$ (3) $\displaystyle
z_{i,\text{foot}}$
$\displaystyle=\begin{cases}z_{off,i}-h+L_{clrnc}\sin(\theta_{i})&\text{if
}\sin(\theta_{i})>0\\\
z_{off,i}-h+L_{pntr}\sin(\theta_{i})&\text{otherwise}\end{cases}$ (4)

where $L_{step}$ denotes the nominal step length, $h$ represents the scaling
factor for body height, $L_{clrnc}$ indicates the maximum ground clearance
during the swing, $L_{pntr}$ signifies the maximum ground penetration during
stance phase, and $x_{off}$ serve as set-point altering the equilibrium point
of oscillation in the $x$ direction. $L_{step}$, $L_{clrnc}$, $L_{pntr}$, $h$,
and $x_{off}$ are parameters that vary between different robots and are
selected based on the relative size of each robot, and these values remain
constant throughout the training process.

##  III Learning Framework

In this section, we present our hierarchical bio-inspired learning framework
for training a single policy to control various quadruped robots. We represent
the supraspinal controller as an ANN, which we train with DRL so the agent can
learn to modulate the intrinsic frequencies and amplitudes of each limb
oscillation to produce stable gaits. We formulate the problem as a Markov
Decision Process (MDP), which consists of observations, actions, and rewards.
The proposed action-space modulates feet trajectories, and we do not include
joint information in the observation space. This leads to a constant size for
action and observation space for all four-limb robots, which facilitates
training robots with various morphologies and DoF with a simple MLP
architecture. We detail the MDP components below.

###  III-A Action Space

We incorporate one RG layer for each limb, as defined by Equations (1) and
(2). The RG output is then utilized in a PF layer to generate spatio-temporal
foot trajectories in task space, as described in Equations (3) and (4).
Notably, we do not include explicit neural coupling, with the intuition that
inter-limb coordination will be controlled by the supraspinal drive. As in
[30, 32], our action space modulates the intrinsic amplitudes and frequencies
of the CPG, by continuously updating $\mu_{i}$ and $\omega_{i}$ for each leg.
Therefore, our action space is represented as
$\mathbf{a}=[\bm{\mu},\bm{\omega}]\in\mathbb{R}^{8}$. The agent selects these
parameters at a rate of 100 Hz, varying them at each step based on sensory
inputs. During training, we impose the following limits on each input:
$\mu\in[0.5,4]$ and $\omega\in[0,5]$ Hz. We emphasize that our action space
modulates the feet trajectories in task-space, and these trajectories are
subsequently mapped to joint space through inverse kinematics. Therefore, the
proposed learning architecture is independent of the robot’s morphology and
DoF, as long as we solve the inverse kinematics separately for each robot.

Figure 3: Characteristics and parameters of 16 diverse quadruped Robots. These
robots exhibit variations in mass, ranging from 2 to 200 kg, nominal height
from 18 to 100 cm, and come in three different morphologies, with two types of
DoF, either 12 or 16.

###  III-B Observation Space

Our observation space includes body orientation, body linear and angular
velocity, foot contact booleans, relative feet positions with respect to the
body frame, the preceding action selected by the policy network, and the CPG
states $\\{\bm{r,\dot{r},\theta,\dot{\theta}}\\}$. It is worth noting that,
unlike most other RL approaches, we omit proprioceptive information like joint
positions and velocities. Instead, we employ a forward kinematics approach
based on the current joint positions to determine relative foot positions with
respect to the body frame, and then we incorporate these foot positions into
the observation space. This design choice simplifies the training process and
ensures it remains independent of specific morphologies and DoFs.

TABLE I: PPO Hyperparameters and neural network architecture used with Isaac Gym. Parameter | Value | Parameter | Value
---|---|---|---
Batch size | 98304 | GAE discount factor | 0.95
Mini-bach size | 24576 | KL-divergence $kl^{*}$ | 0.01
Number of epochs | 5 | Learning rate $\alpha$ | adaptive
Clip range | 0.2 | NN Layers | [512, 256, 128]
Entropy coefficient | 0.01 | Activation | elu
Discount factor | 0.99 | Framework | Torch

###  III-C Reward Function

We use the following reward function to promote viability with forward
progress, minimize changes in body orientation, and encourage energy
efficiency:

$\begin{split}r_{1}=\alpha_{1}\cdot{\min}({f}_{x},{d}_{max})+\alpha_{2}\cdot||\bm{o}_{base}-\bm{o}_{zero}||\\\
+\alpha_{3}\cdot|\bm{\tau}\cdot(\dot{\bm{q}}_{t}-\dot{\bm{q}}_{t-1})|\end{split}$

  * •

Viability in forward progress: In the first term, ${f}_{x}$ represents the
robot’s forward progress in the world (along the $x$-direction). To prevent
the exploitation of simulator dynamics and the attainment of unrealistic
speeds, we constrain this term. The constraint is set to ensure that the robot
is rewarded for moving forward with a certain maximum distance during each
control cycle, where ${d}_{max}$ denotes this maximum distance
($\alpha_{1}=8.0$).

  * •

Base orientation penalty: The second term penalizes non-zero body orientation
($\alpha_{2}=-0.25$).

  * •

Power: The third term penalizes power to encourage energy-efficient gaits,
with $\bm{\tau}$ and $\dot{\bm{q}}$ representing joint torques and velocities,
respectively ($\alpha_{3}=-0.00001$).

##  IV Results

In this section, we present results from learning a single unified policy to
control multiple diverse quadrupeds in both simulation and hardware
experiments. Section IV-A details the implementation settings for both
simulations and experiments. We then delve into the outcomes of training the
policy in Section IV-B. Section IV-C focuses on sim-to-real hardware results.
For clear visualizations of the experiments, we encourage the reader to watch
the supplementary video.

TABLE II: Characteristics and parameters of diverse quadruped robots, these parameters are fixed during training. The bold values represent the highest and lowest mass and dimensions, as well as a morphology that is unconventional for legged robots, resembling the morphology of animals more closely.  Robot | Height $h$ [$\mathrm{c}\mathrm{m}$] |  $L_{step}$ [$\mathrm{c}\mathrm{m}$] |  $L_{clrnc}$ [$\mathrm{c}\mathrm{m}$] |  $L_{pntr}$ [$\mathrm{c}\mathrm{m}$] |  $x_{offset}$ [$\mathrm{c}\mathrm{m}$] | DoF - Morphology | Mass [$\mathrm{k}\mathrm{g}$] | $K_{p}$ | $K_{d}$
---|---|---|---|---|---|---|---|---|---
Little Dog | 19.0 | 5.0 | 4.7 | 0.5 | 1.1 | 12 - 2 | 2.9 | 20.0 | 0.3
Spot-Micro | 18.3 | 5.0 | 3.7 | 0.5 | 1.0 | 12 - 1 | 4.8 | 20.0 | 0.3
Solo | 25.0 | 10.0 | 5.0 | 0.5 | 3.7 | 12 - 2 | 2.5 | 20.0 | 0.3
Mini-Cheetah | 30.0 | 13.0 | 7.0 | 1.0 | 0.0 | 12 - 1 | 8.4 | 100.0 | 2.7
A1 | 30.0 | 13.0 | 7.0 | 1.0 | 0.0 | 12 - 1 | 12.0 | 100.0 | 2.7
Go1 | 30.0 | 13.0 | 7.0 | 1.0 | 0.0 | 12 - 1 | 12.0 | 100.0 | 2.7
Aliengo | 42.0 | 16.0 | 7.0 | 1.0 | 0.0 | 12 - 1 | 20.6 | 100.0 | 2.7
Laikago | 40.0 | 16.0 | 7.0 | 1.0 | 0.0 | 12 - 1 | 25.0 | 100.0 | 2.7
Anymal-B | 48.0 | 17.0 | 7.0 | 0.0 | 10.0 | 12 - 2 | 30.0 | 430.0 | 20.7
Anymal-C | 52.0 | 18.0 | 7.0 | 1.0 | 12.0 | 12 - 2 | 52.1 | 430.0 | 20.7
Spot | 57.0 | 20.0 | 9.0 | 1.0 | 0.0 | 12 - 1 | 30.0 | 430.0 | 20.7
B1 | 57.0 | 18.0 | 9.0 | 1.0 | 0.0 | 12 - 1 | 52.7 | 430.0 | 20.7
HYQ | 63.0 | 20.0 | 9.0 | 1.0 | 8.7 | 12 - 2 | 86.7 | 430.0 | 20.7
Dog1 | 30.0 | 13.0 | 7.0 | 1.0 | 0.0 | 16 - 3 | 13.8 | 100.0 | 2.7
Dog2 | 57.0 | 18.0 | 7.0 | 1.0 | 0.0 | 16 - 3 | 56.0 | 200.0 | 10.7
Dog3 | 100.0 | 36.0 | 9.0 | 2.0 | 0.0 | 16 - 3 | 200.0 | 1400.0 | 140.7
Figure 4: Simulation results for training a single policy for 16 different
quadrupeds: Top: Base locomotion velocity. Middle: CPG Frequencies for Front
Right limb. Bottom: CPG Amplitudes for Front Right limb. We use the FR limb
since the robots exhibit a trot gait where the feet closely repeat the same
pattern.

###  IV-A Implementation Setting

We use Isaac Gym [33, 37] for training and simulating 16 different quadruped
robots. These robots exhibit variations in mass, ranging from 2 to 200 kg,
nominal height from 18 to 100 cm, and come in three different morphologies,
with two types of DoF, either 12 or 16. The robots in our study include the
commercial robots Unitree A1, Go1, Aliengo, Laikago, B1, Boston Dynamics Spot,
ANYbotics ANYmal-B and ANYmal-C, MIT mini-cheetah, Little-Dog, Spot-Micro,
Solo, and HYQ, as well as three customized three-segmented leg quadruped
robots. Characteristics and parameters for each robot are detailed in Table II
and Figure 3. We consider the following three main morphologies in simulation:

Elbow-Up configuration for all limbs (3-DoF): This configuration, used in
robots like Spot, MIT mini-Cheetah, A1, Go1, B1, Laikago, Aliengo, and Spot-
micro, consists of a 2-segmented elbow-up for both front and hind legs. Each
leg has three Degrees of Freedom (DoF): one for adduction-abduction and two
for hip and knee flexion/extension. We solve the inverse kinematics
analytically with constraints to an elbow-up configuration.

Elbow-Up for front and Elbow-Down for hind limbs (3-DoF): Similar to the first
category, the front limbs maintain an elbow-up configuration, but the hind
limbs adopt an elbow-down configuration for the knee. We utilized the same
analytical inverse kinematics solution as the first category but applied an
elbow-down setting for the hind limbs. This configuration is used in robots
like ANYmal-B, ANYmal-C, Solo, HYQ, and Little-Dog.

Quadrupedal Animal-like Configuration (4-DoF): In this morphology, inspired by
quadruped animals, each leg has four DoF: one for adduction-abduction and
three for hip, knee, and foot flexion/extension. We also use an analytical
inverse kinematics solution for this configuration, which is applied to three
animal-like robots with varying sizes and mass properties. For a visual
reference, please see Figure 3.

Figure 5:  Left: Snapshots of Go1 trotting on uneven grass. Right: Snapshots
of A1 carrying $10$ to $15$ $\mathrm{k}\mathrm{g}$. The robot starts trotting
with a $10$ $\mathrm{k}\mathrm{g}$ mass, and then an additional $5$
$\mathrm{k}\mathrm{g}$ is added. The robot successfully walks despite never
having encountered any such disturbances during training.

To train the policies, we use Proximal Policy Optimization (PPO) [38], and
Table I lists the PPO hyperparameters and neural network architecture. The
control frequency of the policy is 100 Hz, and the torques computed from the
desired joint positions are updated at 1 kHz. The equations for each of the
oscillators (Equations 1 and 2) are thus also integrated at 1 kHz. All
policies are trained for $14.0\times 10^{7}$ samples on a NVIDIA GeForce RTX
3070 8GB.

###  IV-B Simulation Results

In this section, we present simulation results from training the 16 quadruped
robots to learn forward locomotion. We consider four parameters for each robot
which are fixed during training: nominal standing height, nominal step length,
ground clearance of feet during the swing phase, and feet penetration into the
ground during stance phase, as shown in Table II. These values are
heuristically scaled for each robot based on its height in the zero joints
position (i.e. when the legs are fully extended). Figure 4 illustrates the
base velocity and CPG frequency and amplitude of the Front Right limb for this
first trained policy across all quadruped robots. In this scenario, the reward
function encourages forward movement as much as possible while penalizing
velocities exceeding 1.5 $\mathrm{m}\mathrm{/}\mathrm{s}$. Small-sized robots,
such as Little-Dog and Spot-Micro, can only reach speeds of 0.8 and 1.2
$\mathrm{m}\mathrm{/}\mathrm{s}$, respectively (Figure 4-A-1). The frequency
trajectory corresponding to small robots (Figure 4-B-1 and 2) shows that, in
comparison to the larger robots (Figure 4-B-3, 4, 5, 6), the policy tends to
use the highest possible frequency to increase the velocity of small-sized
robots. However, increasing velocity for the larger robots does not
necessarily require very high frequencies, as they have greater flexibility to
increase their base velocity by extending their stride length compared to
smaller robots.

Our results show that robots with the second type of configuration such as
Little-Dog and ANYmal-B and ANYmal-C reach smaller velocities (0.8, 1.4 and
1.4 $\mathrm{m}\mathrm{/}\mathrm{s}$) respectively) with respect to similar
sized robots with the first type morphology such as Spot-Micro, B1 and Spot
(1.2, 1.55 and 1.55 $\mathrm{m}\mathrm{/}\mathrm{s}$ respectively).
Furthermore, Figure 4-C-5 and 6 shows that Anymal-B and C locomote with a
smaller average amplitude, while Spot and Unitree B1 tend to have the highest
amplitude possible, resulting in a higher stride length. Despite the similar
mass properties of the compared robots, this observation needs a proper
investigation for different nominal parameters to shed some light on the
effect of morphology and mass properties in agility and efficiency, for future
work.

An interesting observation is that Dog3, the largest simulated robot with a
weight of $200$ $\mathrm{k}\mathrm{g}$ and a nominal height of $100$
$\mathrm{c}\mathrm{m}$, utilizes the highest amplitude. In contrast to other
robots, the amplitude reaches its maximum at the starting point of the
movement and does not change, allowing for the use of the longest possible
stride length, while maintaining a low locomotion frequency. This observation
intuitively suggests that robots with larger and heavier limbs tend to adopt a
gait characterized by a low frequency and the longest possible stride length,
which helps minimize the CoT (penalized in our reward function). At the other
extreme, Spot-Micro, which is the smallest robot, exhibits a similar strategy
for amplitude. It utilizes very high amplitude with small changes but takes
many steps with a high control frequency. This behavior is driven by Spot-
Micro’s priority to increase speed due to its limitations caused by its small
limbs. In contrast, increasing speed for Dog3 is less critical, so it selects
a gait that minimizes energy consumption.

Furthermore, we trained a single policy for 13 quadruped robots, excluding
HYQ, Dog3, and B1, during training. We deliberately selected these robots to
represent the extreme mass and size properties of each morphology to test the
generalization capabilities of the framework. We observed reasonable
locomotion behavior for these robots, even though the policy had not been
specifically trained on them. Please refer to the supplementary video for
visual reference.

###  IV-C Experimental Results

For the experiments, we trained the single policy to locomote with a maximum
velocity of 1 $\mathrm{m}\mathrm{/}\mathrm{s}$, without incorporating any
domain randomization or noise during the training process. We transferred the
trained policy sim-to-real to the Go1 and A1 robots, which are the only
quadruped robots available in the lab. Figures 1 and 5 show both simulation
and experiments snapshots. We tested the control policy on the Go1 robot in
two outdoor environments: one on a concrete pavement, and the other on uneven
grass. In both scenarios, we observed a very smooth trotting gait. Notably,
the grass surface is quite uneven, with holes and bumps that were not
encountered during the training process.

We tested the same policy on the A1 robot in two types of experiments: normal
walking, and load-carrying scenarios. It’s noteworthy that the addition of
loads was not encountered during training. We achieved stable trotting in all
experiments, even with an additional load of 15.0 $\mathrm{k}\mathrm{g}$,
equivalent to $125\%$ of the robot’s nominal mass. To the best of our
knowledge, this accomplishment represents the highest level of robustness
against additional loads ever achieved with the A1 robot. In comparison,
learning-based methods managed a load of up to $12$ $\mathrm{k}\mathrm{g}$
[18], and a model-based approach achieved an $11$ $\mathrm{k}\mathrm{g}$ load
capacity [34].

##  V Conclusion

Biological studies have shown that vertebrates utilize very similar motor
control architectures, despite differences in morphology, size, mass, and DoFs
[2]. Drawing inspiration from this fact, we have presented a biology-inspired
learning framework based on Central Pattern Generators and Deep Reinforcement
Learning (CPG-RL) with the capability to train a single policy for controlling
diverse quadruped robots. These robots vary in the number of DoFs, with
variations of $12$ and $16$, encompass three distinct morphologies, have a
wide mass range of $2$ $\mathrm{K}\mathrm{g}$ to $200$ $\mathrm{K}\mathrm{g}$,
and nominal standing heights ranging from $16$ $\mathrm{c}\mathrm{m}$ to $100$
$\mathrm{c}\mathrm{m}$. The proposed framework acts in task-space to modulate
and control the feet trajectories, and it does not rely on observing joint
information in the observation space, which facilitates training robots with
different DoFs and morphologies. Moreover, we are able to train a single
policy that works for 16 different robots in less than two hours. In our sim-
to-real hardware experiments, we successfully demonstrated stable trotting
even with an additional load of 15.0 $\mathrm{k}\mathrm{g}$ on the Unitree A1
robot, which is equivalent to $125\%$ of the robot’s nominal mass. To the best
of our knowledge, this accomplishment demonstrates the highest level of
robustness against additional loads ever achieved on the A1 robot.

For future work, we plan to expand our framework to include omni-directional
motion planning on uneven terrain. We also aim to further explore how
morphology and DoFs influence the agility and efficiency of quadruped
locomotion in both animals and robots.

## Acknowledgements

We would like to thank Alessandro Crespi for assisting with hardware setup.

## References

  * [1] S. Grillner and A. El Manira, “Current principles of motor control, with special reference to vertebrate locomotion,” _Physiological reviews_ , vol. 100, no. 1, pp. 271–320, 2020.
  * [2] S. Grillner, “Evolution: vertebrate limb control over 420 million years,” _Current Biology_ , vol. 28, no. 4, pp. R162–R164, 2018.
  * [3] A. Spröwitz, A. Tuleu, M. Vespignani, M. Ajallooeian, E. Badri, and A. J. Ijspeert, “Towards dynamic trot gait locomotion: Design, control, and experiments with cheetah-cub, a compliant quadruped robot,” _The International Journal of Robotics Research_ , vol. 32, no. 8, pp. 932–950, 2013.
  * [4] M. Ajallooeian, S. Pouya, A. Sproewitz, and A. J. Ijspeert, “Central pattern generators augmented with virtual model control for quadruped rough terrain locomotion,” in _2013 IEEE International Conference on Robotics and Automation_ , 2013, pp. 3321–3328.
  * [5] S. Aoi, P. Manoonpong, Y. Ambe, F. Matsuno, and F. Wörgötter, “Adaptive control strategies for interlimb coordination in legged robots: a review,” _Frontiers in neurorobotics_ , vol. 11, p. 39, 2017.
  * [6] H. Kimura, Y. Fukuoka, and A. H. Cohen, “Adaptive dynamic walking of a quadruped robot on natural ground based on biological concepts,” _The International Journal of Robotics Research_ , vol. 26, no. 5, pp. 475–490, 2007.
  * [7] A. J. Ijspeert, A. Crespi, D. Ryczko, and J.-M. Cabelguen, “From swimming to walking with a salamander robot driven by a spinal cord model,” _Science_ , vol. 315, no. 5817, pp. 1416–1420, 2007.
  * [8] R. Thandiackal, K. Melo, L. Paez, J. Herault, T. Kano, K. Akiyama, F. Boyer, D. Ryczko, A. Ishiguro, and A. J. Ijspeert, “Emergence of robust self-organized undulatory swimming based on local hydrodynamic force sensing,” _Science Robotics_ , vol. 6, no. 57, 2021.
  * [9] A. J. Ijspeert, “Central pattern generators for locomotion control in animals and robots: A review,” _Neural Networks_ , vol. 21, no. 4, pp. 642–653, 2008, robotics and Neuroscience.
  * [10] L. Righetti and A. J. Ijspeert, “Pattern generators with sensory feedback for the control of quadruped locomotion,” in _2008 IEEE International Conference on Robotics and Automation_. IEEE, 2008, pp. 819–824.
  * [11] X. B. Peng, P. Abbeel, S. Levine, and M. van de Panne, “Deepmimic: Example-guided deep reinforcement learning of physics-based character skills,” _ACM Transactions on Graphics (TOG)_ , vol. 37, no. 4, pp. 1–14, 2018.
  * [12] A. Iscen, K. Caluwaerts, J. Tan, T. Zhang, E. Coumans, V. Sindhwani, and V. Vanhoucke, “Policies modulating trajectory generators,” in _Conference on Robot Learning_. PMLR, 2018, pp. 916–926.
  * [13] J. Hwangbo, J. Lee, A. Dosovitskiy, D. Bellicoso, V. Tsounis, V. Koltun, and M. Hutter, “Learning agile and dynamic motor skills for legged robots,” _Science Robotics_ , vol. 4, no. 26, 2019.
  * [14] J. Lee, J. Hwangbo, L. Wellhausen, V. Koltun, and M. Hutter, “Learning quadrupedal locomotion over challenging terrain,” _Science Robotics_ , vol. 5, no. 47, 2020.
  * [15] T. Miki, J. Lee, J. Hwanbo, L. Wellhausen, V. Koltun, and M. Hutter, “Learning robust perceptive locomotion for quadrupedal robots in the wild,” _Science Robotics_ , 2022.
  * [16] J. Siekmann, K. Green, J. Warila, A. Fern, and J. Hurst, “Blind bipedal stair traversal via sim-to-real reinforcement learning,” _arXiv preprint arXiv:2105.08328_ , 2021.
  * [17] X. B. Peng, E. Coumans, T. Zhang, T.-W. Lee, J. Tan, and S. Levine, “Learning agile robotic locomotion skills by imitating animals,” 2020.
  * [18] A. Kumar, Z. Fu, D. Pathak, and J. Malik, “Rma: Rapid motor adaptation for legged robots,” _arXiv preprint arXiv:2107.04034_ , 2021.
  * [19] G. Ji, J. Mun, H. Kim, and J. Hwangbo, “Concurrent training of a control policy and a state estimator for dynamic and robust legged locomotion,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 4630–4637, 2022.
  * [20] G. B. Margolis, G. Yang, K. Paigwar, T. Chen, and P. Agrawal, “Rapid locomotion via reinforcement learning,” _arXiv preprint arXiv:2205.02824_ , 2022.
  * [21] G. Bellegarda, Y. Chen, Z. Liu, and Q. Nguyen, “Robust high-speed running for quadruped robots via deep reinforcement learning,” in _2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2022, pp. 10 364–10 370.
  * [22] Y. Yang, T. Zhang, E. Coumans, J. Tan, and B. Boots, “Fast and efficient locomotion via learned gait transitions,” in _Conference on Robot Learning_. PMLR, 2022, pp. 773–783.
  * [23] Y. Shao, Y. Jin, X. Liu, W. He, H. Wang, and W. Yang, “Learning free gait transition for quadruped robots via phase-guided controller,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 1230–1237, 2022.
  * [24] W. Yu, C. Yang, C. McGreavy, E. Triantafyllidis, G. Bellegarda, M. Shafiee, A. J. Ijspeert, and Z. Li, “Identifying important sensory feedback for learning locomotion skills,” _Nature Machine Intelligence_ , vol. 5, no. 8, pp. 919–932, 2023.
  * [25] W. Huang, I. Mordatch, and D. Pathak, “One policy to control them all: Shared modular policies for agent-agnostic control,” in _International Conference on Machine Learning_. PMLR, 2020, pp. 4455–4464.
  * [26] V. Kurin, M. Igl, T. Rocktäschel, W. Boehmer, and S. Whiteson, “My body is a cage: the role of morphology in graph-based incompatible control,” _arXiv preprint arXiv:2010.01856_ , 2020.
  * [27] B. Trabucco, M. Phielipp, and G. Berseth, “Anymorph: Learning transferable polices by inferring agent morphology,” in _International Conference on Machine Learning_. PMLR, 2022, pp. 21 677–21 691.
  * [28] J. Whitman, M. Travers, and H. Choset, “Learning modular robot control policies,” _IEEE Transactions on Robotics_ , 2023.
  * [29] G. Feng, H. Zhang, Z. Li, X. B. Peng, B. Basireddy, L. Yue, Z. Song, L. Yang, Y. Liu, K. Sreenath, _et al._ , “Genloco: Generalized locomotion controllers for quadrupedal robots,” in _Conference on Robot Learning_. PMLR, 2023, pp. 1893–1903.
  * [30] G. Bellegarda and A. Ijspeert, “CPG-RL: Learning central pattern generators for quadruped locomotion,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 4, pp. 12 547–12 554, 2022.
  * [31] M. Shafiee, G. Bellegarda, and A. Ijspeert, “Puppeteer and marionette: Learning anticipatory quadrupedal locomotion based on interactions of a central pattern generator and supraspinal drive,” _2023 IEEE International Conference on Robotics and Automation_ , 2023.
  * [32] M. Shafiee, G. Bellegarda, and A. Ijspeert, “Deeptransition: Viability leads to the emergence of gait transitions in learning anticipatory quadrupedal locomotion skills,” _arXiv preprint arXiv:2306.07419_ , 2023.
  * [33] V. Makoviychuk, L. Wawrzyniak, Y. Guo, M. Lu, K. Storey, M. Macklin, D. Hoeller, N. Rudin, A. Allshire, A. Handa, _et al._ , “Isaac gym: High performance gpu-based physics simulation for robot learning,” _arXiv preprint arXiv:2108.10470_ , 2021.
  * [34] M. Sombolestan, Y. Chen, and Q. Nguyen, “Adaptive force-based control for legged robots,” in _2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2021, pp. 7440–7447.
  * [35] I. A. Rybak, N. A. Shevtsova, M. Lafreniere-Roula, and D. A. McCrea, “Modelling spinal circuitry involved in locomotor pattern generation: insights from deletions during fictive locomotion,” _The Journal of physiology_ , vol. 577, no. 2, pp. 617–639, 2006.
  * [36] A. Fukuhara, D. Owaki, T. Kano, R. Kobayashi, and A. Ishiguro, “Spontaneous gait transition to high-speed galloping by reconciliation between body support and propulsion,” _Advanced robotics_ , vol. 32, no. 15, pp. 794–808, 2018.
  * [37] N. Rudin, D. Hoeller, P. Reist, and M. Hutter, “Learning to walk in minutes using massively parallel deep reinforcement learning,” in _Proceedings of the 5th Conference on Robot Learning_ , ser. Proceedings of Machine Learning Research, A. Faust, D. Hsu, and G. Neumann, Eds., vol. 164. PMLR, 08–11 Nov 2022, pp. 91–100. [Online]. Available: https://proceedings.mlr.press/v164/rudin22a.html
  * [38] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, “Proximal policy optimization algorithms,” _CoRR_ , vol. abs/1707.06347, 2017.