Differential geometry – For a curve in 3-space, the tangent always intersects a fixed line, then the curve is planar

This is an old exam question where I do not know how to start.

$$
begin {array} {l} { text {Let} alpha text {is a regular curve in} mathbb {R} ^ {3} text {. Prove that when a tangent line intersects with} alpha text {a solid}} \ { text {line} l in mathbb {R} ^ {3}, text {then} alpha text {is planar. }} end {array}
$$

My first problem is how to actually use the main assumption. I would like to express the condition that these two lines intersect algebraically to deduce that the torsion is $ 0 $, I parametrize $ l = a + b cdot t $but then I do not know if I should get an expression for differentiating (or manipulating) that characterizes the assumption that the tangent and $ l $ to cut. Any hints?

Agal Algebraic Geometry – Variation of the Hodge structure of a complex surface and reference requirement

Imagine a complex, smooth surface that is triple embedded. I'm interested in the limit at which the surface degenerates into a reducible (but reduced) surface. In other words, I have a family of surfaces.

begin {eqnarray}
pi: mathcal {S} rightarrow Delta,
end {eqnarray}

Where $ Delta $ is parameterized by a complex variable $ t $, and $ S_t = pi ^ {- 1} (t) $ is smooth for $ 0 0 $ and, $ S_0 $ is the degenerate surface.

My questions are:

1- What kind of rank one coherent sheaf on $ S_0 $ may be the limit of a bundle of cables $ S_t $, Is it true that any element of $ Pic (S_0) $ are limit of the line bundles $ S_t $?

2- Assume that I select trunk bundles over each component of $ S_0 $ they are not a limit of a line bundle, so it is equivalent $ (1.1) $ drive in $ S_0 $, Then I deform myself $ S_0 $ to something smooth $ S_t $What happens to these (1,1) cycles?

For some physical reasons, I expect the answer to the first question to be yes, and for the second I hope it should become one $ (2.0) $ Cycle after deformation.

Please let me know if there are references to the above questions that are not too abstract (for a physics student to read).

Thanks.

Riemannian geometry – connection between Yamabe invariant and Yamabe equation

I'm trying to understand Lee and Parker's solution to the Yamabe problem. It seems to me that the constant $ lambda $ what appears in the Yamabe equation $$ square varphi = lambda varphi ^ {p-1} $$ and the Yamabe invariant $ lambda = inf_ varphi Q_g ( varphi) $ Where $ Q_g $ is the functional $$ Q_ {g} ( varphi) = frac { int_ {M} left (a | nabla varphi | ^ {2} + S varphi ^ {2} right) d V_ {g}} { | varphi | _ {p} ^ {2}} = frac {E ( varphi)} { | varphi | _ {p} ^ {2}} $$ are indeed the same, that is, if $ varphi $ is an absolute minimum for the function $ Q_g $ then $ varphi $ is a solution to the Yamabe equation with $ lambda $ as the coefficient of $ varphi ^ {p-1} $, It seems to me that Aubin has explicitly demonstrated this in his book "Some Nonlinear Problems in Riemannian Geometry."

An explicit calculation of the Euler-Lagrange equation for $ Q_g $ (This is shown on page 39 of this paper by Lee and Parker and can easily be done explicitly.) This shows that the Euler equation is
$$
a Delta varphi + S varphi | varphi | _ {p} ^ {- p} E ( varphi) varphi ^ {p-1} = 0
$$

that is the Yamabe equation with
$$
lambda = frac {E ( varphi)} { | varphi | _ {p} ^ {p}}.
$$

For the Yamabe invariant to be the same constant that occurs in the Yamabe equation, the exponent on the denominator should be 2 and not $ p $,
So I'm a bit confused: Is the Yamabe invariant the same constant in the Yamabe equation in the presence of a solution? $ varphi $? If so, where do I go wrong?

dg.differential geometry – The effect of the Hodge $ star $ operator on the symplectic structure of a Kahl $ 4 $ distribution

To let $ (M, omega, J, g) $ be a $ 4 $ Dimensional Kahler manifold. Put $ omega & # 39; = star omega $ Where $ star $ Is the hodge operator assigned to the metric? $ g $,

is $ (M, omega & # 39;) $ a symplectic manifold? Is it necessary symplectic equivalent to the original structure $ (M, omega) $Is this a diffeomorphism? $ f $ what carries $ omega & # 39; $ to $ omega $?

Ag.algebraic geometry – Spherical perverse sheaves on the affine Grassmannian and critically twisted $ D $ module

To let $ G $ be a reductive algebraic group and leave $ Gr_G = G ((z)) / G ((z)) $ be his affectionate Grassmannian. Define $ mathcal {D} (Gr_G) _ {crit} -mod $ to be the category of law $ D $Modules $ Gr_G $ twisted by the critical level of $ G $ (which is $ -1 / 2 $ the killing form).

In this work by Frenkel-Gaitsgory is mentioned on page $ 1347 $ that there is an equivalence
$$ mathcal {D} (Gr_G) _ {crit} -mod ^ {G ((z)}} cong Rep (G ^ { vee}) $$
between the category of $ G ((z)) $ Equivariant critically twisted $ D $Modules $ Gr_G $ and representations of the Langlands dual. I'm a bit confused because geometric Satake equivalency is the most commonly used $ G ((z)) $equivariant perverse sheaves on $ Gr_G $, and not $ D $Modules.

From Riemann-Hilbert I know that perverse sheaves are related to regular holonomic sheaves $ D $Modules, but I do not understand how this gives us the equivalence of the categories written above. Does anyone have a quick explanation for that? Thank you in advance.

Agal Algebraic Geometry – Is it possible to embed a group schema into a locally constant one so that the quotient exists?

To let $ S $ be a sufficiently good basic scheme (such as the finite type over an algebraic closed field) and $ G to S $ be a flat group scheme. I would like to ask: Can we always find a closed embedding? $ G to H $ into another flat group scheme $ H $, so that $ H $ is constant over S locally in the Zariski topology, and that $ H / G $ exists as a schema?

dg.differential geometry – expressing the Riemann metric as a pullback metric

For a Riemannian manifold $ M $ with an original metric $ g $for any other metric $ g & # 39; $ on $ M $ Is there a diffeomorphism? $ f: (M, g) rightarrow (M, g) $ so that the pullback metric of $ f $ is $ g & # 39; $? If this is not generally true, what kind of metrics apply, and which properties make it generally inappropriate? I am especially interested in the simple case in which $ M $ is $ mathbb {R} ^ n $ and $ g $ is the usual Euclidean metric, but I'm also interested in the general case.