Monads on projective varieties

TL;DR: In this paper, the existence of monads on the projective space was shown to be irreducible under certain conditions on the cohomology sheaf of a monad.

Abstract: We generalise Floystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.

Summary

1. Introduction

• This is one of the simplest ways of constructing sheaves, after kernels and cokernels.
• The first problem the authors need to tackle when studying monads is their existence.

156 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• The literature, by means of construction of examples of monads over other projective varieties (for instance, blowups of the projective plane [Buchdahl 2004], Abelian varieties [Gulbrandsen 2013], Fano threefolds [Faenzi 2014] and [Kuznetsov 2012], complete intersection, Calabi–Yau threefolds [Henni and Jardim 2013], and Segre varieties [Macias Marques and Soares 2014]).
• One of the most interesting questions to ask is whether this sheaf is stable and this has been established in special cases (see [Ancona and Ottaviani 1994] and [Jardim and Miró-Roig 2008], for instance).
• Since stable sheaves are simple, a common approach is to study simplicity (in [Costa and Miró-Roig 2009] the authors show that any mathematical instanton bundle over an odd-dimensional quadric hypersurface is simple, and in particular that it is stable over a quadric threefold).
• The authors characterize low-rank vector bundles that are the cohomology sheaf of a monad of type (M) (Theorem 5.1).
• There has been much work done on this since the nineties.

2. Monads over ACM varieties

• As well as the results that were the starting point for the present paper, i.e., Fløystad’s work [2000] regarding the existence of monads on projective space.the authors.
• Let us first fix the notation used throughout the paper.
• Notation 2.1. Let RY be the homogeneous graded coordinate ring of Y and IY/PN its ideal sheaf.
• Moreover, the authors note that the notion of ACM variety depends on the embedding.

158 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Observe that the vector bundles which are the cohomology of a monad of the form (2) are the so-called instanton bundles.
• The next construction uses standard techniques of projective geometry and it explains why the authors thought Fløystad’s case could be generalized to other projective varieties.

3. Existence of monads over ACM varieties

• The main ideas of the proof follow Fløystad’s construction, combined with the projective geometry standard results described at the end of the last section.
• Suppose that one of the conditions (i) and (ii) holds.
• Suppose first that condition (i) is satisfied.

160 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Note that the three morphisms are represented by the same matrix.
• Observe that the fact that φ̂ is injective implies that φ is also injective.
• Nevertheless, the authors note that the construction above is far more general.
• The authors next prove the two main results of this section, which generalize Fløystad’s theorem on the existence of monads on projective space.

162 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• The existence of the monad in case conditions (i) or (ii) are satisfied follows from Lemma 3.1.
• Let us show that these conditions are necessary.
• Because p̃ degenerates in the expected codimension the authors have, by [Buchsbaum and Eisenbud 1977, Theorem 2.3], Fitt1(coker p̃)= Ann(coker p̃), MONADS ON PROJECTIVE VARIETIES 163 and so they obtain the following chain of inclusions Fitt1(coker q̃)⊂.
• The proof of the existence of a monad of type (M) follows again from Lemma 3.1.

164 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Using the same approach, the authors can think of similar examples of monads of some varieties that are cut out by quadrics, such as the Grassmannian.
• The simplest case that is not a hypersurface is G(2, 5), the Grassmannian that parametrizes planes in the projective space P5, which is embedded in P19 with Plücker coordinates [X j0 j1 j2]0≤.

166 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Which, as the authors have seen, cannot happen in the Grassmannian variety.
• The authors can reduce the exponent of the middle term in the monad, by using the method they described in the proof of Theorem 3.3, combined with this construction.

168 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Since the linear independence of these linear forms is a key step in the beginning of the proof, the authors see that in this case, any monad of type (6) has a simple cohomology sheaf.
• The next example shows that the statement in Proposition 4.1 is accurate, that is, there are monads of type (6) whose cohomology is not simple.

5. Vector bundles of low rank

• Moreover, the authors will deal with the problem of simplicity and stability of this particular case.
• Generalizing Fløystad’s result, the authors start by proving the following theorem.
• MONADS ON PROJECTIVE VARIETIES 169 Theorem 5.1. Proof.
• Observe that l2k = c2k(L2k) and, by the projection formula, see [Fulton 1998, Theorem 3.2 (c)], this Chern class cannot be zero, contradicting the assertion above.

170 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Minimal rank bundles defined using “many” global sections.
• Moreover, the dimension of the subspace spanned by the entries of these matrices can be bigger as the authors shall see in the following examples.
• In order to use more global sections in the matrices defining the monad, the authors could simply “add another diagonal” whose entries involve an additional global section.
• In the following two examples the authors will achieve such a goal in the particular case of the quadric considered above.
• The technique is easily reproducible for other varieties.

172 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Finally, using once again the projection formula (to the cohomology bundle) as well as commutativity of the tensor product with the pullback, the authors can conclude that the pullback of a simple (respectively stable) bundle E on P2n+1 is a simple (respectively stable) bundle on the projective variety X .
• Nevertheless, the authors always have the following property.
• From the hypotheses, the authors see that X satisfies the conditions in Theorem 3.3 or Theorem 3.4.
• Therefore, the matrix defining g, MONADS ON PROJECTIVE VARIETIES 173 with a suitable change of variables, may be assumed to have δ zero columns.

6. The set of monads and the moduli problem

• The construction therein does not, however, give an answer to the question of “how many” monads of type (M) exist.
• The authors would like to know more about the algebraic structure of the set of pairs of morphisms which define a monad over a projective variety.
• In the case of projective space the authors prove the following: Theorem 6.1.

174 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• The authors can check that the conditions in Theorem 2.4 imply this inequality.
• It was brought to their attention that the Main Theorem in [Jardim et al. 2017] shows that the moduli space of instanton sheaves of rank 2 and charge 3 is reducible, which means that Theorem 6.1 is sharp.
• Hence the isomorphisms of monads of this type correspond bijectively to the isomorphisms of the corresponding cohomology bundles.
• In particular, the two categories are equivalent and the authors will not distinguish between their corresponding objects.

176 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• Let us briefly recall here the definitions of (σ, γ )-globally injective and surjective (see [Jardim and Prata 2015] for more details).
• Set α = N+1∑ i=1 Moreover, by King’s central result [1994, Theorem 4.1], the existence of such a λ guarantees the existence of a coarse moduli space for families of λ-semistable representations up to S-equivalence (two λ-semistable representations are S-equivalent if they have the same composition factors in the full abelian subcategory of λsemistable representations).
• MONADS ON PROJECTIVE VARIETIES 177 Given the equivalences of the categories Mk,c and G gis k,c , and after Remark 6.4, the authors see that they can define a moduli space M(V2k,c) whenever they can construct a moduli space of the abelian category Ggisk,c .

178 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• The only subrepresentations left to consider are the ones of the form (0, b′, 0), but also for these ones, the choice of the triple λ= (−1, 0, 1) satisfies the semistability condition.
• The irreducibility statement follows from Theorem 6.1 and the general setting described above.
• Then the coarse moduli space M(V2k,1) of λ-semistable vector bundles in V2k,1 is irreducible.
• Naturally, irreducibility of the moduli space will be guaranteed in each case where the authors get an irreducible family, as mentioned in the general setting described after Theorem 6.1.
• Therefore, in this case the authors are not able to construct the moduli space M(V2k,2) with the help of King’s construction.

180 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES

• [Jardim et al. 2018] M. Jardim, D. Markushevich, and A. S. Tikhomirov, “New divisors in the boundary of the instanton moduli space”, Moscow Math.
• A. Kuznetsov, “Instanton bundles on Fano threefolds”, Cent. Eur. J. Math.
• SIMONE MARCHESI INSTITUTO DE MATEMÁTICA, ESTATÍSTICA E COMPUTAÇÃO CIENTÍFICA UNIVERSIDADE.

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Full text

Pacific
Journal of
Mathematics
MONADS ON PROJECTIVE VARIETIES
SIMONE MARCHESI
,
PEDRO MACIAS MARQUES
AND
HELENA SOARES
Volume 296 No. 1 September 2018

PACIFIC JOURNAL OF MATHEMATICS
Vol. 296, No. 1, 2018
dx.doi.org/10.2140/pjm.2018.296.155
MONADS ON PROJECTIVE VARIETIES
SIMONE MARCHESI
,
PEDRO MACIAS MARQUES
AND
HELENA SOARES
We generalize Fløystad’s theorem on the existence of monads on projective
space to a larger set of projective varieties. We consider a variety X, a line
bundle L on X, and a basepoint-free linear system of sections of L giving a
morphism to projective space whose image is either arithmetically Cohen–
Macaulay (ACM) or linearly normal and not contained in a quadric. We
give necessary and sufficient conditions on integers a, b and c for a monad
of type
0 → (L
∨
)
a
→ O
b
X
→ L
c
→ 0
to exist. We show that under certain conditions there exists a monad whose
cohomology sheaf is simple. We furthermore characterize low-rank vector
bundles that are the cohomology sheaf of some monad as above.
Finally, we obtain an irreducible family of monads over projective space
and make a description on how the same method could be used on an ACM
smooth projective variety X. We establish the existence of a coarse moduli
space of low-rank vector bundles over an odd-dimensional X and show that
in one case this moduli space is irreducible.
1. Introduction
A monad over a projective variety X is a complex
M
•
: 0 → A
f
−→ B
g
−→ C → 0
of morphisms of coherent sheaves on
X
, where
f
is injective and
g
is surjective. The
coherent sheaf
E := ker g/ im f
is called the cohomology sheaf of the monad
M
•
.
This is one of the simplest ways of constructing sheaves, after kernels and cokernels.
The first problem we need to tackle when studying monads is their existence.
Fløystad [2000] gave sufficient and necessary conditions for the existence of monads
over projective space whose maps are given by linear forms. Costa and Miró-Roig
[2009] extended this result to smooth quadric hypersurfaces of dimension at least
three, and Jardim [2007] made a further generalization to any hypersurface in
projective space. We can find additional partial results on the existence of monads in
MSC2010: primary 14F05; secondary 14J10, 14J60.
Keywords: monads, ACM varieties.
155

156 SIMONE MARCHESI, PEDRO MACIAS MARQUES AND HELENA SOARES
the literature, by means of construction of examples of monads over other projective
varieties (for instance, blowups of the projective plane [Buchdahl 2004], Abelian
varieties [Gulbrandsen 2013], Fano threefolds [Faenzi 2014] and [Kuznetsov 2012],
complete intersection, Calabi–Yau threefolds [Henni and Jardim 2013], and Segre
varieties [Macias Marques and Soares 2014]). In [Jardim and Miró-Roig 2008], the
authors expressed the wish of having a generalization of the results on existence
by Fløystad and by Costa and Miró-Roig to varieties other than projective space
and quadric hypersurfaces. Here we generalize Fløystad’s theorem to a larger set
of projective varieties. We let
X
be a variety of dimension
n
and
L
be a line bundle
on
X
. We consider a linear system
V ⊆ H
0
(L)
, without base points, defining a
morphism
ϕ : X → P(V )
and suppose that its image
X
0
⊂ P(V )
is arithmetically
Cohen–Macaulay (ACM) (see Definition 2.2 and Theorem 3.3) or linearly normal
and not contained in a quadric hypersurface (Theorem 3.4). Then we give necessary
and sufficient conditions on integers
a
,
b
and
c
for the existence of a monad of type
(M) 0 → (L
∨
)
a
→ O
b
X
→ L
c
→ 0.
Once existence of a monad over a variety
X
is proved, we can study its cohomol-
ogy sheaf. One of the most interesting questions to ask is whether this sheaf is stable
and this has been established in special cases (see [Ancona and Ottaviani 1994]
and [Jardim and Miró-Roig 2008], for instance). Since stable sheaves are simple, a
common approach is to study simplicity (in [Costa and Miró-Roig 2009] the authors
show that any mathematical instanton bundle over an odd-dimensional quadric
hypersurface is simple, and in particular that it is stable over a quadric threefold).
We show that under certain conditions, in the case when
X
0
is ACM, there exists a
monad of type (M) whose cohomology sheaf is simple (Proposition 4.1).
As we said, monads are a rather simple way of obtaining new sheaves. When
the sheaf we get is locally free, we may consider its associated vector bundle, and
by abuse of language we will not distinguish between one and the other. There is
a lot of interest in low-rank vector bundles over a projective variety
X
, i.e., those
bundles whose rank is lower than the dimension of
X
, mainly because they are
very hard to find. We characterize low-rank vector bundles that are the cohomology
sheaf of a monad of type (M) (Theorem 5.1).
Finally, we would like to be able to describe families of monads, or of sheaves
coming from monads. There has been much work done on this since the nineties.
Among the properties studied on these families is irreducibility (see for instance
[Tikhomirov 2012; 2013] for the case of instanton bundles over projective space).
Here we obtain an irreducible family of monads over projective space (Theorem 6.1),
and make a description on how the same method could be used on another ACM
projective variety. Furthermore, we establish the existence of a coarse moduli

MONADS ON PROJECTIVE VARIETIES 157
space of low-rank vector bundles over an odd-dimensional, ACM projective va-
riety (Theorem 6.5), and show that in one case this moduli space is irreducible
(Corollary 6.6).
2. Monads over ACM varieties
Let
X
be a projective variety of dimension
n
over an algebraically closed field
k
,
L
be a line bundle on
X
, and
V ⊆ H
0
(L)
yield a linear system without base points,
defining a morphism
ϕ : X →P(V )
. Our main goal is to study monads over
X
of type
0 → (L
∨
)
a
→ O
b
X
→ L
c
→ 0.
In this section we recall the concept of monad, as well as the results that were
the starting point for the present paper, i.e., Fløystad’s work [2000] regarding the
existence of monads on projective space.
Let us first fix the notation used throughout the paper.
Notation 2.1.
Let
Y ⊆ P
N
be a projective variety of dimension
n
over an alge-
braically closed field
k
. Let
R
Y
be the homogeneous graded coordinate ring of
Y
and I
Y/P
N
its ideal sheaf.
If
E
is a coherent sheaf over
Y
we will denote its dual by
E
∨
. We also denote the
graded module H
i
∗
(Y, E ) = ⊕
m∈Z
H
i
(Y, E (m)) and h
i
(E) = dim H
i
(Y, E ).
Given any k-vector space V, we will write V
∗
to refer to its dual.
Definition 2.2.
Let
Y
be a projective variety of dimension
n
over an algebraically
closed field
k
. We say that
Y
is arithmetically Cohen–Macaulay (ACM) if its graded
coordinate ring R
Y
is a Cohen–Macaulay ring.
Remark 2.3.
If
Y ⊆ P
N
is a projective variety then being ACM is equivalent to
the following vanishing:
H
1
∗
(P
N
, I
Y/P
N
) = 0, H
i
∗
(Y, O
Y
) = 0, 0 < i < n.
Moreover, we note that the notion of ACM variety depends on the embedding.
The first problem we will address concerns the existence of monads on projective
varieties (see Section 3) and the generalization of the following result.
Theorem 2.4
[Fløystad 2000, Main Theorem and Corollary 1]. Let
N ≥
1. There
exists a monad of type
(1) 0 → O
P
N
(−1)
a
f
−→ O
b
P
N
g
−→ O
P
N
(1)
c
→ 0
if and only if one of the following conditions holds:
(i) b ≥ a + c and b ≥ 2c + N − 1,
(ii) b ≥ a + c + N.

Frequently asked questions

Q1. What is the morphism of a monad on projective space?

Let X be a variety of dimension n and let L be a line bundle on X. Suppose there is a linear system V ⊆ H 0(L), with no base points, defining a morphism X→ P(V ) whose image X ′ ⊂ P(V ) is a projective ACM variety. 

Q2. What is the morphism of the X ′P(V)?

Let X be a variety of dimension n and let L be a line bundle on X. Suppose there is a linear system V ⊆ H 0(L), with no base points, defining a morphism X→P(V ) whose image X ′⊂P(V ) is linearly normal and not contained in a quadric hypersurface. 

Q3. What is the general setting of monads?

the set of pairs( f, g) ∈ Hom(OP3(−1)5,O12P3)×Hom(O 12 P3 ,OP3(1)5)yielding such a monad is a reducible algebraic variety. 

Q4. What is the simplest case that is not a hypersurface?

The simplest case that is not a hypersurface is G(2, 5), the Grassmannian that parametrizes planes in the projective space P5, which is embedded in P19 with Plücker coordinates [X j0 j1 j2]0≤ 

Q5. What is the morphism of a monad over a projective variety?

A monad over a projective variety X is a complexM• : 0→ A f −→ B g−→C→ 0of morphisms of coherent sheaves on X , where f is injective and g is surjective. 

Q6. What is the cohomology of a monad?

In this section the authors characterize monads whose cohomology is a vector bundle of rank lower than the dimension of X and, in particular, the authors restrict to the case when X is nonsingular. 

Q7. What is the morphism of the monad?

Since the linear independence of these linear forms is a key step in the beginning of the proof, the authors see that in this case, any monad of type (6) has a simple cohomology sheaf. 

Q8. What is the construction of the monad on projective space?

The authors consider a variety X , a line bundle L on X , and a basepoint-free linear system of sections of L giving a morphism to projective space. 

Q9. What was the support for Macias Marques?

Macias Marques and Soares were partially supported by Fundação para a Ciência e Tecnologia (FCT), project “Comunidade Portuguesa de Geometria Algébrica”, PTDC/MAT-GEO/0675/2012. 

Q10. What is the cohomology of a monad of type?

for each odd dimensional variety X with an associated ACM embedding given by a line bundle L and for each c ≥ 1 there exists a vector bundle which is cohomology of a monad of type (9). 

Q11. What is the simplest way to get a monad of type x?

A monad of type (9) over X admits the following display:000 // (L∨)c // K // E // 00 // (L∨)c f // O2k+2cX //gQ // 0LcLc0 0Taking cohomology on the exact sequence0→ IX ′(−1)→OPN (−1)→ ϕ∗L∨→ 0,we get that h0(L∨)=h1(L∨)=0, since h1(IX ′(−1))=h2(IX ′(−1))=0. 

Q12. What is the sabbatical grant for Macias Marques?

Macias Marques was also partially supported by Centro de Investigação em Matemática e Aplicações (CIMA), Universidade de Évora, project PEst-OE/MAT/UI0117/2014, by FCT sabbatical leave grant SFRH/BSAB/1392/2013, and by FAPESP Visiting Researcher Grant 2014/12558-9. 

Q13. What is the codimension of the degeneracy locus of f?

Dualizing this complex, the authors get0→OX ′(−1)a→ObX ′→OX ′(1) c → 0,which is still a monad on X ′, for the codimension of the degeneracy locus of OX ′(−1)a→ObX ′ is at least b− a− c+ 1. 

Q14. What is the g,MONADS ON PROJECTIVE VARIETIES?

the matrix defining g,MONADS ON PROJECTIVE VARIETIES 173with a suitable change of variables, may be assumed to have δ zero columns.