Necháme truhlu zavřenou.

It is the 1st person plural form of nechat: we leave / we will leave / we’ll let.

In this sentence, the most natural meaning is we will leave or let’s leave the chest closed.

A key point: nechat is usually perfective, so its present-looking forms often have future meaning. That is why necháme can mean we will leave.

Because that is normal with perfective verbs in Czech.

With imperfective verbs, Czech forms the future with budu, budeš, bude... + infinitive.
But with perfective verbs, the ordinary present-tense endings usually express a future result.

So:

• necháváme = we are leaving / we leave habitually (imperfective)
• necháme = we will leave (perfective)

Because truhlu is the accusative singular form, and the noun is the direct object of necháme.

• dictionary form: truhla
• direct object form: truhlu

This is very common for feminine nouns ending in -a:

• žena → ženu
• kniha → knihu
• truhla → truhlu

Because zavřenou agrees with truhlu.

The word zavřenou describes the state of the chest, and in this construction it matches the object in:

• gender: feminine
• number: singular
• case: accusative

So:

• nominative: zavřená truhla = a closed chest
• accusative: necháme truhlu zavřenou = we will leave the chest closed

This -ou ending is the normal feminine singular accusative ending here.

For a learner, it is easiest to treat it as an adjective meaning closed/shut.

Historically, it comes from the passive participle of zavřít (to close, shut), but in sentences like this it behaves like a normal adjective and agrees with the noun it relates to.

So you can safely read:

• zavřený / zavřená / zavřené = closed

Because Czech commonly uses the pattern:

verb + object + adjective

to mean leave/keep something in a certain state.

So:

• Necháme truhlu zavřenou. = We’ll leave the chest closed.
• Nechal dveře otevřené. = He left the door open.
• Mám okno zavřené. = I have the window closed.

The adjective expresses the resulting or maintained state of the object.

Not really. Here it means leave/keep in a state, not abandon.

So the idea is not we are leaving the chest somewhere.
The idea is the chest will remain closed.

That is an important difference. In Czech, nechat often means:

• let
• leave
• allow
• have something done
• leave something as it is

In this sentence, it is the leave it as it is meaning.

Yes.

Depending on context and intonation, Necháme truhlu zavřenou. can sound like:

• We will leave the chest closed.
  or
• Let’s leave the chest closed.

That is quite normal in Czech. A 1st person plural form can sometimes express a decision or suggestion, not just a plain future.

Because Czech usually omits subject pronouns when the verb ending already makes the subject clear.

The ending -áme in necháme already tells you it means we.

So my is unnecessary unless you want emphasis or contrast:

• My necháme truhlu zavřenou, oni ji otevřou.
  We will leave the chest closed, but they will open it.

No. Czech word order is fairly flexible.

Necháme truhlu zavřenou. is a normal, neutral order. But other orders are also possible, depending on emphasis:

• Truhlu necháme zavřenou. → more focus on the chest
• Zavřenou necháme truhlu. → marked, stronger focus on closed

The endings still show what goes with what, so case is doing a lot of the grammatical work.

It is an object complement: it describes the condition of the object truhlu after or during the action of the verb.

In other words:

• object: truhlu
• state of that object: zavřenou

So the sentence does not just say we will leave the chest.
It says we will leave the chest closed.

This is a very useful Czech pattern to notice.