# De Rham cohomology for supervarieties

Alexander Polishchuk  Department of Mathematics, University of Oregon, Eugene,
OR 97403, USA; and National Research University Higher School of Economics,
Moscow, Russia<EMAIL_ADDRESS>
###### Abstract.

We study the de Rham cohomology and the Hodge to de Rham spectral sequence for
supervarieties.

Partially supported by the NSF grant DMS-2001224, and within the framework of
the HSE University Basic Research Program and by the Russian Academic
Excellence Project ‘5-100’.

##  1\. Basics

###  1.1. Introduction

Recall (see e.g.,[6]) that for a smooth proper scheme $X_{0}$ over a field
$k$, one defines the de Rham cohomology $H_{dR}(X_{0})$ as the hypercohomology
of the de Rham complex $\Omega^{\bullet}_{X_{0}}$. One has a natural Hodge to
de Rham spectral sequence

$E_{1}^{pq}(X_{0})=H^{q}(X,\Omega^{p}_{X_{0}})\implies H^{p+q}_{dR}(X_{0}).$

In characteristic zero it is known to degenerate at $E_{1}$ (this can be
deduced from the Hodge theory).

In this paper we study the similar spectral sequence for a smooth proper
supervariety $X$. It is easy to see that it often does not degenerate at
$E_{1}$, and its behavior is much more complicated.

We prove that the de Rham cohomology of a smooth supervariety $X$ coincides
with the de Rham cohomology of the underlying variety $X_{0}$ (see Theorem
1.1). We show that the Hodge to de Rham spectral sequence of a smooth proper
supervariety $X$ degenerates at $E_{2}$ if $X$ is split (see Theorem 2.1) or
has dimension $n|2$ (see Theorem 2.4). In the latter case we also give a
formula for dimensions of the terms $E_{2}^{pq}(X)$. On the other hand, we
give an example showing that this spectral sequence does not always degenerate
at the $E_{2}$ page if $X$ has dimension $1|3$ (see Theorem 2.15). This
answers negatively the question raised in [3, Problem 5.14].

An interesting phenomenon is that the limiting terms $(E_{\infty}^{pq}(X))$ of
the Hodge to de Rham sequence can differ from $(E_{\infty}^{pq}(X_{0}))$
(e.g., this happens for the supergrassmannian $G(1|1,2|2)$, see Example 2.13).
Thus, the way the dimension of $H^{i}_{dR}(X_{0})$ gets distributed among
$E_{\infty}^{pq}(X)$ with $p+q=i$ gives interesting numerical invariants of a
supervariety (which cannot be recovered from $X_{0}$ unless $X$ is projected).

We refer to [8] and [2] for basics on algebraic supergeometry. We work with
algebraic supervarieties and Zariski topology, but the analogous picture holds
in the complex analytic case. Note that even though one can introduce analogs
of $(p,q)$-forms on the CS manifold underlying a complex supermanifold, there
seems to be no analog of the Hodge decomposition in the presence of odd
variables. The obvious problem is that the de Rham complex
$\Omega^{\bullet}_{X}$ is unbounded, and the twisted dual
$(\Omega^{\bullet}_{X})^{\vee}\otimes\omega_{X}$ (where $\omega_{X}$ is the
Berezinian) is a different complex, the complex of integral forms.

In many respects the study of supervarieties is similar to the study of
nilpotent extensions of smooth schemes (in the category of usual even
schemes). As a manifestation of this, we prove a result in even algebraic
geometry (inspired by the analogy with supervarieties) that states
surjectivity of the map on cohomology of structure sheaves for nilpotent
extensions of smooth proper schemes in characteristic zero (see Theorem 1.11).

The paper is organized as follows. In Sec. 1.2 we prove the comparison theorem
on de Rham cohomology of a supervariety $X$ (Theorem 1.1). In Sec. 1.3 we
discuss relative de Rham cohomology and the Gauss-Manin connection. In Sec.
1.4 we present some general observations on the Hodge to de Rham spectral
sequence for supervarieties. In Sec. 1.5 we prove a result on even nilpotent
extensions of smooth proper schemes (Theorem 1.11). Next, in Sec. 2 we present
some more specific computations. In Sec. 2.1 we consider split supervarieties
as well as supervarieties in characteristic $2$, in Sec. 2.2— supervarieties
of dimension $n|2$, and in Sec. 2.3—an example showing that in dimension
$1|3$, the differential $d:({\mathcal{O}}_{X})_{-}\to(\Omega^{1}_{X})_{-}$
does not always induce injection on $H^{1}$.

Acknowledgments. I am grateful to Vera Serganova and to Johan de Jong for
useful discussions. Part of this work was done while the author visited the
Simons Center for Geometry and Physics. I’d like to thank this institution for
hospitality and excellent working conditions.

Conventions. For a $\operatorname{\mathbb{Z}}/2$-graded object $X$ we denote
by $X_{+}$ and $X_{-}$ its even and odd parts. We use the parity of
differential forms such that the de Rham differential is even, so the de Rham
complex can be viewed as a complex of $\operatorname{\mathbb{Z}}/2$-graded
sheaves.

###  1.2. De Rham cohomology of a supervariety

Let $X$ be a smooth superscheme over a field $k$. Then we have the de Rham
complex $\Omega_{X}^{\bullet}=(\Omega_{X}^{\bullet},d)$, which we view as a
bounded below complex of $k$-modules in Zariski topology. Therefore, we can
define de Rham cohomology of $X$ as its hypercohomology:

$H^{\bullet}_{dR}(X):=H^{\bullet}(X,\Omega_{X}^{\bullet}).$

Note that as soon as $X$ has nonzero odd dimension, the complex
$\Omega_{X}^{\bullet}$ is not bounded.

Let ${\mathcal{N}}\subset{\mathcal{O}}_{X}$ denote the ideal generated by odd
functions, and let $X_{0}$ denote the bosonization of $X$, so
${\mathcal{O}}_{X_{0}}={\mathcal{O}}_{X}/{\mathcal{N}}$. We have a natural
surjective map of complexes of $k$-vector spaces

$\Omega_{X}^{\bullet}\to\Omega_{X_{0}}^{\bullet}.$

Let us denote by $K$ its kernel.

######  Theorem 1.1.

The complex $K$ is acyclic. Hence, one has a natural isomorphism
$H_{dR}(X)\rTo{\sim}H_{dR}(X_{0})$.

###### Proof.

Let us introduce the decreasing algebra filtration
$(F^{\bullet}_{\mathcal{N}})$ on $\Omega_{X}^{\bullet}$ by letting
$F^{i}_{\mathcal{N}}$ be the ideal (with respect to the algebra structure on
$\Omega_{X}^{\bullet}$) generated by

$({\mathcal{N}}^{i},{\mathcal{N}}^{i-1}d{\mathcal{N}},{\mathcal{N}}^{i-2}(d{\mathcal{N}})^{2},\ldots,(d{\mathcal{N}})^{i}).$

Then $K^{\bullet}=F^{1}_{\mathcal{N}}$.

We claim that for $i\geq 1$, the complex
$\operatorname{gr}^{i}_{F}:=F^{i}_{\mathcal{N}}/F^{i+1}_{\mathcal{N}}$ is
exact. Let us consider the ${\mathcal{O}}_{X_{0}}$-algebra
${\mathcal{A}}:=\operatorname{gr}^{\bullet}_{F}\Omega^{\bullet}$, equipped
with the differential $d_{\mathcal{A}}$ induced by $d$. Note that
$\operatorname{gr}^{0}_{F}=\Omega^{\bullet}_{X_{0}}$. We have an embedding of
${\mathcal{O}}_{X_{0}}$-algebras

${\mathcal{A}}_{0}:=\Omega^{\bullet}_{X_{0}}\otimes{\bigwedge}^{\bullet}({\mathcal{N}}/{\mathcal{N}}^{2})\hookrightarrow{\mathcal{A}}.$

Let us set

${\mathcal{A}}_{n}:={\mathcal{A}}_{0}\cdot\operatorname{gr}^{\leq
n}_{F}\subset{\mathcal{A}}.$

Then $({\mathcal{A}}_{n})$ is an increasing algebra filtration, and we have an
algebra isomorphism

$\overline{{\mathcal{A}}}:=\bigoplus_{n}{\mathcal{A}}_{n}/{\mathcal{A}}_{n-1}\simeq{\mathcal{A}}_{0}\otimes
S^{\bullet}({\mathcal{N}}/{\mathcal{N}}^{2})\simeq\Omega^{\bullet}_{X_{0}}\otimes
K({\mathcal{N}}/{\mathcal{N}}^{2}),$

where
$K({\mathcal{N}}/{\mathcal{N}}^{2}):={\bigwedge}^{\bullet}({\mathcal{N}}/{\mathcal{N}}^{2})\otimes
S^{\bullet}({\mathcal{N}}/{\mathcal{N}}^{2})$. Furthermore, the differential
$d_{\mathcal{A}}$ sends ${\mathcal{A}}_{n}$ to ${\mathcal{A}}_{n+1}$, and the
induced differential
${\mathcal{A}}_{n}/{\mathcal{A}}_{n-1}\to{\mathcal{A}}_{n+1}/{\mathcal{A}}_{n}$
on $\overline{{\mathcal{A}}}$ is induced by the Koszul differential on
$K({\mathcal{N}}/{\mathcal{N}}^{2})$. Finally, we observe that
$\bigoplus_{n}({\mathcal{A}}_{n}\cap\operatorname{gr}^{\geq
1}_{F})/({\mathcal{A}}_{n-1}\cap\operatorname{gr}^{\geq 1}_{F})$ gets
identified with $\Omega^{\bullet}_{X_{0}}\otimes K_{\geq
1}({\mathcal{N}}/{\mathcal{N}}^{2})$, where $K_{\geq
1}({\mathcal{N}}/{\mathcal{N}}^{2})\subset K({\mathcal{N}}/{\mathcal{N}}^{2})$
is the subcomplex of elements of weight $\geq 1$ (i.e., the augmentation
ideal). Now our assertion follows from the exactness of $K_{\geq
1}({\mathcal{N}}/{\mathcal{N}}^{2})$ (in fact, it is homotopic to zero). ∎

######  Remark 1.2.

Note that Theorem 1.1 holds in any characteristic. In the complex analytic
context (and hence, in characteristic zero) Theorem 1.1 can also be deduced
from the fact that $\Omega_{X}^{\bullet}$ is a resolution of the constant
sheaf on $X$.

###  1.3. Relative de Rham cohomology, actions of supergroups and the Gauss-
Manin connection

More generally, if $f:X\to S$ is a smooth proper morphism of superschemes then
we have the relative de Rham complex $\Omega_{X/S}^{\bullet}$, and we can form
the corresponding relative de Rham cohomology sheaves of
${\mathcal{O}}$-modules on $S$,

$H^{i}_{dR}(X/S):=\underline{H}^{i}Rf_{*}(\Omega^{\bullet}_{X/S}).$

Assume that $S$ is Noetherian. Since the sheaves $R^{q}(\Omega^{p}_{X/S})$ are
coherent, it follows that $H^{i}_{dR}(X/S)$ are coherent sheaves.

It is easy to see that the formation of $H^{i}_{dR}(X/S)$ is compatible with
flat base changes $S^{\prime}\to S$. In particular, if we start with $X$,
proper and smooth over the ground field $k$, then for any Noetherian
commutative $k$-superalgebra $A$, we can consider the base change
$X_{A}\to\operatorname{Spec}(A)$, and we have natural isomorphisms

$H^{i}_{dR}(X)\otimes A\simeq H^{i}_{dR}(X_{A}/A).$

This shows that if an algebraic $k$-supergroup $G$ acts on $X$ then
$H^{i}_{dR}(X)$ have natural structure of $G$-representations. Indeed, for $A$
as above, the group $G(A)$ acts on $X_{A}$ and this gives its action on
$H^{i}_{dR}(X_{A}/A)\simeq H^{i}_{dR}(X)\otimes A$.

######  Proposition 1.3.

Assume the characteristic is zero. If the underlying usual algebraic group
$G_{0}$ is connected then $H^{i}_{dR}(X)$ is trivial as a $G$-representation.

###### Proof.

It suffices to show that the corresponding Lie superalgebra acts trivially.
Thus, it is enough to show that for every global vector field $v$ (even or
odd), the automorphism $f\mapsto f+v(f)\epsilon$ of $X_{D}$, where
$D=k[\epsilon]/(\epsilon^{2})$ ($\epsilon$ is either even or odd), induces the
trivial automorphism of $H^{i}_{dR}(X_{D}/D)$. But this immediately follows
from the fact that the corresponding automorphism of the relative de Rham
complex $\Omega^{\bullet}_{X_{D}/D}$, given by
$\eta\mapsto\eta+L_{v}(\eta)\epsilon$ (where $L_{v}$ is the Lie derivative) is
homotopic to the identity via the homotopy $\eta\mapsto i_{v}(\eta)$ (the
contraction by $v$). ∎

In the case when $f:X\to S$ is a smooth proper morphism, where $S$ is smooth
over a field $k$, similarly to the classical case (see e.g., [7, Sec. IV]) one
can construct the Gauss-Manin connection on $H^{i}_{dR}(X/S)$. Namely,
consider the filtration $F^{i}$ on the absolute de Rham complex
$\Omega^{\bullet}_{X}$, where $F^{i}$ is the image of
$f^{*}\Omega^{i}_{S}\otimes\Omega^{\bullet-i}_{X}\to\Omega^{\bullet}_{X}$.
Then we have natural identifications

$\operatorname{gr}^{i}_{F}\Omega^{\bullet}_{X}\simeq
f^{*}\Omega^{i}_{S}\otimes\Omega^{\bullet}_{X/S}[-i].$

The connection on $H^{i}_{dR}(X/S)$ is defined as the connecting homomorphism

$\nabla:H^{i}_{dR}(X/S)\simeq R^{i}f_{*}\operatorname{gr}^{0}_{F}\to
R^{i+1}f_{*}\operatorname{gr}^{1}_{F}\simeq\Omega^{1}_{S}\otimes
H^{i}_{dR}(X/S).$

Its integrability is proved by considering the entire page of the
corresponding spectral sequence.

As in the classical case, this implies that for smooth $S$, the coherent
sheaves $H^{i}_{dR}(X/S)$ on $S$ are in fact locally free (using [9, Sec.
2.2.1]), and that the formation of the $H^{i}_{dR}(X/S)$ is compatible with
arbitrary base change.

Note that the sheaves $R^{q}f_{*}\Omega^{p}_{X}$ are not necessarily locally
free (they are in the even case, in characteristic zero): for example, there
exist families of supercurves $f:X\to S$ such that
$R^{1}f_{*}{\mathcal{O}}_{X}$ is not locally free (see [5, Sec. 3.3]).

###  1.4. Spectral sequence: general observations

From now on we assume that $X$ is proper and smooth over the ground field $k$.

As in the even case, we associate with a complex of sheaves
$\Omega_{X}^{\bullet}$ the spectral sequence

$E_{1}^{pq}(X)=H^{q}(X,\Omega^{p}_{X})\implies H^{p+q}_{dR}(X),$

to which we refer as Hodge to de Rham spectral sequence. By Theorem 1.1, its
limit $H^{\bullet}_{dR}(X)$ is identified with $H^{\bullet}_{dR}(X_{0})$

In the even case, assuming that the characteristic is zero, the Hodge to de
Rham spectral sequence is known to degenerate at the page $E_{1}$. There is
not much we can say about this spectral sequence for general supervarieties,
even in characteristic zero. Already the example of $X$ of dimension $0|n$
shows that it does not have to degenerate at $E_{1}$. We will prove that it
degenerates at $E_{2}$ if $X$ is split or has dimension $n|2$ (see Theorems
2.1 and 2.4). We will also show that there exist $X$ of dimension $1|3$ such
that the Hodge to de Rham spectral sequence of $X$ does not degenerate at
$E_{2}$ (see Theorem 2.15).

At present, we do not know an example of $X$ in characteristic zero such that
$E_{2}(X)$ is not finite-dimensional.

######  Proposition 1.4.

We always have an isomorphism $E_{2}^{00}(X)\simeq H^{0}_{dR}(X_{0})$ and an
injection

$E_{2}^{10}(X)\hookrightarrow H^{0}(\Omega^{1}_{X_{0}}).$ (1.1)

###### Proof.

The first assertion is clear since the spectral sequence degenerates at
$E_{2}^{00}$. It also degenerates at $E_{2}^{10}$, so we have an embedding
$E_{2}^{10}(X)\to H^{1}_{dR}(X)\simeq H^{1}_{dR}(X_{0})$. This map factors as
the composition

$E_{2}^{10}(X)\to H^{0}(\Omega^{1}_{X_{0}})\to H^{1}_{dR}(X_{0}),$

which implies the claimed injection. ∎

For any morphism $f:X\to Y$ between smooth proper varieties, the pull-back
gives a morphism of spectral sequences

$f^{*}:E_{n}^{pq}(Y)\to E_{n}^{pq}(X).$

Applying this to the embedding $i:X_{0}\to X$, we get a morphism of spectral
sequences

$i^{*}:E_{n}^{pq}(X)\to E_{n}^{pq}(X_{0})$

A curious fact is that although the morphism $H_{dR}(X)\to H_{dR}(X_{0})$ is
always an isomorphism, if $X$ is not projected, the morphism
$E_{\infty}^{pq}(X)\to E_{\infty}^{pq}(X_{0})$ is not necessarily an
isomorphism (for example, this happens for $X=G(1|1,2|2)$, see Example 2.13
below).

The next result shows that the distribution of dimensions of
$E_{\infty}^{pq}(X)$ for constant $p+q$, is skewed towards smaller $p$ compare
to $E_{\infty}^{pq}(X_{0})$.

######  Proposition 1.5.

For every $m\geq 0$, $p\geq 0$,

$\sum_{i\geq p}\dim E_{\infty}^{i,m-i}(X)\leq\sum_{i\geq p}h^{i,m-i}(X_{0})$

(and this becomes an equality for $p=0$). Furthermore, for every $m$, the
natural map

$E_{\infty}^{0,m}(X)\to E_{\infty}^{0,m}(X_{0})$

is surjective.

###### Proof.

The first inequality follows immediately from the fact that the filtration
$(F_{p}(X))$ on $H^{n}_{dR}(X)$ induced by the Hodge to de Rham spectral
sequence satisfies

$F_{p}(X)\subset F_{p}(X_{0}).$

The second assertion also follows, as we can identify this map with

$H^{m}_{dR}(X)/F_{1}H^{m}_{dR}(X)\to
H^{m}_{dR}(X_{0})/F_{1}H^{m}_{dR}(X_{0}).$

∎

######  Corollary 1.6.

Assume the characteristic is zero. Then for every $i\geq 0$, the composition

$E_{\infty}^{0,i}(X)\to H^{i}(X,{\mathcal{O}}_{X})\to
H^{i}(X_{0},{\mathcal{O}}_{X_{0}})$

is surjective.

######  Remark 1.7.

Similarly, in characteristic zero, for any $i\geq 0$ and any $m\geq 0$, the
natural map of hypercohomology of truncated de Rham complexes,

$H^{i}(X,[{\mathcal{O}}_{X}\to\Omega_{X}^{1}\to\ldots\to\Omega_{X}^{m}])\to
H^{i}(X_{0},[{\mathcal{O}}_{X_{0}}\to\Omega_{X_{0}}^{1}\to\ldots\to\Omega_{X_{0}}^{m}])$

is surjective.

######  Proposition 1.8.

Assume that $X$ is projected. Then for every $i\geq 1$, the natural maps
$E_{i}^{pq}(X)\to H^{q}(\Omega^{p}_{X_{0}})$ are surjective, and they induce
isomorphisms

$E_{\infty}^{pq}(X)\rTo{\sim}H^{q}(\Omega^{p}(X_{0})).$

In particular, $E_{2}^{10}(X)\simeq H^{0}(\Omega^{1}_{X_{0}})$.

###### Proof.

Let $i:X_{0}\to X$ denote the embedding and let $p:X\to X_{0}$ be a projection
(so $p\circ i=\operatorname{id}_{X_{0}}$). Then we have morphisms of spectral
sequences

$E_{n}^{pq}(X_{0})\rTo{p^{*}}E_{n}^{pq}(X)\rTo{i^{*}}E^{n}_{pq}(X_{0}),$

with the composition equal to the identity. The surjectivity immediately
follows from this. The fact that we get an isomorphism on the infinite page
follows from the equality of dimensions $\dim
E_{\infty}(X)=\sum_{pq}h^{q}(\Omega^{p}_{X_{0}})$. ∎

######  Proposition 1.9.

(i) Let $L$ be an even line bundle on $X$. Then there exists a morphism of
spectral sequences

$c_{1}(L)\cup\cdot:E_{n}^{pq}(X)\to E_{n}^{p+1,q+1}(X),$

such that the map on $E_{1}$ is given by the cup product with $c_{1}(L)\in
H^{1}(\Omega^{1}_{X})$, compatible with the cup product by $c_{1}(L)_{dR}\in
H^{2}_{dR}(X)$ on the limit $H^{\bullet}_{dR}(X)$.

(ii) Assume in addition that the characteristic is zero and $X$ admits an
ample line bundle. Let $d=\dim X_{0}$. Then for any $i\geq j\geq 1$, and any
$n\geq 1$, we have for $p+q=d-i$,

$\operatorname{rk}(E_{n}^{pq}\to
H^{q}(\Omega^{p}_{X_{0}}))\leq\operatorname{rk}(E_{n}^{p+j,q+j}\to
H^{q+j}(\Omega^{p+j}_{X_{0}})).$

###### Proof.

(i) One can realize $c_{1}(L)$ by a Cech cocycle $d\log(f_{ij})$ with values
in $\Omega^{1,cl}$ (sheaf of closed $1$-forms), where $(f_{ij})$ are
transition functions for $L$, and use the multiplication by this cocycle on
the Cech complex of $\Omega^{\bullet}$.

(ii) This follows from (i) together with the fact that the multiplication map

$c_{1}(L)^{j}\cup\cdot:H^{q}(\Omega^{p}_{X_{0}})\to
H^{q+j}(\Omega^{p+j}_{X_{0}})$

is injective under our assumptions. ∎

######  Remark 1.10.

It is natural to ask whether the approach of Deligne-Illusie via the reduction
to finite characteristic (see [4]) sheds any light on the Hodge to de Rham
spectral sequence in the super case. Namely, for $X/k$ in characteristic $p$,
the Frobenius kills odd functions, so it can be viewed as a morphism $F:X\to
X_{0}$. Thus, $F_{*}\Omega_{X}^{\bullet}$ becomes a complex of
${\mathcal{O}}_{X_{0}}$-modules. Now it is easy to see that the analog of the
Cartier isomorphism (given by the same formulas) is

$C^{-1}:\Omega^{i}_{X_{0}}\rTo{\sim}\underline{H}^{i}(F_{*}\Omega_{X}^{\bullet}).$

The same method as in [4] shows that if the characteristic is $>\dim X_{0}$
and $X$ lifts to characterstic zero, then the second spectral sequence for the
hypercohomology of $F_{*}\Omega_{X}^{\bullet}$ (starting from $E_{2}=\bigoplus
H^{q}(\underline{H}^{i}(F_{*}\Omega_{X}^{\bullet}))$) degenerates at $E_{2}$.
This agrees with Theorem 1.1 but does not give any additional information on
the Hodge to de Rham spectral sequence for $X$ (because in the Cartier
isomorphism $\Omega^{i}_{X}$ gets replaced by $\Omega^{i}_{X_{0}}$).

###  1.5. A result on even nilpotent extensions

Corollary 1.6 has the following classical counterpart (for usual schemes),
which does not seem to be widely known.

######  Theorem 1.11.

Let $Y$ be a proper scheme over a field $k$ of characteristic zero, and let
$Y_{0}$ be the corresponding reduced scheme. Assume that $Y_{0}$ is smooth
over $k$. Then for any $i$ the natural map $H^{i}(Y,{\mathcal{O}}_{Y})\to
H^{i}(Y_{0},{\mathcal{O}}_{Y_{0}})$ is surjective.

###### Proof.

The map in question fits into a commutative square of natural maps where in
the first line we consider cohomology for the infinitesimal sites (see [6]).
It is known that the top horizontal arrow is an isomorphism (see [6, Sec.
5.3]). Also, it is well known that
$H^{i}((Y_{0})_{inf},{\mathcal{O}}_{Y_{0}})$ is isomorphic to the de Rham
cohomology of $Y_{0}$, so by the Hodge-to-de Rham degeneration for $Y_{0}$,
the right vertical arrow is surjective. This implies that the composed map
$H^{i}(Y_{inf},{\mathcal{O}}_{Y})\to H^{i}(Y_{0},{\mathcal{O}}_{Y_{0}})$ is
surjective. Since it factors through $H^{i}(Y,{\mathcal{O}}_{Y})$, the
assertion follows. ∎

##  2\. Some computations

###  2.1. Using a special derivation in the split case and in characterstic
$2$

######  Theorem 2.1.

Assume $X$ is split, and the ground field $k$ has characteristic zero. Then
the Hodge to de Rham spectral sequence of $X$ degenerates at $E_{2}$ and there
are natural isomorphisms $E_{2}^{p,q}(X)\rTo{\sim}H^{p,q}(X_{0})$.

###### Proof.

Let ${\mathcal{O}}_{X}=\bigwedge^{\bullet}(E)$, where $E$ is a bundle over
$X_{0}$. Then we have an action of ${\mathbb{G}}_{m}$ on $X$, such that
${\mathcal{O}}_{X_{0}}$ is ${\mathbb{G}}_{m}$-invariant, and $E$ has weight
$1$. Let $\xi$ be the corresponding global (even) vector field on $X$. In
local coordinates $(x_{\bullet},\theta_{\bullet})$ one has
$\xi=\sum_{i}\theta_{i}\partial_{\theta_{i}}$. Now we claim that for each
$n\geq 0$, one has

$\Omega^{n}_{X}=\bigoplus_{i\geq 0}\Omega^{n}_{i},$

where $\Omega^{n}_{i}\subset\Omega^{n}_{X}$ is the subsheaf of $\eta$ such
that $L_{\xi}(\eta)=i\eta$, where $L_{\xi}$ is the Lie derivative with respect
to $\xi$. Indeed, this is a local statement, which is clear in local
coordinates. Furthermore, for each $i\geq 0$, $\Omega^{\bullet}_{i}$ is
preserved by the de Rham differential (since $dL_{\xi}=L_{\xi}d$), and

$\Omega^{\bullet}_{0}\simeq\Omega_{X_{0}}^{\bullet}.$

Finally, the contraction operator $i_{\xi}$ gives a homotopy from
$i\cdot\operatorname{id}$ to $0$ on $\Omega^{\bullet}_{i}$. Hence, each
complex $\Omega^{\bullet}_{i}$ for $i>0$ is contractible. This implies that
the $E_{1}$ page of the spectral sequence associated with each
$\Omega^{\bullet}_{i}$ for $i>0$ is also contractible, so the $E_{2}$ page
corresponding to each $\Omega^{\bullet}_{i}$ with $i>0$ is zero. This implies
our assertion. ∎

######  Remark 2.2.

When $X$ varies in a flat family, there is no semicontinuity for the spaces
$E_{2}^{p,q}(X)$ (since they are obtained by taking cohomology twice), so the
split case does not provide any nontrivial conclusions for the behavior of
$E_{2}^{p,q}(X)$ for general $X$. For example, it is possible that for some
supervariety $X$ one has $E_{2}^{p,q}(\operatorname{gr}X)=0$ for some $(p,q)$,
while $E_{2}^{p,q}(X)\neq 0$ (where $\operatorname{gr}X$ is the split
supervariety associated with $X$), see Example 2.13 below.

When the characteristic of the ground field is $2$ then the vector field
$\xi=\sum_{i}\theta_{i}\partial_{\theta_{i}}$ used in Theorem 2.1 does not
depend on coordinates. Namely, it corresponds to the derivation $\xi$ such
that $\xi(f)=0$ for even $f$ and $\xi(\eta)=\eta$ for odd $\eta$. This
immediately leads to the following result.

######  Proposition 2.3.

Assume that the ground field $k$ has characterstic $2$. Then the odd part of
the de Rham complex $(\Omega^{\bullet}_{X})_{-}$ is homotopic to zero. Hence,
$E_{2}(X)_{-}=0$.

###### Proof.

The homotopy from $\operatorname{id}$ to $0$ on $(\Omega^{\bullet}_{X})_{-}$
is given by the contraction operator $i_{\xi}$. ∎

###  2.2. Odd dimension $2$

In this section we consider the case when $X$ is of dimension $n|2$. We assume
that the characteristic of $k$ is zero.

For each $p,q$, let us set

$\delta_{pq}(X):=\dim\operatorname{coker}(H^{q}(\Omega^{p}_{X})\to
H^{q}(\Omega^{p}_{X_{0}})).$

Note that by Corollary 1.6, we have $\delta_{0q}=0$.

######  Theorem 2.4.

Assume that $X$ has dimension $n|2$, and the characteristic of $k$ is zero.
Then

(i) one has $(E_{2}^{pq})_{-}=0$;

(ii) the Hodge to de Rham spectral sequence of $X$ degenerates at $E_{2}$;

(iii) one has

$\dim E_{2}^{pq}=h^{pq}(X_{0})+\delta_{p+1,q-1}(X)-\delta_{p,q}(X).$

The proof will be based on the following result, where we use the filtration
$F^{\bullet}\Omega^{\bullet}_{X}$ introduced in the proof of Theorem 1.1.

######  Proposition 2.5.

Assume that $X$ has dimension $n|2$, and the characteristic of $k$ is zero.
Then the complexes $(\Omega^{\bullet}_{X})_{-}$ and
$(F^{2}\Omega^{\bullet}_{X})_{+}$ are homotopic to zero.

######  Lemma 2.6.

Assume we have a commutative diagram with exact rows in some abelian category,
Assume that $f_{1}$ and $f_{3}$ are split injections and the second row is
split exact, then $f_{2}$ is also a split injection.

The proof is straightforward and is left to the reader.

Below we will write $(\Omega^{\bullet}_{X})_{\pm}=\Omega^{\bullet}_{\pm}$ for
brevity. For $k\geq 2$, let us set

$G^{k}\Omega^{n}_{\pm}:=({\mathcal{N}}^{2}F^{k-2}\Omega^{n})_{\pm}+F^{k+2}\Omega^{n}_{\pm},$

where the parity of $k$ is the same as the parity of the space. For $k=1$, we
set $G^{1}\Omega^{n}_{-}=F^{3}\Omega^{n}_{-}$.

######  Lemma 2.7.

(i) For every $k\geq 1$ and every $n$, the following sequence is exact:

$0\to G^{k}\Omega^{n}_{\pm}/d(F^{k+2}\Omega^{n-1}_{\pm})\to
F^{k}\Omega^{n}_{\pm}/d(F^{k}\Omega^{n-1}_{\pm})\rTo{\delta}\Omega^{n-k+1}_{X_{0}}\otimes
S^{k}({\mathcal{N}}/{\mathcal{N}}^{2})\to 0,$ (2.1)

where we use $+$ for even $k$ (resp., $-$ for odd $k$), and $\delta$ is given
by

$\delta(\alpha_{n-k+1}\cdot n_{1}dn_{2}\cdot\ldots\cdot
dn_{k})=\alpha_{n-k+1}|_{X_{0}}\otimes(\overline{n}_{1}\cdot\overline{n}_{2}\cdot\ldots\cdot\overline{n}_{k}),$

with $\alpha_{n-k+1}\in\Omega^{n-k+1}$.

(ii) For every $k\geq 1$, the following map induced by $d$ is a split
injection:

$F^{k}\Omega^{n}_{\pm}/d(F^{k}\Omega^{n-1}_{\pm})\to
F^{k}\Omega^{n+1}_{\pm}/{\mathcal{N}}^{2}\cdot F^{k-2}\Omega^{n+1}_{\pm},$
(2.2)

where for $k=1$ we replace ${\mathcal{N}}^{2}F^{-1}\Omega^{n+1}_{-}$ by zero,
and we use $+$ for even $k$ (resp., $-$ for odd $k$),

###### Proof.

We assume that $k\geq 1$ is odd. The case of even $k\geq 2$ is completely
analogous.

(i) We use the induction on $n\geq-1$ (for $n=-1$ the assertion is clear). Let
us write $S^{i}=S^{i}({\mathcal{N}}/{\mathcal{N}}^{2})$ for brevity.

Consider the commutative diagram with exact rows (where the first row is exact
by the induction assumption) It shows that the natural embedding induces an
isomorphism

$({\mathcal{N}}\cdot
F^{k-1}\Omega^{n}_{+}+G^{k}\Omega^{n}_{-})/d(G^{k}\Omega^{n-1}_{-})\rTo{\sim}F^{k}\Omega^{n}_{-}/d(F^{k}\Omega^{n-1}_{-}).$
(2.3)

Next, we have a commutative diagram with exact rows, where $\kappa$ is the
Koszul differential. Since $\kappa$ is injective, we deduce exactness of the
sequence of the cokernels of the vertical maps. Taking into account the
isomorphism (2.3), we get exactness of (2.1).

(ii) We use the decreasing induction on $k$. Note that
$F^{n+3}\Omega^{n}_{-}=0$, so the base of the induction is clear.

Assume first that $k\geq 3$. Consider the commutative diagram with exact rows,

$\begin{diagram}$ (2.4)

Using Lemma 2.6 (with $f_{3}=\operatorname{id}$), we see that it is enough to
prove that the left vertical arrow is a split injection. But the same map
appears as the middle vertical arrow in the commutative diagram with exact
rows,

$\begin{diagram}$ (2.5)

The leftmost vertical arrow is split injective by the induction assumption,
and the rightmost vertical arrow is split injective since the Koszul
differential $\kappa$ is so. By Lemma 2.6, it remains to prove that the second
row is split exact. But the splitting is induced by the well defined map

$\displaystyle\Omega^{n-k+2}_{X_{0}}\otimes{\mathcal{N}}/{\mathcal{N}}^{2}\otimes
S^{k-1}\to{\mathcal{N}}\cdot
F^{k-1}\Omega^{n}_{+}/{\mathcal{N}}^{2}F^{k}\Omega^{n}_{-}:$ $\displaystyle
df_{1}\ldots df_{n-k+2}\otimes n_{1}\otimes(n_{2}\ldots n_{k})\mapsto
d\widetilde{f}_{1}\ldots d\widetilde{f}_{n-k+2}n_{1}dn_{2}\ldots dn_{k},$

where $\widetilde{f}_{i}\in({\mathcal{O}}_{X})_{+}$ are liftings of
$f_{i}\in{\mathcal{O}}_{X_{0}}$.

In the case $k=1$, we modify the above argument as follows: we replace
${\mathcal{N}}^{2}F^{-1}\Omega^{n+1}_{-}$ by zero in the diagram (2.4) and
replace the diagram (2.5) by As before, we see that the bottom row is split
exact, and the leftmost vertical arrow is split injective by the induction
assumption. Hence, the middle vertical arrow is split exact, which implies the
assertion. ∎

###### Proof of Proposition 2.5.

For $k=1$ (resp., $k=2$), Lemma 2.7(ii) implies that the map
$\Omega^{n}_{-}/d\Omega^{n-1}_{-}\to\Omega^{n+1}_{-}$ (resp.,
$F^{2}\Omega^{n}_{+}/dF^{2}\Omega^{n-1}_{+}\to F^{2}\Omega^{n+1}_{+}$) is a
split injection, which immediately gives the result. ∎

###### Proof of Theorem 2.4.

Part (i) immediately follows from Proposition 2.5. To prove parts (ii) and
(iii), it is enough to prove the inequalities

$\dim E_{2}^{pq}=\dim(E_{2}^{pq})_{+}\leq
h^{pq}(X_{0})+\delta_{p+1,q-1}(X)-\delta_{p,q}(X).$

Indeed, summing up these inequalities, we would get

$\dim E_{2}(X)\leq\dim H_{dR}(X_{0})=\dim E_{\infty}(X),$

hence the spectral sequence degenerates at $E_{2}$ and the above inequalites
are in fact equalities.

Since the natural map $(E_{2}^{pq})_{+}\to H^{q}(\Omega^{p}_{X_{0}})$ factors
through a quotient of $H^{q}(\Omega^{p}_{X})$, it has corank
$\geq\delta_{p,q}(X)$. Thus, it suffices to show that

$\dim\ker((E_{2}^{pq})_{+}\to
H^{q}(\Omega^{p}_{X_{0}}))\leq\delta_{p+1,q-1}(X).$

The exact sequence

$0\to F^{2}(\Omega^{p}_{X})_{+}\to(\Omega^{p}_{X})_{+}\to\Omega^{p}_{X_{0}}\to
0$

gives rise to the surjection

$K:=\ker(H^{q}(F^{2}(\Omega^{p}_{X})_{+})\to
H^{q}((\Omega^{p+1}_{X})_{+}))/\operatorname{im}H^{q}((F^{2}\Omega^{p-1}_{X})_{+})\to\ker((E_{2}^{pq})_{+}\to
H^{q}(\Omega^{p}_{X_{0}})).$

Let us consider the morphism of exact sequences Using exactness of the
sequence

$H^{q}(F^{2}(\Omega^{p-1}_{X})_{+})\to H^{q}(F^{2}(\Omega^{p}_{X})_{+})\to
H^{q}(F^{2}(\Omega^{p+1}_{X})_{+}),$

which follows from Proposition 2.5, we get an embedding

$K\hookrightarrow\ker(H^{q}(F^{2}(\Omega^{p+1}_{X})_{+})\to
H^{q}((\Omega^{p+1}_{X})_{+}))\simeq\operatorname{coker}(H^{q-1}((\Omega^{p+1}_{X})_{+})\to
H^{q-1}(\Omega^{p+1}_{X_{0}}))$

so $\dim K\leq\delta_{2,q-1}(X)$. ∎

######  Corollary 2.8.

If $X$ has dimension $n|2$ then $E_{2}^{pq}=0$ for $p>n$ (and clearly for
$q>n$).

Our next goal is to get a lower bound on $\delta_{1q}(X)$ (see Proposition
2.11 below). Set $L:={\mathcal{N}}^{2}$ viewed as a line bundle on $X_{0}$.
Let $Y$ denote the even scheme with the same underlying space as $X_{0}$ and
with the structure sheaf ${\mathcal{O}}_{Y}=({\mathcal{O}}_{X})_{+}$. Then
${\mathcal{O}}_{Y}$ is a square zero extension of ${\mathcal{O}}_{X_{0}}$ by
$L$, so it corresponds to a class $e\in H^{1}(T_{X_{0}}\otimes L)$.

######  Lemma 2.9.

One has a natural exact sequence of coherent sheaves on $X_{0}$,

$0\to L\rTo{\iota}\Omega^{1}_{Y}|_{X_{0}}\to\Omega^{1}_{X_{0}}\to 0$ (2.6)

and the corresponding extension class corresponds to $e$ under the natural
isomorphism $\operatorname{Ext}^{1}(\Omega^{1}_{X_{0}},L)\simeq
H^{1}(T_{X_{0}}\otimes L)$.

###### Proof.

This exact sequence is standard: its exactness on the left follows from the
fact that locally $Y$ is a Cartier divisor in $X_{0}\times{\mathbb{A}}^{1}$.
Let $(U_{i})$ be an open affine covering of $X_{0}$ and let
$\sigma_{i}:{\mathcal{O}}_{X_{0}}\to{\mathcal{O}}_{Y}$ be splittings defined
over $U_{i}$, so that the class of $e$ is represented by the cocylce
$v_{ij}\in T_{X_{0}}\otimes L(U_{ij})$, such that

$(\sigma_{j}-\sigma_{i})(f)=\iota\langle v_{ij},df\rangle.$

We have the induced splittings over $U_{i}$,

$d\sigma_{i}:\Omega^{1}_{X_{0}}\to\Omega^{1}_{Y}|_{X_{0}}:fdg\mapsto
fd\sigma_{i}(g)|_{X_{0}}.$

Now the cocycle $d\sigma_{j}-d\sigma_{i}\in(\Omega^{1})^{\vee}\otimes
L(U_{ij})$ represents the class of the extension (2.6). It remains to observe
that

$\langle
d\sigma_{j}-d\sigma_{i},dg\rangle=d(\sigma_{j}(g)-\sigma_{i}(g))=\iota\langle
v_{ij},df\rangle,$

so this cocycle coincides with $v_{ij}$. ∎

######  Lemma 2.10.

(i) One has a canonical decomposition

$(\Omega^{1}_{X})_{+}/({\mathcal{N}}^{2}\otimes\Omega^{1}_{X_{0}})\simeq\Omega^{1}_{Y}|_{X_{0}}\oplus
S^{2}({\mathcal{N}}/{\mathcal{N}}^{2}).$

(ii) For any $q\geq 0$, one has an embedding

$\operatorname{im}(H^{q}(\Omega^{1}_{X})\to
H^{q}(\Omega^{1}_{X_{0}}))\subset\operatorname{im}(H^{q}(\Omega^{1}_{Y}|_{X_{0}})\to
H^{q}(\Omega^{1}_{X_{0}})).$

###### Proof.

(i) The pull-back map $\Omega^{1}_{Y}\to(\Omega^{1}_{X})_{+}$ induces an
${\mathcal{O}}_{X_{0}}$-linear map
$\Omega^{1}_{Y}|_{X_{0}}\to(\Omega^{1}_{X})_{+}/({\mathcal{N}}^{2}\otimes\Omega^{1}_{X_{0}})$.
On the other hand, the map

${\mathcal{N}}_{-}\otimes{\mathcal{N}}_{-}\to(\Omega^{1}_{X})_{+}:a\otimes
b\mapsto adb+bda$

induces an ${\mathcal{O}}_{X_{0}}$-linear map
$S^{2}({\mathcal{N}}/{\mathcal{N}}^{2})\to(\Omega^{1}_{X})_{+}/({\mathcal{N}}^{2}\otimes\Omega^{1}_{X_{0}})$.
One can check using local coordinates that these maps are embeddings and give
the claimed decomposition.

(ii) Since the map $(\Omega^{1}_{X})_{+}\to\Omega^{1}_{X_{0}}$ factors through
the quotient
$(\Omega^{1}_{X})_{+}/({\mathcal{N}}^{2}\otimes\Omega^{1}_{X_{0}})$, this
follows from the decomposition proved in part (i). ∎

######  Proposition 2.11.

Consider the cup-product map

$\mu^{q}_{e}:H^{q}(\Omega^{1}_{X_{0}})\rTo{\cup e}H^{q+1}(L).$

Then one has

$\delta_{1q}(X)\geq\operatorname{rk}(\mu^{q}_{e}).$

###### Proof.

By Lemma 2.10(ii),

$\delta_{1q}(X)\geq\dim\operatorname{coker}(H^{q}(\Omega^{1}_{Y}|_{X_{0}})\to
H^{q}(\Omega^{1}_{X_{0}})).$

But by Lemma 2.9, we have an identification

$\operatorname{coker}(H^{q}(\Omega^{1}_{Y}|_{X_{0}})\to
H^{q}(\Omega^{1}_{X_{0}}))\simeq\operatorname{im}(\mu^{q}_{e}).$

∎

######  Corollary 2.12.

Assume that $X_{0}$ is a smooth projective curve of genus $g\geq 1$, and
$L\simeq\omega_{X_{0}}$. Then $\delta_{10}(X)\geq 1$.

######  Example 2.13.

Consider $X=G(1|1,2|2)$. Then $X_{0}={\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$,
and ${\mathcal{N}}/{\mathcal{N}}^{2}\simeq{\mathcal{O}}(-1,-1)^{\oplus 2}$,
${\mathcal{N}}^{2}\simeq\omega_{X_{0}}={\mathcal{O}}(-2,-2)$. By Proposition
2.11, we have

$\delta_{11}(X)\geq\operatorname{rk}(\mu^{1}_{e}:H^{1}(\Omega^{1}_{X_{0}})\to
H^{2}(\Omega^{2}_{X_{0}})).$

It is well known that $e\in H^{1}(T_{X_{0}}\otimes\omega_{X_{0}})\simeq
H^{1}(\Omega^{1}_{X_{0}})$ is nonzero, hence $\mu^{1}_{e}\neq 0$, and we get
$\delta_{11}(X)\geq 1$. On the other hand, let ${\mathcal{B}}$ denote the
Berezinian of the tautological bundle on $X=G(1|1,2|2)$. Then
$c_{1}({\mathcal{B}})$ gives an element of $H^{1}(\Omega^{1}_{X})$ projecting
to a nonzero class in $H^{1}(\Omega^{1}_{X_{0}})$ (see e.g., [8, Ch. 4.3]).
Hence, we have $\delta_{11}(X)=1$, and so

$\dim E_{2}^{11}(X)=\dim E_{2}^{02}(X)=1.$

In fact, it is easy to see that $H^{i}(\Omega_{X}^{p})=0$ for $p\geq 2$ and
$i=0,1$, which implies that the only nonzero $E_{2}$ terms are

$E_{2}^{00}=E_{2}^{02}=E_{2}^{11}=E_{2}^{22}=k.$

###  2.3. Example in dimension $1|3$

We will give an example showing that in dimension $1|3$ the map on $H^{1}$
induced by $d:({\mathcal{O}}_{X})_{-}\to(\Omega^{1}_{X})_{-}$ may have a
nontrivial kernel (see Theorem 2.15 below).

We start with a general construction of smooth supervarieties of dimension
$1|3$.

######  Lemma 2.14.

Let $C$ be a smooth curve, $V$ a vector bundle of rank $3$ over $C$,

$0\to{\bigwedge}^{2}V\to{\mathcal{O}}_{Y}\to{\mathcal{O}}_{C}\to 0$ (2.7)

a square zero extension. Then there exists a smooth supervariety $X$ of
dimension $1|3$ with $X_{0}=C$, ${\mathcal{N}}/{\mathcal{N}}^{2}=V$, and
$({\mathcal{O}}_{X})_{+}\simeq{\mathcal{O}}_{Y}$ as a square zero extension of
${\mathcal{O}}_{C}$.

###### Proof.

Let $e\in H^{1}(T_{C}\otimes{\bigwedge}^{2}V)$ be the class of the extension
(2.7). Note that we have a natural isomorphism

${\bigwedge}^{2}V\simeq
V^{\vee}\otimes\det(V)\simeq\underline{\operatorname{Hom}}(V,\det(V)).$

Let $D_{\leq 1}(V,\det(V))$ denote the sheaf of differential operators of
order $\leq 1$ from $V$ to $\det(V)$. Then we have an exact sequence

$0\to\underline{\operatorname{Hom}}(V,\det(V))\to D_{\leq
1}(V,\det(V))\rTo{\sigma}T_{C}\otimes\underline{\operatorname{Hom}}(V,\det(V))\to
0,$

where $\sigma$ is the symbol map. Since
$H^{2}(C,\underline{\operatorname{Hom}}(V,\det(V)))=0$, we can lift $e$ to a
class $\widetilde{e}\in H^{1}(C,D_{\leq 1}(V,\det(V)))$. Let us represent
$\widetilde{e}$ by a Cech cocycle with respect to an affine covering
$C=(U_{1},U_{2})$, $\phi_{12}:V|_{U_{12}}\to\det(V)|_{U_{12}}$. By assumption,
it satisfies

$\phi_{12}(fs)=f\phi_{12}(s)+v_{12}(f)\wedge s,$

for $f\in{\mathcal{O}}$, $s\in V$, where $v_{12}\in
H^{0}(U_{12},T_{C}\otimes{\bigwedge}^{2}V)$ is the Cech cocycle representing
$e$.

Now we are going to glue $X$ from the split supervarieties over $U_{i}$,
$i=1,2$, corresponding to $\bigwedge^{\bullet}(V|_{U_{i}})$. We just need to
construct an automorphism $\alpha$ of the sheaf of
$\operatorname{\mathbb{Z}}/2$-graded rings $\bigwedge^{\bullet}(V|_{U_{12}})$
inducing the identity modulo $\bigwedge^{\geq 1}$. It is enough to specify how
$\alpha$ acts on ${\mathcal{O}}$ and on $V$. We set for $f\in{\mathcal{O}}$,
$s\in V$,

$\alpha(f)=f+v_{12}(f),\ \ \alpha(s)=s+\phi_{12}(s).$

It is easy to check that this gives a well defined automorphism of
$\bigwedge^{\bullet}(V|_{U_{12}})$, and thus, defines the supervariety $X$
with the required properties. ∎

######  Theorem 2.15.

Assume that the ground field $k$ is algebraically closed field of
characteristic $\neq 2,3$. For any smooth projective curve $C$ of genus $2$
over $k$, there exists a smooth supervariety $X$ of dimension $1|3$ with
$X_{0}=C$, and such that

$E_{2}^{01}(X)_{-}=\ker(H^{1}(X,({\mathcal{O}}_{X})_{-})\to
H^{1}(X,(\Omega^{1}_{X})_{-}))\neq 0.$

In particular, the Hodge to de Rham spectral sequence of $X$ does not
degenerate at $E_{2}$.

We start with some auxiliary statements.

######  Lemma 2.16.

Assume the characteristic of $k$ is $\neq 2,3$. For a $k$-vector space $V$,
the linear map

$\tau_{V}:V\otimes{\bigwedge}^{2}V\to
V\otimes{\bigwedge}^{2}V:v_{1}\otimes(v_{2}\wedge v_{3})\mapsto
v_{3}\otimes(v_{1}\wedge v_{2})-v_{2}\otimes(v_{1}\wedge v_{3})$

is invertible.

###### Proof.

Let us consider the projector $\pi$ on $V\otimes{\bigwedge}^{2}V$ (onto a
subspace isomorphic to ${\bigwedge}^{3}V$) given by

$v_{1}\otimes(v_{2}\wedge
v_{3})\mapsto\frac{1}{3}\bigl{(}v_{1}\otimes(v_{2}\wedge
v_{3})+v_{2}\otimes(v_{3}\wedge v_{1})+v_{3}\otimes(v_{1}\wedge
v_{2})\bigr{)}.$

Then we have $\tau_{V}=3\pi-\operatorname{id}$, so it has eigenvalues $2$ and
$-1$. ∎

######  Lemma 2.17.

Assume the characteristic of $k$ is $\neq 2,3$. Let $C$ be a smooth projective
curve over $k$, and let $V$ be a rank $3$ vector bundle on $C$ such that

$H^{1}(\underline{\operatorname{End}}_{0}(V)\otimes\det(V))=H^{0}(V)=H^{0}(\det(V)^{-1})=0,$

while $H^{0}(\omega_{C}\otimes V)\neq 0$ and $H^{1}(\det(V))\neq 0$. Then
there exists a smooth supervariety $X$ of dimension $1|3$ with $X_{0}=C$, and
${\mathcal{N}}/{\mathcal{N}}^{2}=V$, such that $E_{2}^{01}(X)_{-}\neq 0$.

###### Proof.

By Lemma 2.14, for any class $e\in H^{1}(T_{C}\otimes\bigwedge^{2}V)$, there
exists a supervariety $X$ of dimension $1|3$ with $X_{0}=C$, and
${\mathcal{N}}/{\mathcal{N}}^{2}=V$, such that the extension class of
$({\mathcal{O}}_{X})_{+}\to{\mathcal{O}}_{C}$ is $e$. We will show that there
exists a choice of $e$ such that $E_{2}^{01}(X)_{-}\neq 0$.

The exact sequence

$0\to{\mathcal{N}}^{3}\to{\mathcal{O}}_{-}\to V\to 0$

shows that the map $H^{1}({\mathcal{N}}^{3})\to H^{1}({\mathcal{O}}_{-})$ is
injective. Thus, it is enough to prove that the map
$H^{1}({\mathcal{N}}^{3})\to H^{1}(({\mathcal{N}}\Omega^{1})_{-})$ has a
nontrivial kernel. Since
${\mathcal{N}}^{3}\Omega^{1}_{X_{0}}\simeq\det(V)\otimes\omega_{C}$ has
trivial $H^{1}$ (which is dual to $H^{0}(\det(V)^{-1})=0$), it is enough to
prove that the map

$H^{1}({\mathcal{N}}^{3})\to
H^{1}(({\mathcal{N}}\Omega^{1})_{-}/{\mathcal{N}}^{3}\Omega^{1}_{X_{0}})$

has a nontrivial kernel.

We have an exact sequence of ${\mathcal{O}}_{X_{0}}$-modules

$0\to{\mathcal{N}}^{2}d{\mathcal{N}}\to({\mathcal{N}}\Omega^{1})_{-}/{\mathcal{N}}^{3}\Omega^{1}_{X_{0}}\to{\mathcal{N}}/{\mathcal{N}}^{2}\otimes\Omega^{1}_{X_{0}}\to
0.$ (2.8)

Note that if $\sigma_{i}:{\mathcal{O}}_{X_{0}}\to({\mathcal{O}}_{X})_{+}$ are
local splittings of the extension
$({\mathcal{O}}_{X})_{+}\to{\mathcal{O}}_{C}$ then we have induced splittings
of (2.8) given by $n\otimes df\mapsto\widetilde{n}\cdot d\sigma_{i}(f)$ (where
$\widetilde{n}\in{\mathcal{N}}_{-}$ is a lifting of
$n\in{\mathcal{N}}/{\mathcal{N}}^{2}$). The difference between two such
splittings is given by the $1$-cocycle $n\otimes df\mapsto\widetilde{n}\cdot
d\langle v_{ij},df\rangle$, where $v_{ij}$ is the $1$-cocycle representing
$e\in H^{1}(T_{X_{0}}\otimes{\bigwedge}^{2}V)$. Using the Leibnitz rule we
deduce that the extension class of (2.8) in
$\operatorname{Ext}^{1}(V\otimes\Omega^{1}_{X_{0}},V\otimes\bigwedge^{2}V)$ is
equal to the image of $e$ under the natural map

$H^{1}(T_{X_{0}}\otimes{\bigwedge}^{2}V)\simeq\operatorname{Ext}^{1}(\Omega^{1}_{X_{0}},{\bigwedge}^{2}V)\to\operatorname{Ext}^{1}(V\otimes\Omega^{1}_{X_{0}},V\otimes{\bigwedge}^{2}V)\rTo{\tau_{V}\circ}\operatorname{Ext}^{1}(V\otimes\Omega^{1}_{X_{0}},V\otimes{\bigwedge}^{2}V),$

where the last arrow is induced by the automorphism $\tau_{V}$ (see Lemma
2.16).

We have an isomorphism $\bigwedge^{2}V\simeq V^{\vee}\otimes\det(V)$, so we
have a direct sum decomposition

$V\otimes{\bigwedge}^{2}V\simeq V\otimes
V^{\vee}\otimes\det(V)\simeq\det(V)\oplus\underline{\operatorname{End}}_{0}(V)\otimes\det(V).$

By the assumption,
$H^{1}(\underline{\operatorname{End}}_{0}(V)\otimes\det(V))=0$, so the map

$H^{1}({\mathcal{N}}^{3})\simeq H^{1}(\det(V))\to
H^{1}(V\otimes{\bigwedge}^{2}V)$

is an isomorphism. Thus, it is enough to check that the connecting
homomorphism

$H^{0}(\omega_{C}\otimes V)\to H^{1}(V\otimes{\bigwedge}^{2}V)$

associated with (2.8) is nonzero. Since the projection
$V\otimes\bigwedge^{2}V\to\det(V)$ induces an isomorphism on $H^{1}$, this is
equivalent to showing that the homomorphism

$H^{0}(\omega_{C}\otimes V)\rTo{\cup e}H^{1}(\det(V))$

is nonzero, where we view $e$ as an element of $H^{1}(\omega_{C}^{-1}\otimes
V^{\vee}\otimes\det(V))$. Thus, we need to choose $e$ such that this cup
product map is nonzero.

This is possible as long as the map

$H^{0}(\omega_{C}\otimes V)\otimes H^{1}(\omega_{C}^{-1}\otimes
V^{\vee}\otimes\det(V))\to H^{1}(\det(V))$

is nonzero. By Serre duality, we need to check that the map

$H^{0}(\omega_{C}\otimes V)\otimes H^{0}(\omega_{C}\otimes\det(V)^{-1})\to
H^{0}(\omega_{C}^{2}\otimes V\otimes\det(V)^{-1})$

is nonzero. By assumption, there exists a nonzero element $s\in
H^{0}(\omega_{C}\otimes\det(V)^{-1})$ (this space is dual to
$H^{1}(\det(V))\neq 0$). The multiplication by $s$ is an injective map

$H^{0}(\omega_{C}\otimes V)\rTo{s}H^{0}(\omega_{C}^{2}\otimes
V\otimes\det(V)^{-1}).$

Hence, our assertion follows from non-vanishing of $H^{0}(\omega_{C}\otimes
V)$. ∎

We will need some facts about analogs of Prym varieties associated with cyclic
coverings (where we assume that the characteristic is $\neq 3$).

######  Lemma 2.18.

Let $\pi:\widetilde{C}\to C$ be a cyclic covering associated with a line
bundle $\xi$ of order $3$ on a smooth projective curve $C$, and let
$\tau:\widetilde{C}\to\widetilde{C}$ be an action of a nontrivial element of
the Galois group $\operatorname{\mathbb{Z}}/3\operatorname{\mathbb{Z}}$.

(i) The morphisms $\pi^{*}:J_{C}\to J_{\widetilde{C}}$ and
$\operatorname{Nm}_{\widetilde{C}/C}:J_{\widetilde{C}}\to J_{C}$ are dual to
each other with respect to the canonical principal polarizations on the
Jacobians. One has

$\operatorname{Nm}_{\widetilde{C}/C}\circ\pi^{*}=[3]_{J_{C}},\ \
\pi^{*}\circ\operatorname{Nm}_{\widetilde{C}/C}=\tau_{*}+\tau^{-1}_{*}+\operatorname{id}_{J_{\widetilde{C}}}.$

(ii) One has $\ker(\pi^{*})=\langle\xi\rangle\subset J_{C}$, the subgroup of
order $3$ generated by $\xi$.

(iii) Set

$\widetilde{P}=\ker(\operatorname{Nm}_{\widetilde{C}/C}:J_{\widetilde{C}}\to
C),$

and let $P\subset\widetilde{P}$ denote the connected component of $0$ in
$\widetilde{P}$ ($P$ is an analog of the Prym variety). Then
$P=(\tau-\operatorname{id})(J_{\widetilde{C}})$, $\tau-\operatorname{id}:P\to
P$ is an isogeny, and $|\widetilde{P}/P|=3$.

###### Proof.

(i) This is well known (see [1, Sec. 11.4]).

(ii) Recall that
$\pi_{*}{\mathcal{O}}_{\widetilde{C}}\simeq{\mathcal{O}}_{C}\oplus\xi\oplus\xi^{-1}$.
Hence,

$h^{0}(\pi^{*}\xi)=h^{0}(\xi\otimes\pi_{*}{\mathcal{O}})=h^{0}(\pi_{*}{\mathcal{O}})=1,$

so $\pi^{*}\xi$ is a degree $0$ line bundle on $\widetilde{C}$ with a nonzero
section, hence $\pi^{*}\xi\simeq{\mathcal{O}}_{\widetilde{C}}$.

Conversely, assume $\pi^{*}M\simeq{\mathcal{O}}$. Then

$M\otimes\pi_{*}{\mathcal{O}}\simeq\pi_{*}\pi^{*}M\simeq\pi_{*}{\mathcal{O}},$

so $M$ is isomorphic to one of the summands in $\pi_{*}{\mathcal{O}}$.

(iii) Set $A:=J_{\widetilde{C}}/P$. We have an exact sequence of abelian
varieties

$0\to P\to J_{\widetilde{C}}\rTo{p}A\to 0,$ (2.9)

where $p$ is induced by $\operatorname{Nm}_{\widetilde{C}/C}$, and an isogeny
$f:A\to J_{C}$, such that $\ker(f)\simeq\widetilde{P}/P$ and
$\operatorname{Nm}_{\widetilde{C}/C}=f\circ p$. Consider the dual isogeny
$\hat{f}:J_{C}\to\hat{A}$, and the dual sequence to (2.9),

$0\to\hat{A}\to J_{\widetilde{C}}\to\hat{P}\to 0.$

Then the morphism $\pi^{*}:J_{C}\to J_{\widetilde{C}}$, being dual to
$\operatorname{Nm}_{\widetilde{C}/C}=f\circ p$, factors as the composition of
$\hat{f}$ followed by the embedding $\hat{A}\to J_{\widetilde{C}}$. Hence, we
have

$\ker(\hat{f})=\ker(\pi^{*})=\langle\xi\rangle.$

But $\ker(f)$ is Cartier dual to $\ker(\hat{f})$, so we deduce

$|\widetilde{P}/P|=|\ker(f)|=|\ker(\hat{f})|=3.$

Note that
$\operatorname{Nm}_{\widetilde{C}/C}\circ\tau_{*}=\operatorname{Nm}_{\widetilde{C}/C}$,
so $(\tau_{*}-\operatorname{id})(J_{\widetilde{C}})$ is contained in
$\widetilde{P}$, and hence, in $P$. It remains to show that the morphism
$(\tau_{*}-\operatorname{id}):P\to P$ is an isogeny. It is enough to show that
the induced map on the tangent space $T_{0}P$ is nondegenerate. We have

$T_{0}P\simeq\ker(H^{1}(\widetilde{C},{\mathcal{O}}_{\widetilde{C}})\rTo{\operatorname{tr}}H^{1}(C,{\mathcal{O}}_{C})).$

The decomposition
$\pi_{*}{\mathcal{O}}_{\widetilde{C}}={\mathcal{O}}_{C}\oplus\xi\oplus\xi^{-1}$
is compatible with the action of $\tau$: it corresponds to the eigenvalues
$1$, $\zeta$ and $\zeta^{-1}$, where $\zeta$ is a primitive $3$rd root of
unity. Hence $\tau$ acts on $T_{0}P$ with eigenvalues $\zeta$ and
$\zeta^{-1}$, which implies that $\tau-\operatorname{id}$ is nondegenerate. ∎

###### Proof of Theorem 2.15.

It is enough to construct a rank $3$ vector bundle $V$ on $C$ satisfying the
assumptions of Lemma 2.17.

Step 1. A triple covering and a special line bundle of order $3$. We start by
choosing a line bundle $\xi$ of order $3$ on $C$. Let $\pi:\widetilde{C}\to C$
be the corresponding cyclic covering (where $\widetilde{C}$ has genus $4$). We
will use the notations and the results of Lemma 2.18. We claim that there
exists a line bundle $\eta$ of order $3$, such that
$\eta\not\in\langle\xi\rangle$, and $\pi^{*}\eta\in P$.

Indeed, since $\operatorname{Nm}_{\widetilde{C}/C}\circ\pi^{*}=[3]_{J_{C}}$,
we have a well defined homomorphism
$J_{C}[3]/\langle\xi\rangle\to\widetilde{P}$, where
$J_{C}[3]:=\ker([3]_{J_{C}})$. Let us consider the induced homomorphism of
finite groups

$J_{C}[3]/\langle\xi\rangle\to\widetilde{P}/P.$

Since $C$ has genus $2$, we have $|J_{C}[3]/\langle\xi\rangle|=27$, whereas
$|\widetilde{P}/P|=3$. Hence, there exists a nonzero element
$\eta+\langle\xi\rangle$ in the kernel, where $\eta\in
J_{C}[3]\setminus\langle\xi\rangle$. But this means that $\pi^{*}\eta\in P$,
as claimed.

Step 2. Choosing a point. We claim that for a generic point $p\in C$ one has

$H^{*}(\xi(p))=H^{*}(\xi^{-1}(p))=H^{*}(\eta(p))=H^{*}(\eta\otimes\xi(p))=H^{*}(\eta\otimes\xi^{-1}(p)).$

Indeed, for any nontrivial line bundle $M$ of degree $0$ one has
$h^{0}(\omega_{C}\otimes M)=1$, so there exists a unique effective divisor
$D_{M}$ such that $\omega_{C}\otimes M\simeq{\mathcal{O}}_{C}(D_{M})$. Now we
just need to choose $p$ not in the support of $D_{\xi}$, $D_{\xi^{-1}}$,
$D_{\eta\xi}$ and $D_{\eta\xi^{-1}}$. It follows that the line bundle
$\pi^{*}\eta\in P$ satisfies

$H^{0}(\widetilde{C},\pi^{*}\eta(\pi^{-1}(p)))=H^{0}(C,\eta(p)\oplus\eta\xi(p)\oplus\eta\xi^{-1}(p))=0.$
(2.10)

Step 3. Choosing a line bundle in $P$. We claim that there exists a line
bundle $L$ of degree $1$ on $\widetilde{C}$ such that
$\operatorname{Nm}_{\widetilde{C}/C}(L)\simeq{\mathcal{O}}_{C}(p)$,
$H^{0}(\widetilde{C},L)=0$, and

$H^{*}(\widetilde{C},\tau_{*}L\otimes
L^{-1}(\pi^{-1}(p)))=H^{*}(\widetilde{C},\tau^{-1}_{*}L\otimes
L^{-1}(\pi^{-1}(p)))=0.$

Indeed, let us look for $L$ in the form $L=M(q)$, where $p=\pi(q)$ and $M\in
P$. We claim that our conditions are satisfied for generic $M\in P$. Indeed,
let us denote by $\Theta_{p}\subset J_{\widetilde{C}}$ the theta-divisor
associated with $\pi^{-1}(p)$. We need to check that the following conditions
hold for generic $M\in P$:

  1. (1)

$H^{0}(\widetilde{C},M(q))=0$;

  2. (2)

$\phi^{\pm}(M)\not\in\Theta_{p}$, where $\phi^{\pm}=t_{\tau^{\pm
1}(q)-q}\circ(\tau^{\pm 1}_{*}-\operatorname{id})$, where $t_{\tau^{\pm
1}(q)-q}$ is the translation by ${\mathcal{O}}(\tau^{\pm 1}(q)-q)\in P$.

For (1), we observe that this condition holds for $M=\pi^{*}\eta$ by (2.10),
since $H^{0}(\pi^{*}\eta(q))\subset H^{0}(\pi^{*}\eta(\pi^{-1}(p)))$. For (2),
we observe that the maps $\phi^{\pm}:P\to P$ are surjective, hence, it is
enough to prove the existence of $M^{\prime}\in P$ such that
$M^{\prime}\not\in\Theta_{p}$. By (2.10), we can take
$M^{\prime}=\pi^{*}\eta$.

Step 4. Checking the properties of the vector bundle. Now we set
$V=\pi_{*}(L)$. We have
$\det(V)\simeq\operatorname{Nm}_{\widetilde{C}/C}(L)\simeq{\mathcal{O}}_{C}(p)$,
so

$H^{0}(\det(V)^{-1})=H^{0}({\mathcal{O}}_{C}(-p))=0,$

while $H^{1}(\det(V))=H^{1}({\mathcal{O}}_{C}(p))\neq 0$. We also have

$H^{0}(C,V)\simeq H^{0}(\widetilde{C},L)=0,$ $H^{0}(C,\omega_{C}\otimes
V)\simeq H^{0}(\widetilde{C},\pi^{*}\omega_{C}\otimes L)\simeq
H^{0}(\widetilde{C},\omega_{\widetilde{C}}\otimes L)\neq 0,$

since $\deg(L)=1$. Finally, since $\pi$ is unramified, we have an isomorphism
$V^{\vee}\simeq\pi_{*}(L^{-1})$, so

$V^{\vee}\otimes V\simeq\pi_{*}(\pi^{*}\pi_{*}(L)\otimes L^{-1}).$

Furthermore, $\pi^{*}\pi_{*}(L)\simeq L\oplus\tau^{*}L\oplus(\tau^{-1})^{*}L$.
Hence,

$V^{\vee}\otimes V\simeq\pi_{*}{\mathcal{O}}\oplus\pi_{*}(\tau^{*}L\otimes
L^{-1})\oplus\pi_{*}((\tau^{-1})^{*}L\otimes L^{-1}).$

We need to check that

$h^{1}(V^{\vee}\otimes V\otimes\det(V))=h^{1}(V^{\vee}\otimes V(p))=1.$

Since
$\pi_{*}{\mathcal{O}}\otimes{\mathcal{O}}_{C}(p)={\mathcal{O}}_{C}(p)\oplus\xi(p)\oplus\xi^{-1}(p)$,
this follows from the vanishing

$H^{1}(C,\pi_{*}((\tau^{\pm 1})^{*}L\otimes
L^{-1})(p))=H^{1}(\widetilde{C},(\tau^{\pm 1})^{*}L\otimes
L^{-1}(\pi^{-1}(p)))=0.$

∎

######  Remark 2.19.

It is instructive to observe that the construction of Theorem 2.15 breaks in
characteristic $2$ (since Lemma 2.16 needs characteristic $\neq 2$). Indeed,
by Proposition 2.3, in characteristic $2$ we have $E_{2}^{01}(X)_{-}=0$.

## References

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  * [5] G. Felder, D. Kazhdan, A. Polishchuk, Regularity of the superstring supermeasure and the superperiod map, Selecta Math. (N.S.) 28 (2022), no. 1, Paper No. 17, 64 pp.
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  * [8] Yu. I. Manin, Gauge field theory and complex geometry, Springer-Verlag, Berlin, 1988.
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# Suppression of Fast Neutrino Flavor Conversions Occurring at Large Distances
in Core-Collapse Supernovae

Sajad Abbar  Francesco Capozzi

###### Abstract

Neutrinos propagating in dense neutrino media such as core-collapse supernovae
and neutron star merger remnants can experience the so-called fast flavor
conversions on scales much shorter than those expected in vacuum. A very
generic class of fast flavor instabilities is the ones which are produced by
the backward scattering of neutrinos off the nuclei at relatively large
distances from the supernova core. In this study we demonstrate that despite
their ubiquity, such fast instabilities are unlikely to cause significant
flavor conversions if the population of neutrinos in the backward direction is
not large enough. Indeed, the scattering-induced instabilities can mostly
impact the neutrinos traveling in the backward direction, which represent only
a small fraction of neutrinos at large radii. We show that this can be
explained by the shape of the unstable flavor eigenstates, which can be
extremely peaked at the backward angles.

MPP-2021-195

IFIC/21-50

##  1 Introduction

Neutrino emission is a major effect in the most extreme astrophysical
settings, such as core-collapse supernova (CCSN) explosions. The density of
neutrinos in such environments is so high that the neutrino self-interactions
can play an important role in their flavor evolution. In particular, it leads
to a nonlinear and rich phenomenon in which the whole neutrino gas can
oscillate collectively [1, 2, 3, 4, 5, 6, 7].

Collective neutrino oscillations can influence the physics of CCSNe by
modifying the spectra of neutrinos and antineutrinos. On the one hand, they
can influence the nucleosynthesis of heavy elements by shifting the neutron-
to-proton ratio. On the other hand, they can affect the SN dynamics by
changing the neutrino energy deposition into the stalling shock wave. Apart
from these, they can also modify the SN neutrino signal detectable from a
galactic CCSN.

The first studies on collective neutrino oscillations were centered around
what we now call slow flavor conversions. Slow modes occur on scales
determined by the neutrino vacuum frequency, $\omega=\Delta m^{2}/2E$, which
are $\sim\mathcal{O}(1)$ km for a typical SN neutrino energy and atmospheric
mass splitting. However, it was then perceived that a dense neutrino gas can
also experience fast flavor conversions on extremely short scales [8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. Unlike
the slow modes, the fast conversions scale as $\sim
G_{\rm{F}}^{-1}n_{\nu}^{-1}$, with $n_{\nu}$ and $G_{\rm{F}}$ being the
neutrino number density and the Fermi coupling constant, respectively.

The necessary and sufficient condition for the occurrence of fast flavor
instabilities is that the angular distribution of the neutrino electron lepton
number (ELN) defined as,

$G_{\mathbf{v}}=\sqrt{2}G_{\mathrm{F}}\int_{0}^{\infty}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}}{(2\pi)^{3}}[f_{\nu_{e}}(E_{\nu},\mathbf{v})-f_{\bar{\nu}_{e}}(E_{\nu},\mathbf{v})],$
(1.1)

changes its sign at some directions [35]. In other words, fast instabilities
exist provided that the ELN angular distribution possesses a crossing. Here
$E_{\nu}$, $\mathbf{p}$, and $\mathbf{v}$ are the neutrino energy, momentum,
and velocity, respectively, and $f_{\nu}$’s are the neutrino occupation
numbers with $G_{\mathrm{F}}$ being the Fermi constant. Note that here we
assume
$f_{\nu_{\mu}}=f_{\nu_{\tau}}=f_{\bar{\nu}_{\mu}}=f_{\bar{\nu}_{\tau}}$,
otherwise there can exist crossings in the other leptonic channels, which can
similarly lead to fast conversion modes [41, 18].

It has been demonstrated that ELN crossings and their associated fast
instabilities can occur in different SN regions via different mechanisms. The
deepest region where fast instabilities can appear is within the convective
layer of the porto-neutron star (PNS), well below the neutrinospheres of
neutrinos and antineutrinos. These instabilities occur due to a shallow
crossing between the angular distributions of $\nu_{e}$ and $\bar{\nu}_{e}$,
which are almost isotropic [47, 48, 49]. As a matter of fact, large amplitude
modulations in the spatial distributions of $\nu_{e}$ and $\bar{\nu}_{e}$
number densities can be caused by the strong convective activities. This
implies that deep SN zones can always exist for which the number densities of
$\nu_{e}$ and $\bar{\nu}_{e}$ can be extremely close to each other and an ELN
crossing can occur regardless of the isotropic nature of neutrino angular
distributions. Though fascinating, the nearly equal distributions of neutrinos
and antineutrinos of all flavors prevents a significant amount of flavor
conversion resulted from such fast instabilities.

ELN crossings and their associated fast instabilities can also occur within
the neutrino decoupling region, just above the PNS [50, 51, 48, 36]. In the SN
zones above the neutrinospheres, the angular distributions of $\bar{\nu}_{e}$
are more peaked in the forward direction than those of $\nu_{e}$, due to their
decoupling at smaller radii. This leads to a high chance of the occurrence of
ELN crossings in the zones where $\nu_{e}-\bar{\nu}_{e}$ asymmetry is small
[50, 52]. Given this observation, the asymmetry in the neutrino emission
caused by LESA (lepton-emission self-sustained asymmetry) [53, 54] plays a
pivotal role in this context (see also Ref. [55] for other relevant effects).

Finally, a very generic class of ELN crossings and fast instabilities can
occur farther away from the SN core. Though they were first discovered in the
pre-shock SN region [56], it was then understood that they can occur in the
post-shock region, as well [40, 41]. This type of ELN crossings is generated
by the backward scatterings of neutrinos off heavy nuclei or nucleons.
Generally speaking, $\bar{\nu}_{e}$ has a larger cross section than
${\nu}_{e}$. This implies that at larger distances where the backward
direction is almost empty, $\bar{\nu}_{e}$ can become the dominant type due to
the higher scattering rate and an ELN crossing can occur.

In the present work, we study neutrino flavor conversions due to such
scattering-induced fast instabilities. Despite being ubiquitous at large
distances from the SN core, we show that they can not, in general, lead to
significant neutrino flavor conversions if the population of neutrinos in the
backward direction is not large enough. Our results are consistent with the
ones provided in Ref. [57], which appeared while this work was under
preparation. However, we here take a step forward and give an explanation of
this observation based on the shape of unstable eigenstates in the linear
regime. We also provide a rough estimate of the depth of the ELN crossing
below which significant flavor conversions are suppressed.

The paper is organized as follows. In Sec. 2, we introduce our model and
present our results of the linear stability analysis and we develop some
useful insights regarding the form of the unstable wave functions. We then
discuss neutrino flavor evolution in the nonlinear regime before the
conclusion.

##  2 Time-dependent one-dimensional neutrino gas model

In this section, we discuss our time-dependent one-dimensional (1D) neutrino
gas and we present our results in the linear and nonlinear regimes,
respectively.

We consider a time-dependent gas of neutrinos, whose conditions are axially
symmetric around the $z$ axis. We set to zero the neutrino mass splittings so
that the evolution of the system is energy-independent. We also neglect the
matter potential due to neutrino coherent forward scattering on electrons and
also neutrino inelastic collisions. Under these conditions, the flavor
evolution of the neutrino gas is governed by the following Liouville-von
Neumann equation [58, 59, 60, 61, 62] ($c=\hbar=1$)

$i(\partial_{t}+v\partial_{z})\varrho_{v}=\left[\mathsf{H}_{\nu\nu}^{v},\varrho_{v}\right]\,,$
(2.1)

where $\varrho_{v}(t,z)$ represents the usual density matrix for neutrinos
with velocity $v$ along the $z$ axis, and

$\mathsf{H}_{\nu\nu}^{v}=\sqrt{2}G_{\mathrm{F}}\int_{-1}^{+1}\\!\mathrm{d}v^{\prime}(1-v\,v^{\prime})(\varrho_{v^{\prime}}-\bar{\varrho}_{v^{\prime}}).$
(2.2)

is the potential stemming from the neutrino-neutrino forward scattering [63,
64, 65].

###  2.1 Flavor instabilities in the linear regime

In the two-flavor scenario ($e$ and $x$), the density matrix can be written as
[66],

$\displaystyle\varrho=\frac{f_{\nu_{e}}+f_{\nu_{x}}}{2}+\frac{f_{\nu_{e}}-f_{\nu_{x}}}{2}\begin{bmatrix}s&S\\\
S^{*}&-s\end{bmatrix},$ (2.3)

in the weak-interaction basis where the complex and real scalar fields $S$ and
$s$ describe the flavor coherence and the flavor conversion of the neutrino
gas, respectively.

In the linear regime before any significant flavor conversions occur, one has
$|S_{v}|\ll 1$ and $s\approx 1$, and Eq. (2.1) can be written as [66, 67, 11],

$i(\partial_{t}+v\partial_{z})S_{v}=(\epsilon_{0}+v\epsilon)S_{v}-\int\\!\mathrm{d}v^{\prime}(1-v\,v^{\prime})G_{v^{\prime}}S_{v^{\prime}},$
(2.4)

where $\epsilon_{0}=\int\\!\mathrm{d}v^{\prime}G_{v^{\prime}}$,
$\epsilon=\int\\!\mathrm{d}v^{\prime}G_{v^{\prime}}v^{\prime}$, and we have
only retained the terms linear in $S_{v}$. We look for plane wave solutions
obeying

$\displaystyle S_{v}(t,z)=Q_{v}\,e^{-i\Omega t+iKz}\,.$ (2.5)

An eigenmode of flavor conversion can be identified by its frequency and
wavevector, $\Omega$ and $K$, respectively.

The neutrino gas is stable in the flavor space and the flavor mixing amplitude
$S_{v}$ remains small unless for a real wave vector $K$, the corresponding
$\Omega$ has a positive imaginary component, i.e.
$\Omega_{\mathrm{i}}=\mathrm{Im}(\Omega)>0$, referred to as temporal
instabilities which is also the focus of this study. If such a condition is
satisfied, $S_{v}$ can grow exponentially and the neutrino medium can, in
principle, experience non-negligible flavor conversions. As discussed
previously, such instabilities exist if, and only if, the angular distribution
of ELN possesses a crossing.

Figure 1:  Neutrino angular distributions of realistic (dashed lines) and
schematic (solid lines) cases where we have shown the $\delta=10^{-3}$ and
$v_{c}=0.8$ case as an example.  Figure 2:  Upper panels: Growth rate of
temporal fast instabilities as functions of the real wave number $K$. Lower
panel: The normalized angular distributions of the eigenvectors, $Q_{v}$, of
unstable modes with maximum growth rate. Note that the amplitude of $Q_{v}$
can be extremely small in the forward direction where the majority of
neutrinos are traveling. This implies that the backward instabilities should
mainly affect the backward traveling neutrinos, which are a small fraction of
neutrinos at large radii. The left and middle panels show the results obtained
form our schematic ELN distribution (Eq. (2.6)), while the right panels are
the ones from the realistic distribution (Eq. (2.7)). For the sake of
completeness, in the left panels we also study an ELN distribution with two
crossings where the second crossing exists in the tail of the distribution,
namely for $v<-0.9$ we have a second crossing for which $G(v)=+\delta$.

In the following, we focus on two special classes of angular distributions
which mimic crossings in the backward direction. We first consider a pure
$\nu_{e}-\bar{\nu}_{e}$ neutrino gas with schematic distributions of the form:

$f_{\nu_{e}}\propto\left\\{\begin{array}[]{ll}1\quad\text{for}\quad
v>v_{c},\\\ \delta\quad\text{for}\quad
v<v_{c}.\end{array}\right.\,\,,\quad\quad
f_{\bar{\nu}_{e}}\propto\left\\{\begin{array}[]{ll}1\quad\ \text{for}\quad
v>v_{c},\\\ 2\delta\quad\text{for}\quad v<v_{c}.\end{array}\right.\,\,$ (2.6)

Here, $v_{c}$ is the point where the crossing occurs, and $\delta$ is the
depth of the backward scattering. We also consider the more realistic neutrino
angular distributions considered in Ref. [57],

$f_{\nu}\propto g_{b}+g^{\prime}_{b}(e^{v+1}-1)+g_{f}b^{v-1},$ (2.7)

where the parameters $g_{b}$, $g^{\prime}_{b}$, $g_{f}$, and $b$ can be
different for $\nu_{e}$ and $\bar{\nu}_{e}$ and are presented in Table I of
Ref. [57]. Note that we here only take the normalized distributions from Eqs.
(2.6) and (2.7) with fixed $n_{\bar{\nu}_{e}}/n_{\nu_{e}}=0.8$, though the
results are independent of this choice. The neutrino distributions are
indicated in Fig. 1

In the upper panels of Fig. 2, we indicate the growth rate of temporal fast
instabilities, $\Omega_{\mathrm{i}}$, for a number of ELN distributions
introduced in Eqs. (2.6) and (2.7). Here, we use

$\mu=\sqrt{2}G_{\rm{F}}n_{\nu_{e}},$ (2.8)

as a measure of the strength of the weak interactions. Although ELN crossings
in the backward direction can be extremely shallow, their corresponding fast
growth rates may not be necessarily small [40]. For instance, for $\mu=10^{4}$
km-1 and $\delta=10^{-3}$, the growth rate can be as high as $~{}10^{3}$ km-1,
which is much larger than the typical growth rates expected for slow modes.

In spite of such significant growth rates, fast instabilities brought about by
backward ELN crossings can not induce significant neutrino flavor conversions
if the population of backward traveling neutrinos is not large enough. This
can be deduced from the lower panels of Fig. 2, where the eigenvectors of the
unstable modes with maximum growth rates are shown. As can be clearly seen,
$Q_{v}$ is highly peaked in the backward direction. This implies that the
instabilities can mainly impact the backward traveling neutrinos, which can be
a tiny fraction of all neutrinos (at large radii).

###  2.2 Flavor conversions in the nonlinear regime

We solve numerically the partial differential equation system in Eq. (2.1),
assuming periodic boundary conditions. We calculate the spatial derivatives
using the fast fourier transform based differentiation, whereas we solve
temporal evolution using the backward differentiation formulae through the NAG
library. Our initial conditions for the diagonal elements of $\varrho$ are
independent of $z$,

$\varrho_{v,ee}(0,z)=\xi\times\left\\{\begin{array}[]{ll}1\quad\text{for}\quad
v>v_{c},\\\ \delta\quad\text{for}\quad
v<v_{c}.\end{array}\right.\,\,\bar{\varrho}_{v,ee}(0,z)=\xi\times\left\\{\begin{array}[]{ll}0.8\quad\text{for}\quad
v>v_{c},\\\ 2\delta\quad\text{for}\quad
v<v_{c}.\end{array}\right.\,\,\varrho_{v,xx}(0,z)=0,$ (2.9)

which, considering $\alpha=0.8$, allow one to reproduce $G_{v}$ in Eq. (2.6)
with $\xi=4\times 10^{5}$ km-1. Note that $\mu\simeq(1-v_{c})\xi$ for small
$\delta$’s. In terms of initial conditions for the off-diagonal elements, we
choose them to be $v$-independent and we set $\varrho_{ex}(z)=\delta(z)$. This
choice allows to give the same seed to all the $S_{v,K}(t)$ defined below and
provides the necessary perturbation triggering fast flavor conversions.

Figure 3:  The angular distributions of the eigenvectors of unstable modes,
$S_{v,K}$ (Eq. (2.11)), for $v_{c}=0.95$, and $\delta=10^{-3}$ and $10^{-5}$.
The initial perturbations are flat in $v$.

Figure 4:  Time and space evolution of the conversion probability between the
$e$ and $x$ flavors for $\delta=10^{-3}$ (left panel) and $10^{-5}$ (right
panel).

In order to extract the eigenvectors of unstable modes $Q_{v}$, we first
calculate $S_{v}(t,z)$ as

$S_{v}(t,z)=\frac{\varrho_{ex}(t,z)}{\varrho_{ee}(t=0,z)-\varrho_{xx}(t=0,z)}$
(2.10)

and then we perform its Fourier Transform

$S_{v,K}(t)=\frac{1}{L}\int_{0}^{L}\,dz\,S_{v}(t,z)\,e^{iKz},$ (2.11)

$S_{v,K}(t)$ then represents an estimate of the eigenvectors $Q_{v}$
calculated in the linear regime.

Here we focus on the cases of $\delta=10^{-3}$ and $10^{-5}$, and
$v_{c}=0.95$. Similar results are obtained for $v_{c}=0.8$. Fig. 3 shows the
angular distributions of $S_{v,K}(t)$ at different times for $K=2\times
10^{3}$ km-1. We note that the shape of $S_{v,K}$ is similar to that of
$Q_{v}$ reported in Fig. 2, as expected. Moreover, we have checked that we are
already in the nonlinear regime and thus $|S_{v,K}|$ is not growing
exponentially anymore.

Fig. 4 displays the time and space evolution of the conversion probability
between $e$ and $x$ flavor, defined as

$P_{ex}(t,z)=1-\frac{\langle\varrho_{ee}(t,z)\rangle}{\langle\varrho_{ee}(0,z)\rangle},$
(2.12)

where $\langle\varrho_{ee}(t,z)\rangle$ is the angle integrated equivalent of
$\varrho_{ee}$,

$\langle\varrho_{ee}(t,z)\rangle=\int_{-1}^{1}\,\mathrm{d}v\,\varrho_{v,ee}(t,z)\,.$
(2.13)

While large flavor conversions can exist for the case of $\delta=10^{-3}$,
they are significantly suppressed for $\delta=10^{-5}$ with the maximum of
$P_{ex}$ being $\mathcal{O}(10^{-2})$, despite the presence of fast flavor
instabilities with large growth rates. This can be understood by noting that
for the $\delta=10^{-5}$ case, $|Q_{\rm{forward}}|/|Q_{\rm{backward}}|\lesssim
10^{-2}$. Therefore, the amount of flavor conversion of the forward
propagating neutrinos is remarkably small at the onset of the nonlinear
regime. Then as the backward traveling neutrinos enters the nonlinear regime,
the exponential growth ceases and the conversion of forward traveling
neutrinos is suppressed. Nevertheless, the reason for observing significant
conversions for the case of $\delta=10^{-3}$ is twofold. First of all, the
population of the backward traveling neutrinos is relatively large in this
case, namely $\sim 5\%$ of the total neutrinos. In addition and as a
consequence of this, $|Q_{\rm{forward}}|/|Q_{\rm{backward}}|\gtrsim 10\%$.
Hence, the forward traveling neutrinos can experience significant conversions
even at the onset of the nonlinear regime.

This result is indeed consistent with the observation of Ref. [57], where
realistic ELN distributions of Eq. (2.7) were employed. As can be seen in the
lower right panel of Fig. 2, $|Q_{\rm{forward}}|/|Q_{\rm{backward}}|\lesssim
10^{-3}$ for the realistic distributions. Note that these results are obtained
in the absence of slow modes ($\omega=0$) and one can still observe non-
negligible flavor conversions in the presence of slow modes as in Ref. [57],
which has then little to do with the backward ELN crossings and fast modes.

It is also illuminating to note that while for $\delta=10^{-3}$ the flavor
conversion wave travels both in the backward and forward directions, it mostly
affects the forward regions in the $\delta=10^{-5}$ case. This simply comes
from the fact that for latter, there is a smaller number of neutrinos
traveling in the backward direction. We have confirmed that there is
absolutely no backward traveling conversion wave if one completely removes the
backward traveling neutrinos, namely $f_{\nu}=0$ for $v<0$.

Considering the linear stability analysis, our results suggest that the
transition between significant and negligible flavor conversions occurs for
$\delta\lesssim 10^{-4}$, as a very rough estimate. Needless to say, the
precise transition value should depend on the exact angular distribution and
the employed neutrino gas model.

##  3 Conclusions

Fast flavor conversions of SN neutrinos can happen on timescales as short as a
few nanoseconds. A very generic class of fast instabilities is the scattering-
induced ones which are caused by the backward ELN crossings. They actually
occur due to the residual inelastic collisions of neutrinos on nucleons and/or
nuclei when they are nearly free-streaming. Such instabilities can appear
ubiquitously both in the post-shock and pre-shock SN regions at large
distances from the SN core .

Despite their ubiquity in the SN environment, we show that the scattering-
induced fast instabilities can not lead to significant flavor conversions if
the population of the backward traveling neutrinos is not large enough. This
arises from the fact that the unstable flavor states of scattering-induced
fast instabilities are extremely peaked at backward angles, meaning that the
instability should mainly impact the small fraction of neutrinos traveling in
the backward direction. This is quite understandable since backward ELN
crossings are themselves generated by that tiny fraction of neutrinos
traveling in the backward direction.

Though our study is performed in the two-flavor scenario, one should expect
similar results in the three-flavor scenario. This can be simply understood by
noting that the shape of the unstable eigenfunctions should be similar in the
three-flavor scenario where analogous angular crossings can happen in all the
three sectors ($e\mu$, $e\tau$ and $\mu\tau$) [41].

The results we obtained here suggest that one can safely ignore the
scattering-induced fast instabilities in the SN environment when the
population of the backward traveling neutrinos is $\ll 1\%$ 111Unless the tiny
population of the backward traveling neutrinos can lead to some unexpected
non-negligible effects.. Two important questions still remain to be answered.
First and foremost, it is not very clear to us if one can always assume that
for all SN zones where scattering-induced crossings can occur, the depth of
the crossing is such small. If one can find scattering-induced ELN crossings
for which the population of the backward traveling neutrinos is $\sim$ a few
percents, then the next question would be how the flavor content of the
neutrino gas is affected by such instabilities. This is specially interesting
given the fact that the scattering-induced fast instabilities seem to be most
generic/ubiquitous class of fast instabilities in the SN environment.

## Acknowledgments

We are grateful to Georg Raffelt for helpful discussions and his comments on
the manuscript. SA acknowledges support by the Deutsche Forschungsgemeinschaft
through Sonderforschungbereich SFB 1258 Neutrinos and Dark Matter in Astro-
and Particle Physics (NDM). The work of FC at Virginia Tech is supported by
the U.S. Department of Energy under the award number DE-SC0020250. The work of
FC at IFIC is supported by GVA Grant No. CDEIGENT/2020/003.

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