Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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27 views

Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth. We know that ...
2
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0answers
31 views

Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
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1answer
67 views

Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$. Now, let ...
2
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1answer
84 views

Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
5
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2answers
286 views

Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold. Given a smooth $(n-1)$-dimensional smooth ...
3
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2answers
129 views

Definition of the differential of the Cone of a morphism of complexes [on hold]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$. The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...
8
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0answers
128 views

Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$. The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...
2
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180 views

Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such ...
2
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0answers
73 views

Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation: $$ g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F). $$ Here, ...
3
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1answer
157 views

Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
14
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1answer
219 views

Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...
7
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2answers
225 views

Circle Action on Quaternionic Projective Space

Quoting from Wikipedia article on quaternionic projective space: Therefore the quotient manifold $$ \mathbb{HP}^{2}/\mathrm{U}(1) $$ may be taken, writing $U(1)$ for the circle group. It has ...
4
votes
2answers
201 views

Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$ $$\pi^1(H)\to\pi^1(X)$$ is an isomorphism, ...
3
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0answers
71 views

When is a map from a logarithmic tangent bundle to a normal bundle surjective?

Suppose that $X$ is a smooth algebraic variety with free divisor $D$ so that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a ...
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144 views

One-dimension Algebraic groups

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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157 views

Calabi-Yau theorem on Arithmetic Variety

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. ...
7
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91 views

The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals

I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme ...
2
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1answer
136 views

Number of Plücker relations for a Grassmannian

Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: ...
11
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371 views
+200

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
4
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1answer
486 views

Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
3
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0answers
116 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
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265 views
+100

What is Koszul dual of a curve?

Let $X$ be a curve embedded into a projective space $\mathbb P$ such that it is cut out (scheme-theoretically or ideal-theoretically) by quadrics. What is known about the Koszul dual of the ...
5
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1answer
481 views

Is first etale cohomology of a variety always (dual to) a Tate Module?

The two examples I have in mind are curves and abelian varieties. To be precise, if $C$ is a smooth projective algebraic curve over a number field $K$, then, for a prime $l$ ...
2
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110 views

Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation $$ A_1 +...+A_n ...
3
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105 views

Degree formalism for line bundles on Deligne-Mumford stacks

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper ...
2
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0answers
80 views

functor from complex algebraic variety to constructible function

I asked the same question on mathstackexchange but didn't get any response so I guess I should ask here. If you think it is a duplicate and it is not appropriate to post it here, I will delete this ...
9
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1answer
197 views

Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected. If there exists ...
23
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2answers
800 views

$\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...
6
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1answer
496 views

Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field. What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...
3
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139 views

splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...
5
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4answers
3k views

Easiest way to determine the singular locus of projective variety & resolution of singularities

For an affine variety, I know how to compute the set of singular points by simply looking at the points where the Jacobian matrix for the set of defining equations has too small a rank. But what is ...
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0answers
38 views

Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants ...
5
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0answers
137 views

Flat + locally of finite presentation + monomorphism = open immersion

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$: Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then ...
3
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0answers
327 views

Grothendieck, A Place to Begin [on hold]

I'm finishing up an undergrad degree in mathematics and am beginning to think about areas of research. I know that the work of Grothendieck is considered the cornerstone of modern algebraic geometry, ...
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0answers
76 views

A question about Kähler Einstein metric [on hold]

Let $X$, and $Y$ are Kähler manifolds and $f:X\to Y$ is birational and let on $(Y,\omega)$ we have $\text{Ric}(\omega)=-\omega$, then Kähler Einstein metric on $X$ can be of which form? can we say it ...
11
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1answer
314 views
+50

Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
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0answers
24 views

Common Point of Intersection of n-dimensional ellipsoids [on hold]

Suppose we have two ellipses in 2-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have 4 points of intersection. Can we say that in ...
4
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0answers
96 views

Moduli space of complex Tori [duplicate]

Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?
24
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6answers
5k views

Why are local systems and representations of the fundamental group equivalent

My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally ...
1
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1answer
132 views

Direct image for crystals?

If we get a morphism $f : X \to Y$ of schemes over $k$, how should I define the direct image functor $$ f_* : Crys(X) \to Crys(Y)? $$ By a crystal I mean a quasi-coherent sheaf $M$ on $X$ with a ...
3
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0answers
78 views

extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local ...
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0answers
45 views

moduli space of curves under prescribed tangency conditons

We consider an irreducible component of the Hilbert Scheme of curves in $\mathbb P^2$. Denote it as $\mathcal D.$ We fix a line $L$ and a point $A\in L.$ Denote $\mathcal D_0$ as the subscheme of ...
15
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1answer
443 views

Algebraic spaces as locally ringed spaces

Let $S$ be a scheme (although I am more than happy to have $S=\text{Spec}(k)$ for a field $k$) and $\mathsf{AlgSp}/S$ the category of algebraic spaces over $S$. Does there exist an embedding ...
2
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1answer
257 views

Pro-affine varieties over a local field

Let $K$ be a (perfect) local field, and let $S = \lim (\operatorname{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the ...
1
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1answer
131 views

l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic. If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...
3
votes
3answers
508 views

Quotients of complex manifolds

Let $G$ be algebraic group acting on a smooth algebraic variety $M$. Assume the action is proper and free. Is the orbit space $M/G$ an algebraic variety? If so, could someone point to a reference? If ...
14
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2answers
592 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
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1answer
237 views

Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
2
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1answer
98 views

What finite groups are stabilizers in Kirwan's desingularization construction?

Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then ...
5
votes
1answer
114 views

Pythagorean number in Artin's theorem on nonnegative rational fractions

Emil Artin's theorem on nonnegative rational fractions says that a rational fraction $Q$ with $n$ variables with real coefficients which is non-negative on $\mathbb R^n$ is a sum of squares of ...